August 29, 2023

Double Forms, Curvature Integrals and the Gauss–Bonnet Formula Marc
Troyanov††Institut de Mathématiques EPFL, 1015 Lausanne, Switzerland;
<EMAIL_ADDRESS>
###### Abstract

The Gauss–Bonnet Formula is a significant achievement in 19th century
differential geometry for the case of surfaces and the 20th century cumulative
work of H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern for
higher-dimensional Riemannian manifolds. It relates the Euler characteristic
of a Riemannian manifold to a curvature integral over the manifold plus a
somewhat enigmatic boundary term. In this paper, we revisit the formula using
the formalism of double forms, a tool introduced by de Rham, and further
developed by Kulkarni, Thorpe, and Gray. We explore the geometric nature of
the boundary term and provide some examples and applications.

Keywords:  Gauss–Bonnet Formula, Double Forms, Curvature Integrals.

AMS Subject Classification: 58A1, 53C20.

###### Contents

  1. 1 Introduction
  2. 2 An Overview of Hopf’s Problem on the Curvatura Integra
  3. 3 Double Forms and their Geometric Applications
    1. 3.1 Definition and examples
    2. 3.2 Transposition and Symmetric Double Forms
    3. 3.3 The Contraction Operator
    4. 3.4 Integrating Double Forms
    5. 3.5 The Lipschitz–Killing Curvatures
  4. 4 Chern’s Insight and the Tao of Gauss–Bonnet
    1. 4.1 The Emergence of the Pfaffian
    2. 4.2 A Tale of Exactness, featuring a Vector Field
    3. 4.3 Harvesting the ripe Gauss–Bonnet Fruit
    4. 4.4 Demystifying the Gauss–Bonnet Boundary term using Double Forms
  5. 5 Examples and Applications
    1. 5.1 Flat Manifolds.
    2. 5.2 Manifolds with Flat Boundary
    3. 5.3 Space Forms
    4. 5.4 Further results on non Compact Manifolds
    5. 5.5 The direct product of a Ball and a Sphere
    6. 5.6 Rotationally Symmetric Metrics
  6. 6 Miscellaneous Remarks
    1. 6.1 Remarks on Signs and Other Conventions
    2. 6.2 Comparison with Chern’s Papers
    3. 6.3 Double Factorials and the volume of Spheres
  7. 7 Exercises
  8. A Appendix: Some Background in Differential Geometry
    1. A.1 Some Linear Algebra on the (co)tangent Space
    2. A.2 Connection and Curvature
    3. A.3 Moving the Frame with Élie Cartan

##  1 Introduction

The Gauss–Bonnet Formula for Riemannian surfaces is a remarkable achievement
of 19th century differential geometry, resulting from the cumulating work of
K. F. Gauss, P. O. Bonnet, J.P. Binet, and W. von Dyck. The result is well-
covered in a number of textbooks on Riemannian geometry.

Its extension to higher-dimensions is the result of the combined work of H.
Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil, and S. S. Chern, spanning the
period from 1925 to 1945. The formula relates the Euler characteristic of a
compact Riemannian manifold with a curvature integral over the manifold plus a
boundary term involving the curvature of the manifold and the second
fundamental form over the boundary.

In his formulation of the Gauss–Bonnet Formula, Chern uses Élie Cartan’s
moving frame method. His proof is very direct but it does not readily reveal
the geometric nature of the boundary term. In this paper, we revisit and
reframe the higher-dimensional Gauss–Bonnet Formula using double forms, a
formalism introduced by G. de Rham and further developed by R. Kulkarni, J. A.
Thorpe, and A. Gray in the 1960s and 70s. Double forms offer a powerful tool
for manipulating curvature identities and provide us with a different point of
view on the Gaus-Bonnet Formula.

Our objective in this paper is then to present the Gauss–Bonnet Formula within
the framework of double forms. We provide details for Chern’s original proof
and give some examples and applications.

While we do provide some historical perspective, this paper is not meant to be
a historical essay. Further historical remarks and developments can be found
in other sources such as [14], [43], as well as Volumes 3 and 5 of M. Spivak’s
treatise [38]. See also the interviews [4], [24].

Due to space and expertise limitations, we do not explore alternative proofs
(such as based on heat kernel, probabilistic or PDEs methods) or broader
topics such as the theory of characteristic classes, the Hirzebruch signature
theorem and more generally the Atiyah–Singer index theorem. Applications to
physics are also not discussed.

#### Organization of the Paper

The rest of the paper is organized as follows: In Section 2 we discuss the
origin of the problem behind the Gauss–Bonnet Formula, beginning with Hopf’s
famous "Curvatura Integra" problem. Section 3 introduces the notion of double
forms on a manifold. These are tensors that provide a convenient algebraic
framework for manipulating various geometric quantities in Riemannian
geometry.

Section 4 is dedicated to a fully detailed proof of the higher-dimensional
Gauss–Bonnet Formula for compact, even-dimensional Riemannian manifolds with
boundaries, following Chern’s arguments. We also reinterpret the boundary term
using double forms; the formula is stated in three different formulations in
(4.16). Some examples and applications are discussed in Section 5.

Section 6 contains various remarks and comments, and a brief summary on the
notion of double factorial that is often used throughout the paper. Some
exercises are suggested in Section 7; the reader is advised to take a look at
them at some point. The paper concludes with an appendix that offers the
necessary background in Riemannian geometry, including a detailed introduction
to Cartan’s moving frame methods.

As for prerequisites, we assume the reader is familiar with basic Differential
and Riemannian geometry at the graduate level, including tensor fields and
differential forms. Although we have tried to write a clean exposé, the reader
should be aware that a number of algebraic calculations are quite intricate,
and some of the details are left to the reader.

Let us conclude this introduction with a word about our notation. The
manifolds we work with are assumed to be smooth and oriented. Typically, they
are denoted as $M$ or $N$, where $m=\dim(M)$ and $n=\dim(N)$. Throughout most
of the discussion, $m$ will be presumed to be an even number, while $n$ may be
either odd or even. The boundary $\partial M$ of the manifold $M$, if non
empty, will be given the orientation induced by the outer normal.

##  2 An Overview of Hopf’s Problem on the Curvatura Integra

The 2-dimensional Gauss–Bonnet Formula, for a compact Riemannian surface
$(S,g)$ with piecewise smooth boundary, is conventionally stated as follows:

$2\pi\chi(S)=\int_{S}KdA+\int_{\partial S}k_{g}ds+\sum_{j}\alpha_{j},$

where $K$ represents the curvature, $k_{g}$ denotes the geodesic curvature of
its boundary, $\alpha_{j}$ refers to the exterior angles on the boundary and
$\chi(S)$ is the Euler characteristic of the surface.

This formula has been extended to the case of higher-dimensional manifolds in
a series of papers published between 1925 and 1945 by H. Hopf, W. Fenchel, C.
Allendoerfer, A. Weil and S. S. Chern, as we now discuss. The first and
fundamental breakthrough is due to Hopf, who proved in 1925 that for a closed
hypersurface $M$ in Euclidean space $\mathbb{R}^{m+1}$ the following holds if
$m$ is even:111No such formula can hold if $m$ is odd, since in this case we
always have $\chi(M)=0$. See however Corollary 5.2.

$\frac{1}{2}\operatorname{Vol}(S^{m})\chi(M)=\int_{M}K\operatorname{dvol}_{M}.$
(2.1)

Here $K$ is the Gauss–Kronecker curvature of the hypersurface, that is, the
product of its principal curvatures. We shall call Equation (2.1) the
Gauss–Bonnet–Hopf Formula; its right hand side is the total curvature, or
curvatura integra, of the hypersurface.

To prove the Formula, Hopf generalized a result of H. Poincaré that relates
the winding number of a vector field on the boundary of a planar domain to the
indices of its singularities. We recall the definition: given a continuous
vector field $X$ on a smooth manifold $M$ having an isolated zero at $p\in M$,
and choosing a local coordinate system in the neighborhood $U$ of $p$, one may
identify $X$ with a mapping $X:U\to\mathbb{R}^{m}$; for $\varepsilon>0$ small
enough, the following map is well defined:

$\zeta:\partial B(p,\varepsilon)\to
S^{m-1},\quad\zeta(q)=\frac{X_{q}}{\|X_{q}\|},$

where $B(p,\varepsilon)\subset M$ is the ball around $p$ with respect to some
Riemannian metric on $M$. The _index of $X$_ at the point $p$ is defined to be
the degree of that map:

$\operatorname{Ind}(X,p)=\deg\left(\left.\zeta\right|_{\partial
B(p,\varepsilon)}\right).$

The index depends only on the vector field and not on the chosen coordinate
system or the auxiliary Riemannian metric. With this in mind, we state the
Poincaré–Hopf Theorem:

######  Theorem 2.1 (Poincaré–Hopf).

Let $M$ be a smooth compact manifold with (possibly empty) boundary. Let $X$
be a vector field on $M$ that is transverse to the boundary and outward
pointing. If $X$ has isolated zeroes at $x_{1},\ldots,x_{k}\in
M\setminus\partial M$, then the following equality holds:

$\sum_{i=1}^{k}\mathrm{Ind}(X,x_{i})=\chi(M),$

where $\mathrm{Ind}(X,x_{i})$ represents the index of $X$ at the zero $x_{i}$,
and $\chi(M)$ denotes the Euler characteristic of $M$.

Hopf proved this result in [21, 23], mentioning Poincaré for the 2-dimensional
case and earlier partial results by Brouwer and Hadamard in higher-dimensions.
A very readable proof can be found in [31, page 35]. See also references [5,
18, 29] for the case of manifolds without boundary.

######  Corollary 2.2.

Let $D$ be a smooth compact domain in $\mathbb{R}^{m}$. Denote by
$\nu:\partial D\to S^{m-1}$ the exterior pointing Gauss map of the boundary.
The degree of that map is equal to the Euler characteristic of the domain $D$,
that is,

$\deg(\nu)=\chi(D).$

We will give later a more general version of this result, see Corollary 5.2.

###### Proof.

Let us choose a vector field $X$ on $D$ with isolated zeroes at
$x_{1},\dots,x_{k}\in D\setminus\partial D$ and such that $X$ is outward
pointing at the boundary. Let us set
$D_{\varepsilon}=D\setminus\cup_{i=1}^{k}B(x_{i},\varepsilon)\subset M$ for
some $\varepsilon>0$ small enough. We denote by $\omega$ the volume form on
$S^{m-1}$ and by $f:D_{\varepsilon}\to S^{m-1}$ the map defined by
$f(q)=X_{q}/\|X_{q}\|$. Since $\omega$ is a closed form, we have

$0=\int_{D_{\varepsilon}}f^{*}(d\omega)=\int_{\partial
D_{\varepsilon}}f^{*}(\omega)=\int_{\partial
D}f^{*}(\omega)-\sum_{i=1}^{k}\int_{\partial
B(x_{i},\varepsilon)}f^{*}(\omega).$

To complete the proof, observe that

$\int_{\partial
D}f^{*}(\omega)=\operatorname{Vol}(S^{m-1})\deg(\left.f\right|_{\partial
D})\quad\text{and}\quad\sum_{i=1}^{k}\int_{\partial
B(x_{i},\varepsilon)}f^{*}(\omega)=\operatorname{Vol}(S^{m-1})\sum_{i=1}^{k}\operatorname{Ind}(X,x_{i}).$

Moreover $\deg(\left.f\right|_{\partial D})=\deg(\nu)$ because
$\left.f\right|_{\partial D}$ is homotopic to the Gauss map.
∎

We are now in position to prove Hopf’s formula (2.1). Recall first that, by
the Jordan–Brouwer separation Theorem, any closed smooth hypersurface
$M\subset\mathbb{R}^{m+1}$ is the boundary of a compact smooth domain with
$D\subset\mathbb{R}^{m+1}$, see e.g. [18, page 89]. By definition, the
Gauss–Kronecker curvature $K$ of $M$ is the Jacobian of the Gauss map
$\nu:M\to S^{m}$, i.e. we have

$K\operatorname{dvol}_{M}=\nu^{*}\left(\omega\right),$

where $\omega=\operatorname{dvol}_{S^{n}}$ is the volume form of $S^{m}$.
Using the previous corollary, we therefore have

$\int_{M}K\operatorname{dvol}_{M}=\deg(\nu)\int_{S^{m}}\omega=\chi(D)\operatorname{Vol}(S^{m}),$

and (2.1) follows now from the relation

$\chi(M)=\chi(\partial D)=\frac{1}{2}\chi(D).$

∎

In a second paper [22], also published in 1925, Hopf established the validity
of (2.1) for compact space forms, that is, Riemannian manifolds with constant
sectional curvature. A proof will also be given later in §5.3. Motivated by
these results, he asked for an intrinsic version of that result. Let us quote
from Allendoerfer’s paper [1]:

> H. Hopf has repeatedly pointed to the problem of deciding whether one can
> define a curvature scalar in an arbitrary Riemannian manifold of even
> dimension with the following properties:
>
>   1. $\circ$
>
> In the special case of a hypersurface it becomes identical with the
> Gauss–Kronecker curvature mentioned above.
>
>   2. $\circ$
>
> It satisfies a formula of Gauss–Bonnet type, i.e.. its integral over a
> region of the manifold can be expressed by a curvature integral over the
> boundary.
>
>   3. $\circ$
>
> The integral of this curvature scalar is a topological invariant [in the
> case of a closed manifold].
>
>

In short, Hopf is asking for the definition of a locally defined, scalar
function on the manifold, depending only on the metric tensor and whose
integral is the Euler characteristic (plus a boundary term if $\partial
M\neq\emptyset$). We will call this the Hopf Problem on Curvature Integra. A
first observation is that, using Gauss’ Equation relating the second
fundamental form of a hypersurface $M$ in some Euclidean space to its
curvature tensor:

$R_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk},$

one can express the Gauss–Kronecker curvature $K(p)$ at any point $p$ in $M$
as follows

$K=\frac{1}{m!!}\sum_{\sigma\in S_{m}}\sum_{\tau\in
S_{m}}(-1)^{\sigma}(-1)^{\tau}R(e_{\sigma_{1}},e_{\sigma_{2}},e_{\tau_{1}},e_{\tau_{2}})\cdots
R(e_{\sigma_{m-1}},e_{\sigma_{m}},e_{\tau_{m-1}},e_{\tau_{m}}),$ (2.2)

where $\\{e_{1},\dots,e_{m}\\}$ is an orthonormal basis of the tangent space
of $M$ at the point $p$. Here $S_{m}$ is the permutation group on $m$ symbols
and $(-1)^{\sigma}$ is the signature of $\sigma\in S_{m}$. In the above
formula, we have used the double factorial, which for an even number $m$ is
defined as

$m!!=m\cdot(m-2)\cdots 2=2^{\frac{m}{2}}\left(\tfrac{m}{2}\right)!$

we refer to §6.3 for more on this notion. A proof of (2.2) can be found on
page 228 of the book [39] by J.A. Thorpe, but another proof will be given in
Proposition 3.5 below. It is obvious from (2.2) that the Gauss–Kronecker
curvature is intrinsic.

The Hopf problem stated above has been positively resolved in two papers, [1]
and [13], published independently in 1940, by C. B. Allendoerfer in the U.S.A,
and W. Fenchel in Denmark. They proved that (2.2) holds for any closed even-
dimensional manifold $M$ that can be isometrically embedded in some Euclidean
space $\mathbb{R}^{d}$. Note that the embeddability condition is in fact not
restrictive as was proved 16 years later by J. Nash in [32].

In both proofs, the authors consider the boundary of the $\varepsilon$-tubular
neighborhood of $M$, with $\varepsilon>0$ small enough; let us denote it by
$N$, and observe that it is a hypersurface in $\mathbb{R}^{d}$. Moreover, the
natural projection $\pi:N\to M$ is a fiber bundle with typical fiber the
sphere $S^{d-1-m}$.

The scalar $K_{p}$, defined at any point $p$ of $M$ by integrating the
Gauss–Kronecker curvature of the hypersurface $N$ over the
$\varepsilon$-sphere $\pi^{-1}(p)\subset N$, is the generalized curvature
function answering Hopf’s Problem. This is verified by applying (2.1) to the
hypersurface $N$ and computing. Fenchel mentions that R. Lipschitz and W.
Killing already proved in the 1880s that $K_{p}$ depends solely on the
Riemannian metric of $M$ and not on its embedding in Euclidean space.
Alternatively, Allendoerfer used calculations from H. Weyl’s 1938 paper on the
tube formula [41] to establish that (2.2) still holds for the averaged
Gauss–Kronecker curvature as defined.

In short, the Hopf Problem for a closed submanifold $M\subset\mathbb{R}^{d}$
is resolved by applying the Gauss–Bonnet–Hopf Formula (2.1) to a tubular
neighborhood $N$ of the manifold, then integrating the Gauss–Kronecker
curvature over the fibers of the fibration $\pi:N\to M$, and proving that the
resulting function is an intrinsic quantity attached to the Riemannian
manifold.

The two aforementioned papers did not provide a complete solution to the Hopf
Problem as stated earlier, since the second requirement explicitly called for
a version of the Gauss–Bonnet formula applicable to manifolds with boundary.
Also the condition that the manifold should be isometrically embedded in some
Euclidean space was seen as a serious restriction (Nash’s Theorem not being
available at that time). The joint paper [2] published in 1943 by Carl B.
Allendoerfer and André Weil addresses both issues with the following method:
They consider a sufficiently fine triangulation of the manifold and use a
result proved by E. Cartan in 1927 that guarantees the possibility to locally
embed a Riemannian manifold isometrically into some $\mathbb{R}^{d}$, provided
it is analytic. Using Cartan’s Theorem, one can isometrically embed each cell
of the triangulation in some Euclidean space $\mathbb{R}^{d}$, then consider a
tubular neighborhood of that cell and repeat the previous construction. The
result is a Gauss–Bonnet Formula, first for piecewise analytic Riemannian
polyhedra, and then, invoking an approximability result by Whitney, for
general Riemannian polyhedra of class $\operatorname{C}^{2}$.

The result achieved by Allendoerfer and Weil indeed solved Hopf’s problem, but
it was not entirely satisfactory. The proof involved a high level of
complexity and, furthermore, it relied on local embeddings in Euclidean space,
which is clearly not the most natural approach for constructing an
intrinsically defined generalized curvature function on a given Riemannian
manifold.

According to the story, as told say in [43], Chern visited the Institute for
Advanced Study in Princeton in August 1943. He was 31 years old. There, he met
André Weil, who discussed with him the Hopf problem and the need for an
intrinsic proof. Within two weeks, Chern solved the problem using the moving
frame method that had been recently introduced by Élie Cartan, and some
innovative ideas of his own. He published his proof in two papers in the
Annals of Mathematics in 1944 and 1945. The first paper, titled A Simple
Intrinsic Proof of the Gauss–Bonnet Formula for Closed Riemannian Manifolds,
contains a short intrinsic proof of the Gauss–Bonnet Formula for closed even-
dimensional Riemannian manifolds. The second paper, titled On the Curvatura
Integra in a Riemannian Manifold, extends the investigation to manifolds of
either odd or even dimension with boundary. The 1945 paper also gives
additional formulas expressing some characteristic classes of sphere bundles
using curvature integrals. Chern’s work laid the foundation for the
development of Chern–Weil theory, which relates characteristic classes to
differential forms.

##  3 Double Forms and their Geometric Applications

The notion of double form on a manifold $N$ has been introduced by G. de Rham
who defined them as differential forms on a manifold $N$ with values in the
Grassmann algebra of $TN$, see [12, §7]. Double forms have been used in the
1960s and 70s as a convenient tool in the computation of various curvature
formulae in Riemannian geometry by R. Kulkarni, J. Thorpe and A. Gray. In this
section we introduce the algebra of double forms following the book [15] by
Gray and we refer to [27] for further developments.

###  3.1 Definition and examples

  1. (i)

Let $N$ be a smooth $n$-dimensional manifold. A _double-form_ $A$ of type
$(k,l)$ on $N$ is a smooth covariant tensor field of degree $k+l$, which is
alternating in the first $k$ variables and also in the last $l$ variables.
This condition can be expressed as follows: If
$X_{1},\dots,X_{k},Y_{1},\dots,Y_{l}$ are $(k+l)$ vector fields on $N$ and
$\sigma,\tau\in S_{k}$, then

$A(X_{1},\dots,X_{k};Y_{1},\dots,Y_{l})=(-1)^{\sigma}(-1)^{\tau}A(X_{\sigma_{1}},\dots,X_{\sigma_{k}};Y_{\tau_{1}},\dots,Y_{\tau_{l}}).$

We denote by $\mathcal{D}^{k,l}(N)$ the set of all $(k,l)$-double forms on
$N$. This is a module over the algebra of smooth functions $C^{\infty}(N)$
that can be expressed as the tensor product

$\mathcal{D}^{k,l}(N)=\mathcal{A}^{k}(N)\otimes_{C^{\infty}(N)}\mathcal{A}^{l}(N),$

where $\mathcal{A}^{k}(N)$ is the usual space of differential $k$-forms on
$N$. The direct sum of these spaces is denoted as

$\mathcal{D}(N)=\bigoplus_{k,l=0}^{n}\mathcal{D}^{k,l}(N).$

  2. (ii)

A natural product, denoted by $\owedge$, is defined on $\mathcal{D}(N)$ as
follows: If $\psi_{1}=\alpha_{1}\otimes\beta_{1}\in\mathcal{D}^{k,l}(N)$ and
$\psi_{2}=\alpha_{2}\otimes\beta_{2}\in\mathcal{D}^{p,q}(N)$ then
$\psi_{1}\owedge\psi_{2}$ is the following double-form of type $(k+p,l+q)$:

$\psi_{1}\owedge\psi_{2}=(\alpha_{1}\wedge\alpha_{2})\otimes(\beta_{1}\wedge\beta_{2})\in\mathcal{D}^{k+p,l+q}(N).$

The product is then extended to general elements of $\mathcal{D}(N)$ by
linearity.; it can also be defined by the formula:

$\displaystyle\left(\psi_{1}\owedge\psi_{2}\right)(X_{1},\dots,X_{k+p};Y_{1},\dots,Y_{l+q})=$
$\displaystyle\frac{1}{k!\,l!\,p!\,q!}\sum_{\sigma\in S_{k+p}}\sum_{\tau\in
S_{l+q}}(-1)^{\sigma}(-1)^{\tau}\psi_{1}(X_{\sigma_{1}},\dots,X_{\sigma_{k}};Y_{\tau_{1}},\dots,Y_{\tau_{l}})\cdot\psi_{2}(X_{\sigma_{k+1}},\dots,X_{\sigma_{k+p}};Y_{\tau_{l+1}},\dots,Y_{\tau_{l+q}})$

We shall call this operation the double wedge product of $\psi_{1}$ and
$\psi_{2}$.

  3. (iii)

The pair $\left(\mathcal{D}(N),\owedge\right)$ is the _algebra of double-
forms_ on the manifold $N$, it is a supercommutative bigraded algebra, meaning
that if $\psi_{1}\in\mathcal{D}^{p,q}(N)$ and
$\psi_{2}\in\mathcal{D}^{r,s}(N)$, then
$\psi_{1}\owedge\psi_{2}\in\mathcal{D}^{k+r,l+s}(N)$ and

$\psi_{1}\owedge\psi_{2}=(-1)^{pr+qs}\psi_{2}\owedge\psi_{1}.$

######  Remark 3.1.

The product in the algebra $\mathcal{D}(N)$ is simply denoted as
$\psi_{1}\cdot\psi_{2}$ by various authors such as in [26] and [27], and it is
denoted by $\psi_{1}\wedge\psi_{2}$ in the book [15]. The notation $\owedge$
is used in [3] for symmetric double forms of type $(1,1)$. We find it
convenient to extend this notation to the whole algebra of double forms.

Examples.

  1. (a)

The double forms of type $(0,0)$ are the smooth functions on $N$, that is,
$\mathcal{D}^{0,0}(N)=C^{\infty}(N)$.

  2. (b)

To any differential $k$-form $\alpha\in\mathcal{A}^{k}(N)$ we can associate
the double forms $1\owedge\alpha\in\mathcal{D}^{0,1}(N)$ and $\alpha\owedge
1\in\mathcal{D}^{1,1}(N)$. This gives two natural embeddings of the exterior
algebra into the algebra of double forms.

  3. (c)

The double forms of type $(1,1)$ are the covariant tensor fields of order $2$
on $M$. In particular the metric tensor $g$ of a Riemannian manifold is an
element in $\mathcal{D}^{1,1}(N)$.

  4. (d)

If $N$ is a (cooriented) hypersurface in a Riemannian manifold $(M,g)$, then
the second fundamental form $h$ of $N\subset M$ is another important double
forms of type $(1,1)$.

  5. (e)

The curvature tensor $R$ of a Riemannian manifold is a double form of type
$(2,2)$. It is furthermore symmetric in the following sense:

$R(X_{1},X_{2}:Y_{1},Y_{2})=R(Y_{1},Y_{2};X_{1},X_{2}).$

  6. (f)

The double wedge product of two symmetric $(0,2)$ tensors fields
$h^{\prime},h^{\prime\prime}\in\mathcal{D}^{1,1}(N)$ is usually called their
_Kulkarni–Nomizu product_. It is explicitly given by

$h_{1}\owedge
h_{2}(X_{1},X_{2};Y_{1},Y_{2})=h_{1}(X_{1},Y_{1})h_{2}(X_{2},Y_{2})-h_{1}(X_{2},Y_{1})h_{2}(X_{1},Y_{2})\\\
+h_{1}(X_{2},Y_{2})h_{2}(X_{1},Y_{1})-h_{1}(X_{1},Y_{2})h_{2}(X_{2},Y_{1}).$

  7. (g)

A covariant tensor field $S$ of degree $3$ can be seen as an element in
$\mathcal{D}^{2,1}(N)$ if and only if it is skew symmetric in its first two
variables: $S(X,Y;Z)+S(Y,X;Z)=0$.

  8. (h)

Given $S\in\mathcal{D}^{2,1}(N)$ and a $1$-form $\alpha$ seen as an element in
$\mathcal{D}^{0,1}(N)$, the product $S\owedge\alpha\in\mathcal{D}^{2,2}(N)$ is
given by

$S\owedge\alpha(X,Y;Z,W)=S(X,Y;Z)\alpha(W)-S(X,Y;W)\alpha(Z).$

  9. (i)

A smooth field $A\in\mathrm{End}(\Lambda^{k}TN)$ of endomorphisms of the
exterior power $\Lambda^{k}TN$ can be seen as the double form in
$\mathcal{D}^{k,k}(N)$ defined by

$(X_{1},\dots X_{k};Y_{1},\dots Y_{k})\mapsto\left\langle X_{1}\wedge
X_{2},\wedge\dots\wedge X_{k},A(Y_{1}\wedge Y_{2},\wedge\dots\wedge
Y_{k})\right\rangle,$

where we have used the natural scalar product $\langle\;,\;\rangle$ on
$\Lambda^{k}TN$ induced by $g$.

###  3.2 Transposition and Symmetric Double Forms

The transpose of a double form $\psi\in\mathcal{D}^{k,l}(N)$ is the double
form $\psi^{\top}\in\mathcal{D}^{l,k}(N)$ defined by

$\psi^{\top}(X_{1},\dots,X_{l};Y_{1},\dots,Y_{k})=\psi(Y_{1},\dots,Y_{k};X_{1},\dots,X_{l}).$

We trivially have $({\psi^{\top}})^{\top}=\psi$, hence transposition defines
an involution of the algebra of double forms Furthermore we have

$(\psi_{1}\owedge\psi_{2})^{\top}=\psi_{1}^{\top}\owedge\psi_{2}^{\top}.$

A double form $\psi\in\mathcal{D}^{k,k}(N)$ is termed symmetric if
$\psi^{\top}=\psi$. Note that the product of two symmetric double forms is
itself a symmetric double form.

The symmetric elements of the double algebra $\mathcal{D}(N)$ form a
commutative subalgebra, denoted by Kulkarni as $\mathcal{C}$. He refers to it
as the Ring of Curvature Structures. Moreover, he considers sseveral
subalgebras of $\mathcal{C}$, characterized by the Bianchi identities.

###  3.3 The Contraction Operator

Given a Riemannian metric $g$ on the manifold $N$, and $1\leq k,l\leq
n=\dim(N)$, we inductively define the _contraction operator of order $p$:_

$\operatorname{C}^{p}:\mathcal{D}^{k,l}(N)\to\mathcal{D}^{k-p,l-p}(N)$

for any integer $p$ such that $0\leq p\leq\min(k,l)$ as follows. For
$\psi\in\mathcal{D}^{k,l}(N)$ we set $\operatorname{C}^{0}(\psi)=\psi$ and at
any point $x\in N$ we set

$\operatorname{C}^{p}(\psi)(X_{1},\ldots,X_{k-p};Y_{1},\ldots,Y_{l-p})=\sum_{i=1}^{n}\operatorname{C}^{p-1}(\psi)(X_{1},\ldots,X_{k-p},e_{i};Y_{1},\ldots,Y_{l-p},e_{i}),$

for $1\leq p\leq\min\\{l,k\\}$, where $e_{1},\ldots,e_{n}\in T_{x}M$ is an
orthonormal basis and $X_{i},Y_{j}\in T_{x}M$ are arbitrary tangent vectors.
This operation is independent of the chosen orthonormal basis.

As an example, the first and second contractions of the curvature tensor $R$
of a Riemannian manifold are respectively its Ricci and scalar curvatures.

If $\psi\in\mathcal{D}^{p,p}(N)$, its full contraction is defined to be
$\operatorname{C}^{p}(\psi)$. This is a scalar quantity (i.e. an element of
$C^{\infty}(M)$) and the following result is useful in computations:

######  Proposition 3.2.

Let $\\{e_{1},\dots,e_{n}\\}$ be a positively oriented orthonormal moving
frame defined in a domain $U$ of an oriented $n$-dimensional Riemannian
manifold $(N,g)$, and let $\\{\theta^{1},\dots\theta^{n}\\}$ be the dual
coframe. Then the following holds: If $\mu_{1},\ldots,\mu_{p}$ and
$\nu_{1},\ldots,\nu_{p}$ are two lists of pairwise distinct indices in
$\\{1,\ldots,m\\}$, then

$\displaystyle\frac{1}{p!}\;\operatorname{C}^{p}\left(\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{p}}\otimes\theta^{\nu_{1}}\wedge\cdots\wedge\theta^{\nu_{p}}\right)$
$\displaystyle=\delta_{\mu_{1},\ldots,\mu_{p}}^{\nu_{1},\ldots,\nu_{p}}$

is equal to $+1$ or $-1$, depending on whether ${\nu_{1},\ldots,\nu_{p}}$ is
an odd or even permutation of ${\mu_{1},\ldots,\mu_{p}}$, and equal to zero
otherwise.

###### Proof.

By definition we have

$\displaystyle\operatorname{C}^{p}\left(\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{p}}\otimes\theta^{\nu_{1}}\wedge\cdots\wedge\theta^{\nu_{p}}\right)$
$\displaystyle\qquad=\sum_{i_{1},\dots,i_{m}=1}^{m}\left(\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{p}}(e_{i_{1}},\dots,e_{i_{p}})\right)\cdot\left(\theta^{\nu_{1}}\wedge\cdots\wedge\theta^{\nu_{p}}(e_{i_{1}},\dots,e_{i_{p}})\right)$
$\displaystyle\qquad=\sum_{i_{1},\dots,i_{m}=1}^{m}\det\left(a^{j}_{k}\right)\cdot\det\left(b^{j}_{k}\right)$
$\displaystyle\qquad=p!\;\delta_{\mu_{1},\ldots,\mu_{p}}^{\nu_{1},\ldots,\nu_{p}},$

where we have used the notation

$a^{j}_{k}=\left(\theta^{\mu_{j}}(e_{i_{k}})\right)\quad\text{and}\quad
b^{j}_{k}=\left(\theta^{\nu_{j}}(e_{i_{k}})\right).$

∎

######  Corollary 3.3.

If $\alpha,\beta\in\mathcal{A}^{p}(N)$, then

$\langle\alpha,\beta\rangle=\frac{1}{p!}\operatorname{C}^{p}\left(\alpha\otimes\beta\right),$

where $\langle\ ,\ \rangle$ is the natural scalar product defined on the
Grassmann algebra (see (A.1)).

###### Proof.

This follows immediately from the previous proposition and the fact that at
any point $x$ of the manifold, the family
$\\{\theta^{i_{1}}\wedge\cdots\wedge\theta^{i_{p}}\\}_{i_{1}<\cdots<i_{p}}$ is
orthonormal in $\Lambda^{p}T_{x}^{*}N$.
∎

In the next result we gather several useful contraction formulas:

######  Proposition 3.4.

With the same notation as above, we have

  1. (1)

Any $\alpha\in\mathcal{A}^{p}(N)$ can be written as

$\alpha=\frac{1}{p!}\sum_{i_{1}<\dots<i_{p}=1}^{n}\operatorname{C}^{n}\left(\makebox{\Large$\alpha$}\otimes\theta^{i_{1}}\wedge\cdots\wedge\theta^{i_{p}}\right)\theta^{i_{1}}\wedge\cdots\wedge\theta^{i_{p}}.$

  2. (2)

If $\makebox{\Large$\alpha$}\in\mathcal{A}^{n}(N)$ is a differential of degree
$n$, then for any permutation $\sigma\in S_{n}$ we have

$n!\;{\makebox{\Large$\alpha$}}=(-1)^{\sigma}\operatorname{C}^{n}\left(\makebox{\Large$\alpha$}\otimes\theta^{\sigma_{1}}\wedge\cdots\wedge\theta^{\sigma_{m}}\right)\operatorname{dvol}_{N,g}.$

  3. (3)

For any $\psi\in\mathcal{D}^{p,p}(N)$ we have

$\displaystyle\operatorname{C}^{p+q}\left(\psi\owedge g^{q}\right)$
$\displaystyle=\prod_{j=1}^{q}(p+j)(n-p+1-j)\operatorname{C}^{p}\left(\psi\right)$
$\displaystyle=(q!)^{2}\binom{p+q}{q}\binom{n-p}{q}\operatorname{C}^{p}\left(\psi\right)$

for any integer $1\leq q\leq(n-p)$.

In particular, we have

$\operatorname{C}^{q}(g^{q})=\dfrac{n!q!}{(n-q)!}=\binom{n}{q}(q!)^{2}.$ (3.1)

###### Proof.

The first formula follows immediately from the corollary and the expansion

$\alpha=\sum_{i_{1}<\dots<i_{p}=1}^{n}\
\left\langle\makebox{\Large$\alpha$},\theta^{i_{1}}\wedge\cdots\wedge\theta^{i_{p}}\right\rangle\theta^{i_{1}}\wedge\cdots\wedge\theta^{i_{p}},$

and the second identity is a special case of the first.

We now prove the third formula for $q=1$. Using the first identity, it
suffices to consider the tensor
$\psi=\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{p}}\otimes\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{p}}$
where the $\mu_{i}$s are pairwise distinct. We then have

$\displaystyle\operatorname{C}^{p+1}\left(\psi\owedge g\right)$
$\displaystyle=\sum_{\nu=1}^{n}\operatorname{C}^{p+1}\left(\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{p}}\wedge\theta^{\nu}\otimes\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{p}}\wedge\theta^{\nu}\right)$
$\displaystyle=(n-p)(p+1)!$

because there are exactly $(n-p)$ indices
$\nu\in\\{1,\dots,n\\}\setminus\\{\mu_{1},\ldots,\mu_{p}\\}$. Since
$\operatorname{C}^{p}\left(\psi\right)=p!$, we have proved that

$\operatorname{C}^{p+1}\left(\psi\owedge
g\right)=(n-p)(p+1)\operatorname{C}^{p}\left(\psi\right).$

The general case $q\geq 2$ easily follows by induction. ∎

We can now prove that the Gauss–Kronecker Curvature of a hypersurface in
$\mathbb{R}^{m+1}$ is an intrinsic invariant of the underlying Riemannian
manifold if $m$ is even. This observation can be seen as a generalization of
Gauss’ Theorema Egregium.

######  Proposition 3.5.

The Gauss–Kronecker curvature of a hypersurface $M$ of even dimension $m$ in
Euclidean space $\mathbb{R}^{m+1}$ can be written in terms of its Riemann
curvature tensor as follows:

$K=\frac{2^{{\frac{m}{2}}}}{(m!)^{2}}\operatorname{C}^{m}(R^{{\frac{m}{2}}})=\frac{1}{m!!}\sum_{\sigma\in
S_{m}}\sum_{\tau\in
S_{m}}(-1)^{\sigma}(-1)^{\tau}R_{\sigma_{1}\sigma_{2}\tau_{1}\tau_{2}}\cdots
R_{\sigma_{m-1}\sigma_{m}\tau_{m-1}\tau_{m}}.$ (3.2)

In this formula, $R_{ijkl}=R(e_{i},e_{j};e_{k},e_{l})$, where
$e_{1},\dots,e_{m}$ is a given orthonormal frame. If the basis is not
orthonormal, the right hand side must be divided by ${\det(g_{ij})}$.

###### Proof.

From Lemma A.2 we know that

$R=\frac{1}{4}\sum_{i,j,k,l=1}^{m}R_{ijkl}\
\theta^{i}\wedge\theta^{j}\otimes\theta^{k}\wedge\theta^{l},$

therefore

$R^{{\frac{m}{2}}}=\frac{1}{4^{{\frac{m}{2}}}}\sum_{\mu_{1},\dots,\mu_{n}=1}^{m}\sum_{\nu_{1},\dots,\nu_{n}=1}^{m}R_{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}\cdots
R_{\mu_{m-1}\mu_{m}\nu_{m-1}\nu_{m}}\
\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{m}}\otimes\theta^{\nu_{1}}\wedge\cdots\wedge\theta^{\nu_{m}},$

and using Proposition 3.2, we obtain

$\frac{1}{m!}\operatorname{C}^{m}(R^{{\frac{m}{2}}})=\frac{1}{4^{{\frac{m}{2}}}}\sum_{\sigma\in
S_{m}}\sum_{\tau\in
S_{m}}(-1)^{\sigma}(-1)^{\tau}R_{\sigma_{1}\sigma_{2}\tau_{1}\tau_{2}}\cdots
R_{\sigma_{m-1}\sigma_{m}\tau_{m-1}\tau_{m}}.$

The second fundamental form $h$ of $M\subset\mathbb{R}^{m+1}$ is a double form
$h\in\mathcal{D}^{1,1}(M)$, and by Gauss’ Equation we have

$R=\frac{1}{2}h\owedge h=\frac{1}{2}\sum_{i,j,k,l=1}^{m}\ h_{ik}h_{jl}\
\theta^{i}\wedge\theta^{j}\otimes\theta^{k}\wedge\theta^{l}.$

Repeating the previous argument we have

$\displaystyle\frac{1}{m!}\operatorname{C}^{m}(R^{{\frac{m}{2}}})$
$\displaystyle=\frac{1}{m!}\frac{1}{2^{{\frac{m}{2}}}}\operatorname{C}^{m}(h^{m})$
$\displaystyle=\frac{1}{2^{{\frac{m}{2}}}}\sum_{\sigma\in S_{m}}\sum_{\tau\in
S_{m}}(-1)^{\sigma}(-1)^{\tau}h_{\sigma_{1}\tau_{1}}h_{\sigma_{2}\tau_{2}}\cdots
h_{\sigma_{m}\tau_{m}}$
$\displaystyle=\frac{m!}{2^{{\frac{m}{2}}}}\det\left(h_{ij}\right),$

and we conclude that

$\displaystyle K=\det\left(h_{ij}\right)$
$\displaystyle=\frac{2^{{\frac{m}{2}}}}{(m!)^{2}}\operatorname{C}^{m}(R^{{\frac{m}{2}}})$
$\displaystyle=\frac{1}{2^{{\frac{m}{2}}}m!}\sum_{\sigma\in
S_{m}}\sum_{\tau\in
S_{m}}(-1)^{\sigma}(-1)^{\tau}R_{\sigma_{1}\sigma_{2}\tau_{1}\tau_{2}}\cdots
R_{\sigma_{m-1}\sigma_{m}\tau_{m-1}\tau_{m}}.$

∎

###  3.4 Integrating Double Forms

Some interesting geometric quantities can be obtained by considering a
geometrically defined double form $\psi\in\mathcal{D}^{p,p}(M)$ and
integrating its full contraction $\operatorname{C}^{p}(\psi)$ over the
manifold. In particular, the following functional, which is defined on
$\mathcal{D}^{2,2}(N)\times\mathcal{D}^{1,1}(N)$, will play an important role
in the sequel:

######  Definition 3.6.

Given a Riemannian manifold $(N,g)$ of dimension $n$, and double forms
$A\in\mathcal{D}^{2,2}(N)$ and $b\in\mathcal{D}^{1,1}(N)$, we define for any
integer $0\leq k\leq\frac{n}{2}$

$\mathcal{Q}_{k}(A,b\mid
N,g)=\frac{1}{n!\,k!\,(n-2k)!}\int_{N}\operatorname{C}^{n}(A^{k}\owedge
b^{n-2k})\operatorname{dvol}_{g},$ (3.3)

provided $\operatorname{C}^{n}(A^{k}\owedge b^{n-2k})$ is integrable on $N$.

The special cases where $k=0$ or $2k=n$ will also be denoted as

$\mathcal{Q}_{0}(b\mid
N,g)=\frac{1}{(n!)^{2}}\int_{N}\operatorname{C}^{n}(b^{n})\operatorname{dvol}_{g}=\int_{N}\det\left(b_{ij}\right)\operatorname{dvol}_{g},$

and, when $n$ is even,

$\mathcal{Q}_{\frac{n}{2}}(A\mid
N,g)=\frac{1}{(n!)^{2}}\int_{N}\operatorname{C}^{n}(A^{\frac{n}{2}})\operatorname{dvol}_{g}.$

The notation $\mathcal{Q}_{k}(A,b\mid N,g)$ is somewhat heavy but it precisely
indicates what is being integrated on which manifold. When there is no risk of
ambiguity, we will simplify it to $\mathcal{Q}_{k}(A,b\mid N)$, or even
$\mathcal{Q}_{k}(A,b)$.

If either $A$ or $b$ (or both) are curvature invariants of the manifold, the
integral (3.3) will be referred to as a curvature integral. We will need the
special cases where $b$ is either the metric tensor $g$ or the second
fundamental form $h$ in the case of a hypersurface, and $A$ is a linear
combination of the curvature tensor $R$ and the double forms $g\owedge g$ or
$h\owedge h$.

######  Example 3.7.

As a first important example, the index of a smooth vector field $\xi$ on an
$m$-dimensional Riemannian manifold $(M,g)$ with an isolated singularity at
$p\in M$ can be expressed as

$\operatorname{Ind}(X,p)=\lim_{\varepsilon\to
0}\frac{1}{\operatorname{Vol}(S^{m-1})}\int_{S_{\varepsilon}}\det(L)\operatorname{dvol}_{S_{\varepsilon}},$

where $L:TM\to TM$ is defined as $L(Y)=\nabla_{Y}\zeta$ with
$\zeta=\xi/\|\xi\|$. Introducing the double form $\psi\in\mathcal{D}^{1,1}$
defined by $\psi(X,Y)=\langle X,L(Y)\rangle$, one can write the index as

$\operatorname{Ind}(X,p)=\lim_{\varepsilon\to
0}\tfrac{1}{\operatorname{Vol}(S^{m-1})}\mathcal{Q}_{0}(\psi\mid{S_{\varepsilon}},g).$
(3.4)

###  3.5 The Lipschitz–Killing Curvatures

An important class of examples of curvature integrals is given by the
_Lipschitz–Killing Curvatures_ , which we define now:

######  Definition 3.8.

The _total Lipschitz–Killing curvatures_ of order $j$ of an $n$-dimensional
Riemannian manifold $(N,g)$ are defined as $\mathcal{K}_{j}(N,g)=0$ if $j$ is
odd and

$\mathcal{K}_{2k}(N,g)=\mathcal{Q}_{k}(R,g\mid N),$ (3.5)

if $j=2k\leq n$ is even.

Remark. Using Proposition 3.2, we have

$\operatorname{C}^{n}\left(R^{k}\owedge
g^{n-2k}\right)=\dfrac{(n-2k)!\;n!}{(2k)!}\operatorname{C}^{2k}\left(R^{k}\right),$

therefore the Lipschitz–Killing curvatures can also be written as

$\mathcal{K}_{2k}(N,g)=\frac{1}{k!\,(2k)!}\int_{N}\operatorname{C}^{2k}(R^{k})\operatorname{dvol}_{N},$
(3.6)

which is the definition given in [15].

######  Example 3.9.

(a) The Lipschitz–Killing curvature of order $0$ is integrable if and only if
$(N,g)$ has finite volume. In this case we have

$\mathcal{K}_{0}(N,g)=\text{Vol}_{g}(N).$

(b) The total Lipschitz–Killing curvature of order $2$ on a compact manifold
$(N,g)$ is half the integral of its scalar curvature:

$\mathcal{K}_{2}(N,g)=\frac{1}{2}\int_{N}S_{g}\operatorname{dvol}_{N}.$

(c) If $(N,g)$ is a compact space form with constant sectional curvature $a$,
then using $R=\frac{a}{2}g\owedge g$ we obtain

$\mathcal{K}_{2k}(N,g)=\frac{n!\
a^{k}}{2^{k}k!(n-2k)!}\cdot\mathrm{Vol}(N,g).$ (3.7)

A notable application of the Lipschitz–Killing curvatures is illustrated by
the celebrated H. Weyl Tube Formula proved in 1938:

######  Theorem 3.10.

If $N$ is a smooth closed submanifold of dimension $n$ in $\mathbb{R}^{d}$,
then the volume of a (small enough) $\varepsilon$-tubular neighborhood
$U_{\varepsilon}(N)=\\{x\in\mathbb{R}^{d}\mid\mathrm{dist}(x,N)\leq\varepsilon\\}$
is given by

$\operatorname{Vol}\left(U_{\varepsilon}(N)\right)=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}\alpha_{d,n,k}\,\mathcal{K}_{2k}(N)\,\varepsilon^{d-n+2k},$

where

$\alpha_{d,n,k}=\frac{\pi^{\frac{d-n}{2}}}{2^{k+1}\Gamma\left(\frac{d-n}{2}+1+k\right)}.$

Here $\lfloor{\frac{n}{2}}\rfloor$ denotes the nearest integer that is less
than or equal to $\frac{n}{2}$.

We refer to the original paper [41] by H. Weyl or the book [15] for a proof of
the Tube Formula.

The other important result is the Gauss–Bonnet Formula itself, which, for a
closed Riemannian manifold $(M,g)$ of even dimension $m$, can be written as

$\mathcal{K}_{m}(M)={(2\pi)^{\frac{m}{2}}}\chi(M).$ (3.8)

As mentioned in Section 2 and explained in detail in [15, §5.5]), Allendoerfer
used some arguments and calculations from Weyl’s paper in a crucial step of
his proof.

We will now give a new proof of the Gauss–Bonnet–Hopf Formula (2.1), this time
an intrinsic one, as a special case of the Gauss–Bonnet Formula (3.8). To this
aim, it will be convenient to introduce the following definition:

######  Definition 3.11.

The _Gauss–Bonnet density_ of the Riemannian manifold $(M,g)$ of even
dimension $m$ is the function $\mathcal{R}:M\to\mathbb{R}$ defined as

$\mathcal{R}=\frac{1}{m!\left(\frac{m}{2}\right)!}\operatorname{C}^{m}(R^{{\frac{m}{2}}}).$
(3.9)

Observe that the top Lipschitz–Killing curvature is the integral of the
Gauss–Bonnet density:

$\mathcal{K}_{m}(M)=\int_{M}\mathcal{R}\operatorname{dvol}_{g}.$ (3.10)

Note also that the proof of Proposition 3.5 shows that the Gauss–Bonnet
density of an oriented even-dimensional hypersurface in Euclidean space is
equal, up to a multiplicative constant, to the Gauss–Kronecker curvature. More
precisely we have

$K=\frac{2^{{\frac{m}{2}}}}{(m!)^{2}}\operatorname{C}^{m}(R^{{\frac{m}{2}}})=\frac{m!!}{m!}\,\mathcal{R}.$
(3.11)

Assuming the Gauss–Bonnet Formula (3.8) has been established, we now easily
deduce (2.1):

$\int_{M}Kd\mathrm{vol}_{N}=\frac{{2}^{\frac{m}{2}}\,\left(\frac{m}{2}\right)!}{m!}\,\int_{M}\mathcal{R}\operatorname{dvol}_{M}=\frac{{2}^{\frac{m}{2}}\,\left(\frac{m}{2}\right)!}{m!}\,(2\pi)^{\frac{m}{2}}\chi(M)=\frac{1}{2}\operatorname{Vol}(S^{m})\chi(M),$
(3.12)

because the volume of an even-dimensional sphere is

$\operatorname{Vol}(S^{m})={2^{m+1}\pi^{\frac{m}{2}}\,\frac{\left(\frac{m}{2}\right)!}{m!}}.$

The proof of (3.8) will be given in next section.

######  Remark 3.12.

The Gauss–Bonnet density $\mathcal{R}$ that we have just defined is also
referred to as the Gauss–Bonnet Integrand. Fenchel proposed calling it the
Lipschitz–Killing curvature, and some authors do adopt that terminology.
However, we find it problematic considering our Definition 3.8. Note that
$\mathcal{R}$ is precisely the intrinsically defined curvature function that
was asked for in Hopf’s Curvatura Integra problem (up to a possible
multiplicative constant).

######  Remark 3.13.

The Gauss–Bonnet density, as well as its integral, also made an appearance in
the notable 1938 paper [28] by C. Lanczos. His work’s context was the
formulation of the fundamental laws of nature based on the principle of least
action. Lanczos observed that the integral $\int\mathcal{R}$ remains invariant
under variations of the metric $g$, establishing it as a differential
invariant of the manifold. He concluded that, within the framework of field
theory, the Gauss–Bonnet density doesn’t bring any additional information.
Lanczos’ work was carried out independently of the concurrent efforts by
mathematicians addressing the Hopf problem and the Gauss–Bonnet formula.

##  4 Chern’s Insight and the Tao of Gauss–Bonnet

For a compact even-dimensional manifold with boundary $(M,g)$, one needs to
add a boundary term to (3.8). This additional term is a sum of curvature
integrals involving the curvature tensor $R$ of $(M,g)$ and the second
fundamental form $h$ of the boundary $\partial M$. The Gauss–Bonnet Formula
takes the following form:

$(2\pi)^{\frac{m}{2}}\chi(M)=\mathcal{K}_{m}(M)+\sum_{k=0}^{\frac{m}{2}-1}b_{m,k}\,\mathcal{Q}_{k}({R},h\mid\partial
M,{g}),$

where the coefficient $b_{m,k}$ will be identified in due time. We will also
obtain a similar formula, where the tensor $R$ is being replaced by the
intrinsic curvature $\bar{R}$ of the boundary.

Our goal in this section is to present a comprehensive and intrinsic proof of
the above formula. Our approach closely follows the works of Chern [9, 10],
with a few minor adjustments, which are discussed in §6.2. In what follows,
the formalism of moving frames is extensively used; we refer the reader to the
appendix, which contains all the necessary definitions and formulas. Also, a
number of expressions below involve double factorials, see §6.3 for the
definition ans some basic properties.

###  4.1 The Emergence of the Pfaffian

The ingredients in Chern’s proof are differential forms and specifically E.
Cartan’s moving frame method. Referring to (3.10), we see that one needs to
closely consider the differential form

$\mathcal{R}\operatorname{dvol}_{g},$

where $\mathcal{R}$ is the Gauss–Bonnet density defined in (3.11). Invoking
Lemma A.2, we see that the double form $R^{m}\in\mathcal{D}^{m,m}(M)$ can be
written as

$R^{{\frac{m}{2}}}=\frac{1}{2^{{\frac{m}{2}}}}\sum_{\mu_{1},\dots,\mu_{m}=1}^{m}{\Omega}_{\mu_{1},\mu_{2}}\wedge\ldots\wedge{\Omega}_{\mu_{m-1},\mu_{m}}\otimes\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{m}},$

and Proposition 5.4 implies

$\frac{1}{m!}\operatorname{C}^{m}(R^{{\frac{m}{2}}})\theta^{1}\wedge\cdots\wedge\theta^{m}=\frac{1}{2^{{\frac{m}{2}}}}\sum_{\sigma\in
S_{m}}(-1)^{\sigma}{\Omega}_{\sigma_{1},\sigma_{2}}\wedge\ldots\wedge{\Omega}_{\sigma_{m-1},\sigma_{m}}.$

This relation can be written as

$\mathcal{R}\operatorname{dvol}_{g}=\operatorname{Pf}(\Omega),$ (4.1)

where the $m$-form

$\operatorname{Pf}(\Omega)=\frac{1}{m!!}\sum_{\sigma\in
S_{m}}(-1)^{\sigma}{\Omega}_{\sigma_{1},\sigma_{2}}\wedge\ldots\wedge{\Omega}_{\sigma_{m-1},\sigma_{m}}$

is the Pfaffian222The Pfaffian of a skew-symmetric matrix $A=(a_{ij})\in
M_{m}(\mathbb{R})$ is defined for $m$ even to be
$\operatorname{Pf}(A)=\frac{1}{m!!}\sum_{\sigma\in
S_{m}}(-1)^{\sigma}\prod_{i=1}^{{\frac{m}{2}}}a_{\sigma_{2i-1},\sigma_{i}}$.
The notion was introduced by A. Cayley. of the curvature form. Observe that
(4.1) implies that the Pfaffian $\operatorname{Pf}(\Omega)$ does not depend on
the chosen moving frame.

Examples. Recall that in dimension $2$, there is only one non-zero curvature
form (up to sign), namely $\Omega_{12}=-\Omega_{21}$. Therefore, for a
Riemannian surface $(S,g)$, the Pfaffian of its curvature form reduces to:

$\operatorname{Pf}(\Omega)=\frac{1}{2}(\Omega_{12}-\Omega_{21})=\Omega_{12}=\frac{1}{2}R_{12ij}\theta^{i}\wedge\theta^{j}=R_{1212}\operatorname{dvol}_{S}=K\operatorname{dvol}_{S},$

where $K$ is the Gauss curvature of the surface $S$.

If $M$ is a 4-dimensional manifold, then its Pfaffian is given by:

$\operatorname{Pf}(\Omega)=\frac{1}{8}\sum_{\sigma\in
S_{4}}(-1)^{\sigma}{\Omega}_{\sigma_{1},\sigma_{2}}\wedge{\Omega}_{\sigma_{3},\sigma_{4}}$

is a sum of $24$ terms. But using the symmetries $\Omega_{ab}=-\Omega_{bc}$
and $\Omega_{ab}\wedge\Omega_{cd}=\Omega_{cd}\wedge\Omega_{ab}$, it can be
reduced to

$\operatorname{Pf}(\Omega)={\Omega}_{1,2}\wedge{\Omega}_{3,4}+{\Omega}_{2,3}\wedge{\Omega}_{1,4}+{\Omega}_{3,1}\wedge{\Omega}_{2,4}.$

See exercise 2 in §7 for a generalization in higher dimension.

###  4.2 A Tale of Exactness, featuring a Vector Field

The next result lies at the core of Chern’s proof of the Gauss–Bonnet Formula.
It states that when the manifold carries a non-vanishing vector field, the
Pfaffian of the curvature form is exact.

######  Proposition 4.1 (Chern).

Let $(M,g)$ be an oriented Riemannian manifold of even dimension $m$. If $M$
admits an everywhere non vanishing vector field $\xi$, then
$\operatorname{Pf}(\Omega)$ is an exact differential form.

We can rephrase this Proposition by expressing that the Gauss–Bonnet density
$\mathcal{R}$ is a divergence. An immediate consequence is the following
corollary, which is a special case of the Gauss–Bonnet Formula:

######  Corollary 4.2.

Let $M$ be a compact oriented manifold without boundary of even dimension $m$.
If $M$ admits an everywhere non vanishing vector field $\xi$, then the top
global Lipshitz–Killing curvature of $M$ vanishes.

Indeed, by Stoke’s Theorem we have
$\displaystyle\mathcal{K}_{m}(M,g)=\int_{M}\mathcal{R}\operatorname{dvol}_{{g}}=0$.

The rest of his section is devoted to proving the above Proposition.

Proof. Following Chern’s ideas, we use the non-vanishing vector field $\xi$ to
construct an $(m-1)$-form $\Phi$ that satisfies
$\operatorname{Pf}(\Omega)=d\Phi$. To achieve this, we first introduce the
double form $h=h_{\zeta}\in\mathcal{D}^{1,1}(M)$ of type $(1,1)$ on $M$,
defined as

$h(X,Y)=\langle\nabla_{X}\zeta,Y\rangle,$ (4.2)

where $\zeta$ is the unit vector field $\zeta=\xi/|\xi|$. Next we define a
collection of $(m-1)$-forms $\Phi_{k}$, for $0\leq k<\frac{m}{2}$, as follows:

$\Phi_{k}=-\frac{2^{k}}{(m-1)!}\cdot\operatorname{C}^{m-1}\left(R^{k}\owedge
h^{m-1-2k}\right)\varpi,$ (4.3)

where $\varpi=\varpi_{\zeta}$ is defined in (A.2). Additionally, we set
$\Phi_{-1}=0$ for convenience.

We will now build a primitive of $\operatorname{Pf}(\Omega)$ as a linear
combination of the $\Phi_{k}$s. To perform the necessary computations, we
choose a positively oriented moving frame $e_{1},\dots,e_{m}$ such that
$e_{m}=\zeta$. This is always possible in a neighborhood $U\subset M$ of any
point. Referring to Definition A.1, we then have for any $1\leq i,j\leq m-1$

$h(e_{i},e_{j})=g(\nabla_{e_{i}}e_{m},e_{j})=\omega_{jm}(e_{i}),$

that is,

$h=\sum_{j=1}^{m-1}\omega_{jm}\otimes\theta^{j}.$ (4.4)

We will now use some formulas established in the appendix. The first one is
(A.3), which tells us that

$\varpi=-\theta^{1}\wedge\cdots\wedge\theta^{m-1},$

because $m=\dim(M)$ is even. The second formula is stated in Lemma A.2:

$R=\frac{1}{2}\sum_{i,j=1}^{m}\Omega_{ij}\otimes\theta^{i}\wedge\theta^{j}.$

We therefore have

$\displaystyle R^{k}\owedge h^{m-1-2k}=$
$\displaystyle\frac{1}{2^{k}}\sum_{\mu_{1},\ldots,\mu_{m-1}=1}^{m-1}\Omega_{\mu_{1},\mu_{2}}\wedge\dots\wedge\Omega_{\mu_{2k-1}\mu_{2k}}\wedge\omega_{\mu_{2k+1},m}\wedge\cdots\wedge\omega_{\mu_{m-1},m}\otimes\theta^{\mu_{1}}\wedge\cdots\wedge\theta^{\mu_{m-1}}.$

Applying now Proposition 3.4, we obtain the following local expression for the
$(m-1)$-form $\Phi_{k}$:

$\Phi_{k}=\sum_{\sigma\in
S_{m-1}}(-1)^{\sigma}{\Omega}_{\sigma_{1},\sigma_{2}}\wedge\ldots\wedge{\Omega}_{\sigma_{2k-1},\sigma_{2k}}\wedge\omega_{\sigma_{2k+1},m}\wedge\ldots\wedge\omega_{\sigma_{m-1},m}.$
(4.5)

This is how the form $\Phi_{k}$ appears in Chern’s paper [9, 10], and we will
now closely follow Chern’s computation of its exterior derivative. Because a
2-form commutes with any other differential form, we can write the exterior
derivative of $\Phi_{k}$ as

$\displaystyle d\Phi_{k}$ $\displaystyle=k\sum_{\sigma\in
S_{m-1}}(-1)^{\sigma}d{\Omega}_{\sigma_{1},\sigma_{2}}\wedge\Omega_{\sigma_{3},\sigma_{4}}\wedge\ldots\wedge{\Omega}_{\sigma_{2k-1},\sigma_{2k}}\wedge\omega_{\sigma_{2k+1},m}\wedge\ldots\wedge\omega_{\sigma_{m-1},m}$
$\displaystyle+(m-2k-1)\sum_{\sigma\in
S_{m-1}}(-1)^{\sigma}{\Omega}_{\sigma_{1},\sigma_{2}}\wedge\ldots\wedge{\Omega}_{\sigma_{2k-1},\sigma_{2k}}\wedge
d\omega_{\sigma_{2k+1},m}\wedge\omega_{\sigma_{2k+2},m}\wedge\ldots\wedge\omega_{\sigma_{m-1},m}.$

We next use the second Structure Equation (A.4) and the Bianchi identity
(A.5). These identities allow us to rewrite $d\Omega_{\sigma_{1},\sigma_{2}}$
and $d\omega_{\sigma_{k+1},m}$ as

$d\omega_{\sigma_{k+1},m}=\Omega_{\sigma_{k+1},m}-\sum_{s=1}^{m}\omega_{\sigma_{k+1},s}\wedge\omega_{s,m},$

and

$d\Omega_{\sigma_{1},\sigma_{2}}=\sum_{s=1}^{m}\left(\Omega_{\sigma_{1},s}\wedge\omega_{s,\sigma_{2}}-\omega_{\sigma_{1},s}\wedge\Omega_{s,\sigma_{2}}\right)=\sum_{s=1}^{m}\left(\Omega_{\sigma_{2},s}\wedge\omega_{\sigma_{1},s}-\Omega_{\sigma_{1},s}\wedge\omega_{\sigma_{2},s}\right).$

Carefully computing,333See [30, Lemma 3.14] for a detailed explanation. one
then finds that

$d\Phi_{k}=\frac{m-2k-1}{2k+2}\Psi_{k}+\Psi_{k-1}+\Xi_{k},$

where $\Psi_{k}$ is the $m$-form

$\displaystyle\Psi_{k}=(-1)^{k+1}(2k+2)\cdot\sum_{\sigma\in
S_{m-1}}(-1)^{\sigma}{\Omega}_{\sigma_{1},\sigma_{2}}\wedge\ldots\wedge{\Omega}_{\sigma_{2k-1},\sigma_{2k}}\wedge\Omega_{\sigma_{2k+1},m}\wedge\omega_{\sigma_{2k+2},m},\wedge\ldots\wedge\omega_{\sigma_{m-1},m}$

and $\Xi_{k}$ is a linear combinations of products of curvature and connection
forms containing some $\omega_{ij}$ with $1\leq i,j\leq(m-1)$. The forms
$\Psi_{k}$ are defined for $0\leq k\leq\frac{m}{2}-1$, but it is convenient to
also define $\Psi_{k-1}=0$.

A key fact about the previous formula, observed by Chern, is that we must have
$\Xi_{k}=0$. The reason is that the forms $\Phi_{k}$, and hence $d\Phi_{k}$,
do not depend on the chosen moving frame $e_{1},\dots,e_{m}$, except for the
constraint $e_{m}=\zeta$. Lemma A.5 implies therefore that all terms in
$d\Phi_{k}$ containing some $\omega_{ij}$ with $1\leq i,j\leq(m-1)$ must
cancel each other. We have thus established the following relation:

$d\Phi_{k}=\frac{m-2k-1}{2(k+1)}\Psi_{k}+\Psi_{k-1}.$ (4.6)

This identity gives us $\Psi_{k}$ as a linear combination of the $d\Phi_{r}$s.
Let us indeed write

$\Psi_{k}=\sum_{r=0}^{k}\lambda_{m,k,r}\,d\Phi_{r},$ (4.7)

where the unknown coefficients $\lambda_{m,k,r}$ are to be determined. From
(4.6) we have

$d\Phi_{k}=\sum_{r=0}^{k}\left(\frac{m-2k-1}{2(k+1)}\lambda_{m,k,r}+\lambda_{m,k-1,r}\right)d\Phi_{r}.$

The condition on $\lambda_{m,k,r}$ is then

$\frac{m-2k-1}{2(k+1)}\lambda_{m,k,r}+\lambda_{m,k-1,r}=\delta_{k,r},\ \text{
with $\lambda_{m,k,r}=0$ for $r>k$,}$

which is equivalent to

$\lambda_{m,k,r}=(-1)^{k-r}\prod_{j=r}^{k}\frac{2j+2}{m-2j-1}.$ (4.8)

We have established that each $\Psi_{k}$ is an exact form. Considering the
particular case $k=\frac{m}{2}-1$, we have

$\displaystyle\Psi_{\frac{m}{2}-1}$ $\displaystyle=m\cdot\sum_{\sigma\in
S_{m-1}}(-1)^{\sigma}{\Omega}_{\sigma_{1},\sigma_{2}}\wedge\ldots\wedge{\Omega}_{\sigma_{m-1},\sigma_{m}}$
$\displaystyle=\sum_{\sigma\in
S_{m}}(-1)^{\sigma}{\Omega}_{\sigma_{1},\sigma_{2}}\wedge\ldots\wedge{\Omega}_{\sigma_{m-1},\sigma_{m}}$
$\displaystyle=m!!\,\operatorname{Pf}(\Omega),$

and we conclude that $\operatorname{Pf}(\Omega)$ is an exact form.

∎

Remarks 1. The differential forms $\Psi_{k}$ have been defined using the
connection and curvature forms of a particular moving frame, however Formula
(4.7) implies that the $\Psi_{k}$ do not depend on the chosen frame, provided
$e_{m}=\zeta$.

2. The proof gives us an explicit formula for the primitive of the Pfaffian. We have

$\operatorname{Pf}(\Omega)=d\Phi,\quad\text{with}\quad\Phi=\sum_{r=0}^{\frac{m}{2}-1}a_{m,r}\Phi_{r},$
(4.9)

where the $a_{m,r}$ can be computed from (4.8):

$a_{m,r}=\frac{1}{m!!}\,\prod_{j=r}^{\frac{m}{2}-1}\frac{2j+2}{m-2j-1}={\frac{1}{(2r)!!\,\left(m-2\,r-1\right)!!}}.$
(4.10)

The coefficients $a_{m,r}$ can be written in several ways, which all appear in
the literature,. For instance444These identities hold for $m$ even and
$r<{\frac{m}{2}}$.

$a_{m,r}={\frac{1}{{2}^{r}r!\,\left(m-2\,r-1\right)!!}}=\frac{2^{\frac{m}{2}-2r}\left(\frac{m}{2}-r\right)!}{(m-2r)!\,r!}={\frac{\sqrt{\pi}}{2^{\frac{m}{2}}\,r!\,\Gamma\left(\frac{m-2\,r+1}{2}\right)}}.$
(4.11)

###  4.3 Harvesting the ripe Gauss–Bonnet Fruit

With the hard work behind us, we are now in a position to state and prove the
Gauss–Bonnet Formula, as originally formulated by Chern (see Equation (19) in
[10]).

######  Theorem 4.3.

The Euler characteristic of a compact Riemannian manifold $(M,g)$ of even
dimension $m$ with (possibly empty) boundary has the following integral
representation:

$(2\pi)^{\frac{m}{2}}\chi(M)=\int_{M}\operatorname{Pf}(\Omega)-\int_{\partial
M}\Phi.$

###### Proof.

For the proof, we choose a smooth vector field $\xi$ on $M$ with the following
properties:

  1. (i)

$\xi$ has only isolated singularities at $x_{1},\dots,x_{q}\in
M\setminus\partial M$.

  2. (ii)

At any boundary point, $\xi$ is non vanishing, outward pointing and orthogonal
to $\partial M$.

On the manifold $M^{\prime}=M\setminus\\{x_{1},\dots,x_{q}\\}$ we define the
normalized vector field $\zeta=\xi/\|\xi\|$ and the double form
$h(X,Y)=\langle X,\nabla_{Y}\zeta\rangle$.

Let us then denote by $B_{i}(\varepsilon)$ the open ball of radius
$\varepsilon$ centered at $x_{i}$. We assume $\varepsilon>0$ small enough so
that the balls $B_{i}(\varepsilon)$ are pairwise disjoint and have empty
intersection with $\partial M$, and we set
$M_{\varepsilon}=M\setminus\cup_{i=1}^{q}B_{i}(\varepsilon).$ The unit vector
field $\zeta=\xi/|\xi|$ is well defined on $M_{\varepsilon}$ and we have by
Stokes Theorem:

$\int_{M_{\varepsilon}}\operatorname{Pf}(\Omega)=\int_{\partial M}\Phi\ -\
\sum_{i=1}^{q}\int_{\partial B_{i}(\varepsilon)}\Phi,$

where the form $\Phi$ has been defined in (4.9), see also (4.3). Chern
observes then that

$\lim_{\varepsilon\to 0}\int_{\partial B_{i}(\varepsilon)}\Phi_{k}=0$

for $k\geq 1$. Furthermore, from the definition (4.3) of $\Phi_{0}$ and the
formula (3.4) representing the index, we see that

$\displaystyle\lim_{\varepsilon\to 0}\left(-\int_{\partial
B_{i}(\varepsilon)}\Phi_{0}\right)$ $\displaystyle=\lim_{\varepsilon\to
0}\frac{1}{(m-1)!}\cdot\mathcal{Q}_{0}(h\mid{\partial
B_{\varepsilon}},g)=(m-1)!\operatorname{Vol}(S^{m-1})\operatorname{Ind}(\xi,x_{i}).$

We have thus established that

$\displaystyle\int_{M}\operatorname{Pf}(\Omega)-\int_{\partial M}\Phi$
$\displaystyle=\lim_{\varepsilon\to
0}\,\sum_{i=1}^{q}\left(-a_{m,0}\right)\int_{\partial
B_{i}(\varepsilon)}\Phi_{0}$ $\displaystyle=\lim_{\varepsilon\to
0}\,\sum_{i=1}^{q}\frac{a_{m,0}}{(m-1)!}\cdot\mathcal{Q}_{0}(h\mid{\partial
B_{\varepsilon}},g)$
$\displaystyle=a_{m,0}(m-1)!\operatorname{Vol}(S^{m-1})\sum_{i=1}^{q}\operatorname{Ind}(\xi,x_{i}).$

Recall that $a_{m,0}=\dfrac{1}{(m-1)!!}$. Furthermore, since $m$ is even, we
have $\operatorname{Vol}(S^{m-1})=\dfrac{(2\pi)^{{\frac{m}{2}}}}{(m-2)!!}$,
which leads to the following relationship:

$a_{m,0}(m-1)!\,\operatorname{Vol}(S^{m-1})=(2\pi)^{\frac{m}{2}}.$

Applying the Poincaré–Hopf Theorem, we finally conclude that

$\displaystyle\int_{M}\operatorname{Pf}(\Omega)-\int_{\partial
M}\Phi=(2\pi)^{\frac{m}{2}}\sum_{i=1}^{q}\operatorname{Ind}(\xi)=(2\pi)^{\frac{m}{2}}\chi(M).$

∎

Let us pause and take a short moment to listen to Chern’s own account of his
discovery, as he recalls it in an interview with Allyn Jackson in 1998 (see
[24]).

> The Gauss–Bonnet formula is one of the important, fundamental formulas, not
> only in differential geometry, but in the whole of mathematics. Before I
> came to Princeton [in 1943] I had thought about it, so the development in
> Princeton was in a sense very natural. I came to Princeton and I met André
> Weil. He had just published his paper with Allendoerfer. Weil and I became
> good friends, so we naturally discussed the Gauss–Bonnet formula. And then I
> got my proof. I think this is one of my best works, because it solved an
> important, a fundamental, classical problem, and the ideas were very new.
> And to carry out the ideas you need some technical ingenuity. It’s not
> trivial. It’s not something where once you have the ideas you can carry it
> out. It is subtle. So I think this a very good piece of work.

###  4.4 Demystifying the Gauss–Bonnet Boundary term using Double Forms

In this section, we use the formalism of double forms to introduce two
alternative formulations of the Gauss–Bonnet Formula for a compact, oriented
Riemannian manifold $(M,g)$ with non-empty boundary and even dimension $m$.
The first reformulation involves the curvature tensor $R$ of the oriented
Riemannian manifold $(M,g)$, coupled with the second fundamental form $h$ of
its boundary. The formula can be expressed as follows:

$(2\pi)^{\frac{m}{2}}\chi(M)=\mathcal{K}_{m}(M)+\sum_{k=0}^{\frac{m}{2}-1}b_{m,k}\,\mathcal{Q}_{k}(R,h\mid\partial
M,\bar{g}).$ (4.12)

Here $\bar{g}$ is the first fundamental form of $\partial M$ as a hypersurface
in $(M,g)$. We will prove this formula and determine the coefficients
$b_{m,k}$ below.

We work with the orientation of the boundary $\partial M$ induced by the unit
outer normal field $\nu\in T(\partial M)^{\perp}$. This means that a basis
$v_{1},\ldots,v_{m-1}$ of $T_{p}(\partial M)$ is positively oriented if and
only if $\nu,v_{1},\ldots,v_{m-1}$ is a positively oriented basis in $T_{p}M$.
Referring to (A.3), we see that, under this orientation, the volume form on
the boundary is given by

$\operatorname{dvol}_{\partial M}=\varpi_{\nu}.$

We next choose a vector field $\zeta$ of length $1$, defined in some
neighborhood of $\partial M\subset M$ and such that $\zeta$ is outward
pointing and orthogonal to $\partial M$. Observe that $\zeta=\nu$ at every
boundary point $p\in\partial M$. We then define $\Phi_{k}$ and $h$ as
previously, that is,

$\displaystyle h(X,Y)=\langle X,\nabla_{Y}\zeta\rangle\quad\text{and
}\quad\Phi_{k}=-\frac{2^{k}}{(m-1)!}\cdot\operatorname{C}^{m-1}\left(R^{k}\owedge
h^{m-1-2k}\right)\varpi_{\zeta},$

and observe that the restriction of $h$ to the boundary coincides with its
second fundamental form.555In the literature, the second fundamental form is
frequently defined using the opposite sign convention. The present convention
yields positive principal curvatures for the boundary of the unit ball in
Euclidean space.

From Definition 3.6, we then have

$\displaystyle\mathcal{Q}_{k}(R,h\mid\partial M)$
$\displaystyle=\frac{1}{(m-1)!\,k!\,(m-2k-1)!}\int_{\partial
M}\operatorname{C}^{m-1}({R}^{k}\owedge
h^{m-2k-1})\operatorname{dvol}_{\partial M}$
$\displaystyle=\frac{-1}{2^{k}\,k!\,(m-2k-1)!}\int_{\partial M}\Phi_{k}.$

So if we now define the coefficients $b_{m,k}$ by the following relation:

$b_{m,k}\,\mathcal{Q}_{k}(R,h\mid\partial M)=-a_{m,k}\,\int_{\partial
M}\Phi_{k},$

we conclude that

$\displaystyle\mathcal{K}_{m}(M)+\sum_{k=0}^{\frac{m}{2}-1}b_{m,k}\,\mathcal{Q}_{k}(R,h\mid\partial
M,\bar{g})=\int_{M}\operatorname{Pf}(\Omega)\operatorname{dvol}_{M}-\int_{\partial
M}\Phi=(2\pi)^{\frac{m}{2}}\chi(M).$

Formula (4.12) is proved. Furthermore, the coefficients are calculated as
follows:

${b_{m,k}}=a_{m,k}\cdot{2}^{k}\,k!\,\left(m-2\,k-1\right)!=\frac{\left(m-2\,k-1\right)!}{\left(m-2\,k-1\right)!!}=(m-2k-2)!!$
(4.13)

The second reformulation of the Gauss–Bonnet Formula expresses the boundary
term using the intrinsic curvature tensor $\bar{R}$ of the boundary
hypersurface

$\mathcal{K}_{m}(M)+\sum_{k=0}^{\frac{m}{2}-1}c_{m,k}\,\mathcal{Q}_{k}(\bar{R},h\mid\partial
M,\bar{g})=(2\pi)^{\frac{m}{2}}\chi(M),$ (4.14)

where, again, the coefficients $c_{m,k}$ are to be determined. Using the Gauss
Equation

$R=\bar{R}-\frac{1}{2}h\owedge h,$

together with the linearity of the contraction, the fact that $\bar{R}$ and
$h\owedge h$ commute and the binomial formula, we see that the following
relation holds on the boundary:

$R^{p}\owedge
h^{m-1-2p}=\sum_{k=0}^{p}\binom{p}{k}\left(-\tfrac{1}{2}\right)^{p-k}\bar{R}^{k}\owedge
h^{m-1-2k},$

therefore

$\operatorname{C}^{m-1}\left(R^{p}\owedge
h^{m-1-2p}\right)=\sum_{k=0}^{p}\binom{p}{k}\left(-\tfrac{1}{2}\right)^{p-k}\,\operatorname{C}^{m-1}\left(\bar{R}^{k}\owedge
h^{m-1-2k}\right).$

From (3.3) and the above relation we obtain:

$\mathcal{Q}_{p}\left(R,h\mid\partial
M,\bar{g}\right)=\sum_{k=0}^{p}w_{m,p,k}\mathcal{Q}_{k}(\bar{R},h\mid\partial
M,\bar{g}),$

where

$w_{m,p,k}=\left(-\tfrac{1}{2}\right)^{p-k}\frac{(m-1-2k)!}{(m-1-2p)!\,(p-k)!}$

if $0\leq k\leq p$. It will be convenient to also define $w_{m,p,k}=0$ if
$k>p$. We then have

$\sum_{p=0}^{\frac{m}{2}-1}b_{m,p}\mathcal{Q}_{p}\left(R,h\mid\partial
M,\bar{g}\right)=\sum_{k=0}^{\frac{m}{2}-1}c_{m,k}\mathcal{Q}_{p}\left(\bar{R},h\mid\partial
M,\bar{g}\right)),$

where $c_{m,k}$ is defined as

$c_{m,k}=\sum_{p=k}^{\frac{m}{2}-1}w_{m,p,k}b_{m,p}.$

We have thus proved that (4.12) holds with

$\displaystyle c_{m,k}$
$\displaystyle=(m-2k-1)!\sum_{p=k}^{\frac{m}{2}-1}(-1)^{p-k}\frac{(m-2p-2)!!}{2^{p-k}(m-2p-1)!\,(p-k)!}$
$\displaystyle=(-1)^{\frac{m}{2}-k-1}(m-2k-3)!!$ (4.15)

Synthesizing our findings, we arrive at the following formulations of the
Gauss–Bonnet–Chern Formula:

$\boxed{\begin{aligned}
(2\pi)^{\frac{m}{2}}\chi(M)&=\int_{M}\operatorname{Pf}(\Omega)-\sum_{k=0}^{\frac{m}{2}-1}a_{m,k}\int_{\partial
M}\Phi_{k}\\\
&=\mathcal{K}_{m}(M)+\sum_{k=0}^{\frac{m}{2}-1}b_{m,k}\,\mathcal{Q}_{k}(R,h\mid\partial
M,\bar{g})\\\
&=\mathcal{K}_{m}(M)+\sum_{k=0}^{\frac{m}{2}-1}c_{m,k}\,\mathcal{Q}_{k}(\bar{R},h\mid\partial
M,\bar{g}),\end{aligned}}$ (4.16)

where the coefficients are

$\displaystyle a_{m,k}$
$\displaystyle=\frac{1}{(2k)!!\,\left(m-2\,k-1\right)!!}$ $\displaystyle
b_{m,k}$ $\displaystyle=(m-2k-2)!!$ $\displaystyle c_{m,k}$
$\displaystyle=(-1)^{\frac{m}{2}-k-1}(m-2k-3)!!$

We also remind the reader that

$\int_{M}\operatorname{Pf}(\Omega)=\int_{M}\mathcal{R}\operatorname{dvol}_{M}=\mathcal{K}_{m}(M).$

######  Remark 4.4.

Perhaps one ought to assign a name to the tensors $R^{k}\owedge h^{m-1-2k}$
appearing in the preceding formulas. Although the term Generalized
Gauss–Kronecker Curvature comes to mind, this terminology has already been
adopted in [26] for the tensor $R^{k}$. An alternative suggestion could be the
Gauss–Kronecker–Chern Curvatures.

It is noteworthy that the boundary term in the Gauss–Bonnet Formula appears as
the integral of a weighted homogeneous polynomial of degree $m-1$ in $R$ (the
curvature tensor) and $h$ (the second fundamental form), with weights of $1$
assigned to $h$ and $2$ assigned to $R$.

Another point that might be worth the effort would be to try and find a more
intuitive explanation for the values of the coefficients $b_{m,k}$ and
$c_{m,k}$ (in particular the presence of double factorials here is no
coincidence).

##  5 Examples and Applications

###  5.1 Flat Manifolds.

Recall that a Riemannian manifold $(M,g)$ is flat if its curvature tensor
satisfies $R=0$. Equivalently, every point $x\not\in\partial M$ admits a
neighborhood isometric to an open subset of $\mathbb{R}^{n}$.

######  Theorem 5.1.

If $(M,g)$ is a compact flat oriented Riemannnian manifold with boundary of
dimension $m$, then

$\int_{\partial M}K\operatorname{dvol}_{\partial
M}=\operatorname{Vol}(S^{m-1})\,\chi(M),$

where $K=\det(h_{ij})$ is the Gauss–Kronecker curvature of the boundary
$\partial M$.

###### Proof.

The proof splits in two cases according to whether $m$ is even or odd.
Consider first the case where $m$ is even,. We have $\mathcal{K}_{m}(M)=0$
since $R=0$. We also have $\mathcal{Q}_{k}(R,h\mid\partial M,\bar{g})=0$ for
any $k\geq 1$. Because $b_{m,0}=(m-2)!!$, the second formula in (4.16) reduces
thus to

$\int_{\partial M}K\operatorname{dvol}_{\partial
M}=\mathcal{Q}_{0}(h\mid\partial
M)=\frac{(2\pi)^{{\frac{m}{2}}}}{(m-2)!!}\,\chi(M)=\operatorname{Vol}(S^{m-1})\chi(M).$

If $m$ is odd, then $N=\partial M$ is an even-dimensional manifold and we know
from Equation (3.12) that

$\int_{N}K\operatorname{dvol}_{N}=\frac{1}{2}\operatorname{Vol}(S^{m-1})\,\chi(N)$

(see also Proposition 3.5). We conclude the proof with the relation
$\chi(\partial M)=2\chi(M)$, which holds when $m=\dim(M)$ is odd. ∎

An interesting outcome of this theorem is the following generalization of
Corollary 2.2. Suppose that an immersion $f:M\to\mathbb{R}^{m}$ is given,
where $M$ is a smooth compact manifold of (even or odd) dimension $m$. We can
then prescribe an orientation on $M$ by requiring $f$ to be a positively
oriented map and then define the Gauss map $\nu:\partial M\to S^{m-1}$.

######  Corollary 5.2.

The degree of the Gauss map $\nu:\partial M\to S^{m-1}$ is equal to the Euler
characteristic of $M$:

$\deg(\nu)=\chi(M).$

###### Proof.

Observe first that the pullback $g$ of the Euclidean metric by the map
$f:M\to\mathbb{R}^{m}$ is a flat metric on $M$. The proof is now obvious from
Theorem 5.1 since the Gauss–Kronecker curvature is the Jacobian of the Gauss
map. ∎

This result was independently discovered in 1960 by A. Haefliger and H.
Samelson, see [19, Theorem 3bis] and [37, Theorem 1]. In the special case
where $M$ is a domain in $\mathbb{R}^{n}$, H. Hopf had already established it
in [23, Satz VI.]. Some historical context is discussed in Gotlieb’s essay
[14].

###  5.2 Manifolds with Flat Boundary

Consider now the case of a compact oriented Riemannian manifold $(M,g)$, of
even dimension $m$, whose boundary is flat, that is, $\bar{R}~=0$. The second
formula in (4.16) tells us that

$\int_{M}\mathcal{R}\,\operatorname{dvol}_{M}+\int_{\partial
M}K\operatorname{dvol}_{\partial M}=(2\pi)^{{\frac{m}{2}}}\chi(M),$ (5.1)

where $\mathcal{R}$ is the Gauss–Bonnet density of $(M,g)$ and
$K=\det(h_{ij})$ is the Gauss–Kronecker curvature of the boundary $\partial
M$.

When $m=2$, this recovers the classical Gauss–Bonnet Formula for compact
surfaces with smooth boundary (up to a change in notation). In this case, we
have indeed $\bar{R}=0$ because the boundary is a one-dimensional manifold.

###  5.3 Space Forms

The Gauss–Bonnet density of a manifold $(M,g)$ of constant sectional curvature
$a$ and even dimension $m$ is a constant, equal to

$\mathcal{R}=\left(\tfrac{a}{2}\right)^{\frac{m}{2}}\frac{m!}{\left(\frac{m}{2}\right)!}=\frac{(2\pi
a)^{\frac{m}{2}}}{\frac{1}{2}\operatorname{Vol}(S^{m})}.$ (5.2)

If $M$ is compact, then the Gauss–Bonnet Formula immediately implies the
following identity, which was already proved in 1925 by Hopf in [22]:

$a^{\frac{m}{2}}\operatorname{Vol}(M)=\frac{1}{2}\operatorname{Vol}(S^{m})\chi(M).$
(5.3)

In particular, when $a>0$, the Euler characteristic $\chi(M)$ is positive, it
vanishes in the flat case ($a=0$), and its sign is $(-1)^{\frac{m}{2}}$ when
$a<0$. Note that Equation (5.3) holds trivially for flat manifolds, while it
becomes almost self-evident for spherical manifolds due to the fact that the
universal cover of such a manifold is a sphere. A geometric proof, based on a
geodesic triangulation, is presented in [35, Theorem 11.3.2]. Let us add a few
comments on this result:

  1. (1)

The formula (5.2) also works for the case of non compact, complete hyperbolic
manifolds of finite volume of even dimension. This was first proved by R.
Kellerhals and T. Zehrt [25] using an ideal geodesic triangulation of the
manifold. It can also be seen as a consequence of (5.1). Here is a brief
explanation of why this holds: such a manifold is the union of a compact
manifold with horospherical boundary and a finite collection of cusps. Each
connected component of the boundary is a compact flat manifold with constant
Gauss–Kronecker curvature $K=+1$ (in fact we have $h=\bar{g}$, equivalently
all the principal curvature are equal to $+1$). Progressively chopping off the
manifold at increasing distances, it becomes evident that $\int_{\partial
M}K\operatorname{dvol}_{\partial M}\to 0$ and we conclude by (5.1).

  2. (2)

Our considerations prove that the volume of a complete hyperbolic manifold of
even dimension $m$ must be an integer multiple of
$\frac{1}{2}\operatorname{Vol}(S^{m})$. Furthermore, if the manifold has
dimension $4$, we know from the work of J. Ratcliffe and T. Tschantz [36] that
every positive multiple of $\frac{4}{3}\pi^{2}$ is realized as the volume of a
complete 4-dimensional hyperbolic manifold.

  3. (3)

The situation is very different for odd-dimensional hyperbolic manifolds. A
fundamental result by W. Thurston and O. Jørgensen, based on a construction
involving hyperbolic Dehn surgeries, shows that the set of volumes of complete
hyperbolic 3-manifolds is a closed and non-discrete subset of $\mathbb{R}$,
containing infinitely many accumulation points, see [40] and [17].

###  5.4 Further results on non Compact Manifolds

An almost immediate consequence of (4.14) is the following

######  Theorem 5.3.

Let $(M,g)$ be a non-compact Riemannian manifold of even dimension $m$ and
finite topological type. Assume $g$ has an integrable Gauss–Bonnet density,
and that there exists an exhausting sequence

$M_{1}\subset\cdots\subset M_{j}\subset M_{j+1}\subset
M=\bigcup_{j=1}^{\infty}M_{j},$

such that each $M_{j}$ is a compact $m$-dimensional manifold with smooth
boundary. If

$\int_{\partial
M_{j}}\left(\|\bar{R}_{x}\|^{k}\|h_{x}\|^{m-1-2k}\right)\operatorname{dvol}_{g}\to
0\quad\text{as }j\to\infty$ (5.4)

for any integer $k$ such that $0\leq k\leq\frac{m}{2}-1$, then the
Gauss–Bonnet Formula holds; that is, we have

$(2\pi)^{\frac{m}{2}}\chi(M)=\mathcal{K}_{m}(M,g).$ (5.5)

By an observation of M. Gromov, this result applies to the case of complete
Riemannian manifolds with pinched negative sectional curvature, that is, whose
sectional curvature satisfies $-a\leq K\leq-b<0$, see [16, appendix 3]. When
the manifold has non-positive curvature, the control of the geometry at
infinity is more delicate but some explicit asymptotic conditions have been
given by S. Rosenberg in [34]; see also [20] for the case of arithmetic
manifolds. The paper [8] by Cheeger and Gromov contains a strikingly stronger
result.

###  5.5 The direct product of a Ball and a Sphere

Our next example will illustrate that, even in a very simple case, keeping
track of the boundary terms in the Gauss–Bonnet–Chern Formula can be tedious.
Nevertheless, this example holds significance as it validates the accurate
values of the coefficients $b_{m,k}$ in Formula (4.16).

Let us denote the standard metric on $\mathbb{R}^{p+1}$ as $g_{1}$ and the
standard metric on $\mathbb{R}^{q+1}$ as $g_{2}$. Their respective
restrictions to the unit spheres $S^{p}$ and $S^{q}$ are denoted by
$\bar{g}_{1}$ and $\bar{g}_{2}$. The Riemannian metric on the manifold $M$ is
given by $g=g_{1}+\bar{g}_{2}$, and its restriction to $\partial M$ is
$\bar{g}=\bar{g}_{1}+\bar{g}_{2}$. It is important to note that, in the
algebra of double forms, $\bar{g}_{1}^{j}=0$ if $j>p$ and $\bar{g}_{2}^{j}=0$
if $j>q$.

With this notation, one can write the curvature tensor $R$ of $M$ as
$R=\frac{1}{2}\bar{g}_{2}\owedge\bar{g}_{2}$ (because $B^{p+1}$ is flat, all
the curvature comes from $S^{q}$). It follows immediately that the
Gauss–Bonnet density defined in (3.9) vanishes on $M$:

$\mathcal{R}=\tfrac{1}{m!\left(\frac{m}{2}\right)!}\,\operatorname{C}^{m}(R^{{\frac{m}{2}}})=\tfrac{1}{2^{m}m!\left(\frac{m}{2}\right)!}\,\operatorname{C}^{m}(\bar{g}_{2}^{m})=0,$

because $m>q$. Furthermore, the second fundamental form of the boundary
$\partial M=S^{p}\times S^{q}$ is given by $h=\bar{g}_{2}$. We therefore have
for any $0\leq k\leq\frac{1}{2}(p+q)$

$R^{k}\owedge
h^{m-1-2k}=\left(\tfrac{1}{2}\right)^{k}\bar{g}_{1}^{2k}\owedge\bar{g}_{2}^{m-1-2k},$

which vanishes if either $2k>p$ or $(m-1-2k)>q$. This implies that

$R^{k}\owedge
h^{m-1-2k}=\begin{cases}\left(\tfrac{1}{2}\right)^{\frac{p}{2}}\bar{g}_{1}^{p}\owedge\bar{g}_{2}^{q},&\text{
if }p=2k,\vspace{0.14cm}\\\ \;0&\text{ else. }\end{cases}$

Note also that

$\bar{g}^{m-1}=\bar{g}^{p+q}=\sum_{j=0}^{p+q}\binom{p+q}{j}\,\bar{g}_{1}^{j}\,\owedge\bar{g}_{2}^{p+q-j}=\frac{(p+q)!}{p!\,q!}\bar{g}_{1}^{p}\,\owedge\bar{g}_{2}^{q}.$

We thus have $\mathcal{Q}_{k}(R,h\mid\partial M,\bar{g})=0$ if
$k\neq\frac{p}{2}$ and

$\displaystyle\mathcal{Q}_{\frac{p}{2}}(R,h\mid\partial M,\bar{g})$
$\displaystyle=\frac{1}{(p+q)!\,\left(\frac{p}{2}\right)!\,q!}\,\int_{\partial
M}\operatorname{C}^{p+q}\left(\left(\tfrac{1}{2}\right)^{\frac{p}{2}}\bar{g}_{1}^{p}\owedge\bar{g}_{2}^{q}\right)\operatorname{dvol}_{\partial
M}$
$\displaystyle=\frac{1}{(p+q)!\,\left(\frac{p}{2}\right)!\,q!}\,\int_{\partial
M}\operatorname{C}^{p+q}\left(\frac{p!\,q!}{2^{\frac{p}{2}}(p+q)!}\,g^{p+q}\right)\operatorname{dvol}_{\partial
M}$
$\displaystyle=\frac{p!}{2^{\frac{p}{2}}\left(\frac{p}{2}\right)!}\,\operatorname{Vol}(S^{p})\operatorname{Vol}(S^{q}).$

The latter quantity can be explicitly computed. If $p$ is even and $q$ odd, we
have

$\operatorname{Vol}(S^{p})\operatorname{Vol}(S^{q})=\frac{2^{\frac{p}{2}+1}\left(\frac{p}{2}\right)!}{p!\,(q-1)!!}\cdot(2\pi)^{\frac{p+q+1}{2}}.$
(5.6)

We finally conclude that

$\displaystyle\mathcal{K}_{m}(M)$
$\displaystyle+\sum_{k=0}^{\frac{m}{2}-1}b_{m,k}\,\mathcal{Q}_{k}(R,h\mid\partial
M,\bar{g})$
$\displaystyle=0+b_{p+q+1,\frac{p}{2}}\cdot\mathcal{Q}_{\frac{p}{2}}(R,h\mid\partial
M,\bar{g})$ $\displaystyle=(q-1)!!\
\frac{p!}{2^{\frac{p}{2}}\left(\frac{p}{2}\right)!}\,\operatorname{Vol}(S^{p})\operatorname{Vol}(S^{q})$
$\displaystyle=2\,(2\pi)^{\frac{p+q+1}{2}}$
$\displaystyle=(2\pi)^{\frac{m}{2}}\chi(M).$

######  Remark 5.4.

We emphasize that this calculation is independent of the proof of Proposition
4.1. Consequently, it offers an alternative approach to determining the exact
value of the coefficients $b_{m,k}$. This example is inspired by the closing
sentence in Volume 5 of Spivak’s treatise (page 391). Spivak suggests working
out the above computation and adds that “after performing the calculation, it
should be fun to compare with Chern’s paper”.

###  5.6 Rotationally Symmetric Metrics

In our last example, we consider the case of a rotationally symmetric smooth
Riemannian metric $g$ in a closed ball $B_{r}$ of radius $r$. Denoting by
$o\in B_{m}(r)$ its center, we may identify ${B}^{m}\setminus\\{o\\}$ with
$(0,r]\times S^{m-1}$, and the metric tensor can be represented as a warped
product

$g=dt^{2}+f(t)^{2}\,g_{0},$

where $g_{0}$ is the standard metric on the unit sphere $S^{m-1}$. The second
fundamental form and the intrinsic curvature tensor of the boundary sphere
$S_{r}=\partial B_{r}$ are then given by

$h=\frac{f^{\prime}(r)}{f(r)}\bar{g}=f^{\prime}(r)f(r)\,g_{0}\quad\text{and}\quad\bar{R}=\frac{g\owedge
g}{2\,f(r)^{2}}=\frac{1}{2}\,g_{0}\owedge g_{0}.$

The boundary contribution to the Gauss–Bonnet Formula (4.12) appears as a sum
of terms of the following type:

$c_{m,k}\,\mathcal{Q}_{k}(\bar{R},h\mid
S_{r},\bar{g})=\gamma_{m,k}\operatorname{Vol}\left(S^{m-1}\right)\,(f^{\prime}(r))^{m-2k-1},$

for some coefficient $\gamma_{m,k}$. The courageous reader will compute and
find the value

$\gamma_{m,k}={\frac{(m-1)!\,c_{m,k}}{{2}^{k}k!\,\left(m-1-2\,k\right)!}}=\frac{(-1)^{\frac{m}{2}-k-1}\left(m-1\right)!}{{2}^{\frac{m}{2}-k}k!\,\left(m-2k-1\right)\left(\frac{m}{2}-k-1\right)!}.$

The Gauss–Bonnet Formula for $(B_{r},g)$ is then the following identity:

$\int_{B_{r}}\mathcal{R}\operatorname{dvol}_{g}+\operatorname{Vol}(S^{m-1})\sum_{k=0}^{\frac{m}{2}-1}\gamma_{m,k}\,(f^{\prime}(r))^{m-2k-1}=(2\pi)^{m}.$
(5.7)

Let us look more closely at the cases of the Euclidean, spherical and
hyperbolic balls. In the Euclidean case, we have $\mathcal{R}=0$ and $f(t)=t$.
Then $f^{\prime}(t)=1$ and (5.7) reduced to the following summation of the
coefficients $\gamma_{m,k}$, which holds when $m$ is an even integer

$\sum_{k=0}^{\frac{m}{2}-1}\gamma_{m,k}\
=\frac{(2\pi)^{\frac{m}{2}}}{\operatorname{Vol}(S^{m-1})}=(m-2)!!$ (5.8)

Consider next the ball of radius $r$ in the hyperbolic space
$B_{r}\subset\mathbb{H}^{m}$ (with $m$ even). We have $f(r)=\sinh(r)$, and
therefore (5.7) gives us

$\int_{B_{r}}\mathcal{R}\operatorname{dvol}_{g}=\frac{(-2\pi)^{\frac{m}{2}}}{\frac{1}{2}\operatorname{Vol}(S^{m})}\operatorname{Vol}_{{}_{\mathbb{H}^{m}}}(B_{r})=\frac{(-2\pi)^{\frac{m}{2}}\operatorname{Vol}(S^{m-1})}{\frac{1}{2}\operatorname{Vol}(S^{m})}\int_{0}^{r}\left(\sinh(t)\right)^{m-1}dt.$

Using now (5.2), we see that the Gauss–Bonnet Formula (5.7) for the hyperbolic
ball is equivalent to the identity

${\frac{(-1))^{\frac{m}{2}}m!}{{2}^{\frac{m}{2}}\left(\frac{m}{2}\right)!}}\,\int_{0}^{r}\\!\left(\sinh\left(t\right)\right)^{m-1}\,{\rm
d}t\ +\
\sum_{k=0}^{\frac{m}{2}-1}\gamma_{m,k}\left(\cosh\left(r\right)\right)^{m-2\,k-1}\
=\ (m-2)!!$

A direct derivation of this identity (without invoking the Gauss–Bonnet
Formula) can be achieved through an induction-based argument, starting from
the relation:

$(m-1)\int_{0}^{r}(\sinh(t))^{m-1}(t)dt=\cosh(r)\left(\cosh(r)^{2}-1\right)^{\frac{m}{2}-1}-(m-2)\int_{0}^{r}(\sinh(t))^{m-3}(t)dt,$

which is easily verified with an integration by parts.

The argument and the calculations are similar in the spherical case. The
corresponding integral identity being

${\frac{m!}{{2}^{\frac{m}{2}}\left(\frac{m}{2}\right)!}}\,\int_{0}^{r}\\!\left(\sin\left(t\right)\right)^{m-1}\,{\rm
d}t\ +\
\sum_{k=0}^{\frac{m}{2}-1}\gamma_{m,k}\left(\cos\left(r\right)\right)^{m-2\,k-1}\
=\ (m-2)!!$

We leave the detailed verification to the interested reader.

##  6 Miscellaneous Remarks

###  6.1 Remarks on Signs and Other Conventions

In the literature, including in Spivak’s books, the curvature tensor $R$ is
often defined with the opposite sign. Nonetheless, we chose to follow the
convention as in Gray’s book [15], as it aligns well with the formalism of
double forms. However, the crucial point to note is that the signs in our
Lemma A.2 match those in [38] and [42].

These references have also a different convention on indices. They write the
connection and curvatures forms as

$\omega^{i}_{j}=\omega_{ij}\quad\text{and}\quad\Omega^{i}_{j}=\Omega_{ij}.$
(6.1)

Our convention is justified by our choice to work with the covariant curvature
tensor $R_{ijkl}$ rather than the mixed one $R^{i}_{\>ijkl}$. Observe however
that both index conventions yield the same quantities, as the general
transformation rule

$\Omega^{i}_{j}=\sum_{k}g_{ik}\,\Omega^{k}_{j}$

reduces to (6.1) due to the orthonormality of the moving frame.

###  6.2 Comparison with Chern’s Papers

We strongly encourage the reader to look into Chern’s two original articles.
Although not easily accessible, they are concise, elegant, and remarkably
insightful. Both papers start with a succinct review of the moving frame
method, in Élie Cartan’s style. In this approach, given a moving frame
$Pe_{1}\cdots e_{m}$, the following equations are posed:

$\displaystyle dP=\sum_{i}\omega_{i}e_{i},\hskip 56.9055pt$
$\displaystyle\Omega_{ij}=d\omega_{ij}-\sum_{k}\omega_{ik}\omega_{kj},$
$\displaystyle de_{i}=\sum_{j}\omega_{ij}e_{j},\hskip 56.9055pt$
$\displaystyle d\Omega_{ij}=\omega_{ik}\Omega_{kj}-\omega_{jk}\Omega_{ki}.$

If, following Cartan, we interpret the symbol $P$ (the “point $P$ itself”) as
representing the identity map of the manifold to itself, then $dP$ is the
identity map in the tangent space. The first equation tells us that
$\omega_{i}=\theta^{i}$ is the dual coframe. The second and third equations
are the structure equations, with $de_{i}$ interpreted as $\nabla e_{i}$. The
last equation represents the Bianchi identity. Cartan, and Chern after him,
use these equations as definitions of the connection and curvature forms. Note
however that these definitions yield the opposite sign compared to what
appears in the present paper:

$\omega_{ij}^{\text{Chern}}=-\omega_{ij},\qquad\Omega_{ij}^{\text{Chern}}=-\Omega_{ij}.$

This differences in sign are reflected in several equations. Another important
difference is that Chern does not state our Proposition 4.1 concerning
manifolds with a vector field. Instead, he works with the unit tangent bundle
$SM$ of the manifold $M$ and proves that the pullback of the Pfaffian
$\operatorname{Pf}(\Omega)$ is an exact form on $SM$. This operation is called
a transgression of the corresponding differential form. The calculation is
essentially the same as the one presented in our proof of Proposition 4.1.

Our presentation of the proof shows that working on the sphere bundle and
transgressing the Pfaffian are not required to work out Chern’s argument and
prove the intrinsic Gauss–Bonnet Formula. However, the concept of
transgression has played a central role in the theory of characteristic
classes.

###  6.3 Double Factorials and the volume of Spheres

Here we briefly discuss some quantities that have been used throughout the
paper. The double factorial is inductively defined for any integer $n\geq-1$
by the conditions $(-1)!!=0!!=1$ and

$n!!=n(n-2)!!,$

for $n\geq 1$. We thus have for even integers:

$n!!=n\cdot(n-2)\cdot(n-4)\cdots
2=2^{\frac{n}{2}}\cdot\left(\tfrac{n}{2}\right)!$

and for odd integers

$n!!=n\cdot(n-2)\cdot(n-4)\cdot
1=\frac{(n+1)!}{2^{\frac{n+1}{2}}\cdot\left(\frac{n+1}{2}\right)!}=\frac{2^{\frac{n+1}{2}}\,\Gamma\left(\tfrac{n+1}{2}\right)}{\sqrt{\pi}},$

where $\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}\,dt$ is the Euler Gamma
function.

The double factorial has the following simple combinatorial interpretation:
For a set with $n$ elements, the number of different ways to form
$\tfrac{n}{2}$ unordered pairs if $n$ is even, or $\tfrac{n-1}{2}$ unordered
pairs with one element left unmatched if $n$ is odd, is given by

$\begin{cases}(n-1)!!&\text{ if $n$ is even,}\\\ \quad n!!&\text{ if $n$ is
odd.}\end{cases}$

The other ubiquitous quantity is the volume (or should we call it the area?)
of the unit sphere $S^{d}\subset\mathbb{R}^{d+1}$. It can be written in
several different formalisms, starting with

$\operatorname{Vol}(S^{d})=2\,{\frac{{\pi}^{\frac{d+1}{2}}}{\Gamma\left(\frac{d+1}{2}\right)}}.$

We see that for $d$ odd we have

$\operatorname{Vol}(S^{d})=2\,{\frac{{\pi}^{\frac{d+1}{2}}}{\left(\frac{d-1}{2}\right)!}}={\frac{\left(2\,\pi\right)^{\frac{d+1}{2}}}{\left(d-1\right)!!}},$

and for even $d$:

$\operatorname{Vol}(S^{d})={\frac{{2}^{d+1}{\pi}^{\frac{d}{2}}\left(\frac{d}{2}\right)!}{d!}}={\frac{2\left(2\,\pi\right)^{\frac{d}{2}}}{\left(d-1\right)!!}}.$

##  7 Exercises

In conclusion of this paper, we propose a few exercises.

1. Verify that, in the case of a surface, each of the three formulations (4.16) yields the classical 2-dimensional Gauss–Bonnet Formula (this amounts essentially to identifying the boundary term as the integral of the geodesic curvature, with the correct orientation).

2. Recall that the Pfaffian of an $m\times m$ matrix $A$, where $m$ is even, is the following sum involving $m!$ terms:

$\operatorname{Pf}(A)=\frac{1}{m!!}\sum_{\sigma\in
S_{m}}(-1)^{\sigma}a_{\sigma(1)\sigma(2)}\cdot a_{\sigma(3)\sigma(4)}\cdots
a_{\sigma(m-1)\sigma(m)}.$

In this excercice we will show that the Pfaffian can be rewritten as a sum of
only $(m-1)!!$ terms, each involving a different partition of the index set
$\\{1,\dots,m\\}$ in disjoint pairs. To this aim, we consider the following
subset

$\mathcal{P}_{m}=\\{\sigma\in
S_{m}\mid\sigma_{2i+2}>\max\\{\sigma_{2i},\sigma_{2i+1}\\},\;1\leq
i<\tfrac{m}{2}\\}$

of $S_{m}$. Your task is to first verify the following identity:

$\frac{1}{m!!}|S_{m}|=(m-1)!!=|\mathcal{P}_{m}|.$

Then prove that

$\operatorname{Pf}(A)=\sum_{\sigma\in\mathcal{P}_{m}}(-1)^{\sigma}a_{\sigma(1)\sigma(2)}\cdot
a_{\sigma(3)\sigma(4)}\cdots a_{\sigma(m-1)\sigma(m)}.$

3. Use the Gauss–Bonnet formula to prove that if $M_{1}$ and $M_{2}$ are two closed manifolds, then $\chi(M_{1}\times M_{2})=\chi(M_{1})\chi(M_{2})$. Extend this to the case where one of the manifolds has a non-empty boundary.

4. Generalize the Gauss–Bonnet Formula to the case of a closed non-orientable manifold. Can this be done for manifolds with non-empty boundaries?

5. Verify equation (4.4) in the proof of Proposition 4.1.

6. Prove that if $N$ is a hypersurface in a Riemannian manifold $(M,g)$, then the Gauss Equation can be written as

$\bar{R}=R+\frac{1}{2}g\owedge g,$

where $R$ is the curvature tensor of the ambient manifold $(M,g)$, $\bar{R}$
is the intrinsic curvature tensor of the submanifold $N$ and $h$ is the second
fundamental form of $N$.

Two proofs (at least) are possible. One proof is to reduce the above formula
to the Gauss Equation expressed in a formalism familiar to the reader. The
other is to try and write a direct proof using the moving frame formalism.

7. Let $M$ be a 4-dimensional manifold. Use formula (4.5) to calculate the forms $\Phi_{0}$, $\Phi_{1}$, $\Psi_{0}$, and $\Psi_{1}$. Find then an explicit primitive of the Pfaffian in this case (follow the steps in the proof of Proposition 4.1).

8. Our proof of the Gauss–Bonnet–Chern Formula uses the Poincaré–Hopf theorem. Show that, in fact, these two results are equivalent.

The argument will proceed in two steps. First, reverse the proof of Theorem
4.3 to demonstrate that the sum of the indices of a vector field is an
invariant of the manifold. Then, identify this invariant by selecting a well-
chosen vector field (for example, the gradient of a Morse function).

9. Let $h_{1},h_{2}\in\mathcal{D}^{1,1}(M)$ be two double forms of type $(1,1)$ and set $A=h_{1}\owedge h_{2}$. Show that if $h_{1}$ and $h_{2}$ are symmetric, then $A$ satisfies the _first Bianchi identity:_

$A(X,Y;U,V)+A(Y,U;X,V)+A(U,X;Y,V)=0.$

10. With the help of a computer algebra system such as Maple, Mathematica, or SageMath, verify the validity of some, or all, of the following identities appearing in the present paper: (4.10), (4.11), (4.13), (4.4), (5.6) and (5.8). After mastering this task, you should be able to navigate through the computational details of Example 5.6.

11. In Example 5.5 we have discussed the Gauss–Bonnet Formula for the manifold $M=B^{p+1}\times S^{q}$ assuming $p$ is even and $q$ is odd. What happens if, instead, $p$ is odd and $q$ is even?

##  Appendix A Appendix: Some Background in Differential Geometry

In this appendix, we review some of the geometric concepts employed in this
paper and establish our notation. Additionally, we provide a detailed
introduction to the moving frame method.

###  A.1 Some Linear Algebra on the (co)tangent Space

By its very definition, a Riemannian metric $g$ on a smooth manifold $M$
endows each tangent space with a Euclidean structure. The scalar product of
two tangent vectors $X$ and $Y$ in $T_{p}M$ will be denoted interchangeably as
$g_{p}(X,Y)$ or $\langle X,Y\rangle_{p}$, and the Riemannian norm will be
represented by $\|X\|_{p}=\sqrt{\langle X,Y\rangle_{p}}$. The subscript $p$
will be omitted when there is no ambiguity. The same notation will be used
when $X$ and $Y$ are smooth vector fields over $M$; in this case, the scalar
product becomes a smooth function on $M$.

To any tangent vector $X\in T_{p}M$, we associate a covector $X^{\flat}\in
T_{p}^{*}M$ defined by the relation $X^{\flat}(Y)=\langle X,Y\rangle$ for any
$Y\in T_{p}M$. This yields a linear isomorphism from the tangent space
$T_{p}M$ to the cotangent space $T_{p}^{*}M$, and the inverse isomorphism maps
a covector $\varphi\in T_{p}^{*}M$ to the unique tangent vector
$\varphi^{\sharp}\in T_{p}M$ such that
$\langle\varphi^{\sharp},Y\rangle=\varphi(Y)$, again for any $Y\in T_{p}M$.
The operators $\sharp$ and $\flat$ are sometimes called the musical
isomorphisms.666The name and notation have apparently been introduced by
Marcel Berger in the 1970s. A scalar product is then defined on
$\Lambda^{k}T^{*}_{p}M$, first for $k=1$ by

$\langle\alpha,\beta\rangle=\langle\alpha^{\sharp},\beta^{\sharp}\rangle$

for any $\alpha,\beta\in T^{*}_{p}M$, then by the formula:

$\langle\alpha_{1}\wedge\cdots\wedge\alpha_{k},\beta_{1}\wedge\cdots\wedge\beta_{k},\rangle=\det\left(\langle\alpha_{i},\beta_{j}\rangle\right),$
(A.1)

where $\alpha_{i},\beta_{j}\in T^{*}_{p}M$. The “volume element” at the point
$p$ is the unique element $\Theta=\Theta_{p}\in\Lambda^{m}T_{p}^{*}M$
representing the natural orientation and such that $\|\Theta\|=1$. Its
existence and uniqueness follows from the fact that
$\dim\left(\Lambda^{m}T_{p}^{*}M\right)=1$. It is characterized by the
condition

$\Theta_{p}(e_{1},\dots,e_{m})=1,$

for any positively oriented orthonormal basis $\\{e_{1},\dots,e_{m}\\}$ of
$T_{p}M$. The volume element is independent of the chosen moving frame, making
it a globally defined $m$-form on the manifold. We will commonly use the
notation:

$\operatorname{dvol}_{g}=\Theta.$

This notation, however, does not assume that $\Theta$ is a closed form.
Instead, we view it as a measure on the manifold. In local coordinates, it can
be expressed as:

$\operatorname{dvol}_{g}=\sqrt{\det(g_{ij})}\,dx_{1}\cdots dx_{m},$

where the $g_{ij}$ are the components of the metric tensor.

We recall that the Hodge star operator is the isomorphism:
$*:\Lambda^{k}T^{*}_{p}M\to\Lambda^{m-k}T^{*}_{p}M$ which is defined by the
condition

$\alpha\wedge(*\alpha)=\|\alpha\|^{2}\Theta.$

The dual basis of a positively oriented orthonormal basis
$\\{e_{1},\dots,e_{m}\\}\subset T_{p}M$ will be denoted by
$\\{\theta^{1},\dots,\theta^{m}\\}\subset T_{p}^{*}M$ of $T_{p}M$. It is given
by $\theta^{i}=e_{i}^{\flat}$ and we have

$\Theta=\theta^{1}\wedge\cdots\wedge\theta^{m}.$

It is easy to check that

$*\theta^{i_{1}}\wedge\cdots\wedge\theta^{i_{k}}=\varepsilon\;\theta^{j_{1}}\wedge\cdots\wedge\theta^{j_{n-k}},$

where $\\{i_{1},\dots,i_{k},j_{1},\dots,j_{m-k}\\}$ is any permutation of
$\\{1,\dots,m\\}$ and $\varepsilon=\pm 1$ is its signature. Furthermore, the
collection

$\\{\theta^{i_{1}}\wedge\cdots\wedge\theta^{i_{k}}\\}_{i_{1}<i_{2}<\cdots<i_{k}}$

forms an orthonormal basis of $\Lambda^{k}T^{*}_{p}M$. Using these operators,
we define the isomorphism $\varpi:T_{p}M\to\Lambda^{n-1}T^{*}_{p}M$ by777Using
the interior product, one can also define it as
$\varpi_{{}_{X}}=\iota_{X}(\Theta)$.

$\varpi_{{}_{X}}=*(X^{\flat}).$ (A.2)

This isomorphism will play an important role in the sequel; note that
$\varpi_{{}_{X}}$ is also characterized by the condition

$X^{\flat}\wedge\varpi_{{}_{X}}=\|X\|^{2}\,\Theta.$

As an important example, observe that if $\\{e_{1},\dots,e_{m}\\}$ is a
positively oriented orthonormal basis of $T_{p}M$, then

$\varpi_{{e_{m}}}=(-1)^{m-1}\,\theta^{1}\wedge\cdots\wedge\theta^{m-1}.$ (A.3)

###  A.2 Connection and Curvature

Riemannian geometry is of course much more than linear algebra in tangent
spaces. To compare nearby tangent spaces effectively, we need the concept of
connection. In Kozsul’s formalism, the Levi-Civita connection of $(M,g)$ is
seen as the unique operator that associates to any pair of smooth vector
fields $X,Y$ on $M$ a new vector field, denoted as $\nabla_{X}Y$, and
satisfying the following conditions:

  1. (i)

$\nabla_{X}(Y_{1}+Y_{2})=\nabla_{X}Y_{1}+\nabla_{X}Y_{2}$ and
$\nabla_{X_{1}+X_{2}}Y=\nabla_{X_{1}}Y+\nabla_{X_{2}}Y$.

  2. (ii)

$\nabla_{fX}Y=f\nabla_{X}Y$ and $\nabla_{X}(fY)=f\nabla_{X}Y+X(f)Y$ for any
$f\in C^{\infty}(M)$.

  3. (iii)

$\nabla_{X}Y-\nabla_{Y}X=[X,Y]$ (the Lie bracket of $X$ and $Y$).

  4. (iv)

$Z\langle X,Y\rangle=\langle\nabla_{Z}X,Y\rangle+\langle
X,\nabla_{Z}Y\rangle$.

We will denote by $R$ the associated covariant curvature tensor. It is the
$(0,4)$-tensor field defined by

$R(X,Y;Z,W)=\langle\nabla_{[X,Y]}Z+\nabla_{Y}\nabla_{X}Z-\nabla_{X}\nabla_{Y}Z,W\rangle.$

Its basic properties are:

  1. (a)

$R$ is $C^{\infty}(M)$-multilinear.

  2. (b)

$R(X,Y,Z,W)=R(Z,W,X,Y)=-R(Y,X,Z,W)=-R(X,Y,W,Z)$.

  3. (c)

$R(X,Y,Z,W)+R(Y,Z,X,W)+R(Z,X,Y,W)=0$ (the first Bianchi Identity).

A consequence of property (a) is that the value of $R(X,Y,Z,W)$ at a point
$p\in M$ only depends on the values of the vectors $X_{p},Y_{P},Z_{p},W_{p}\in
T_{p}M$ at this point and not on the global, or even local, behavior of the
vector fields. A consequence of (b) is that if $v,w\in T_{p}M$ are linearly
independent, then the scalar

$K(v,w)=\frac{R(v,w;v,w)}{\|v\|^{2}\|w\|^{2}-\langle u,v\rangle^{2}}$

only depends on the $2$-plane $\sigma=\mathrm{Span}(u,v)\subset T_{p}M$
spanned by these vectors. This value is the sectional curvature of $\sigma$.

###  A.3 Moving the Frame with Élie Cartan

The calculus of differential forms was developed in the 1920s by Élie Cartan,
who employed it as a convenient formalism in Riemannian geometry.888Chern
became acquainted with Cartan’s calculus first through his interactions with
E. Kähler in Hamburg, and later when he met Cartan in Paris. See his comments
on Cartan in [24]. Helpful references include Chapter 7 in [38, Vol II],
Chapter 4 in T.J. Willmore’s book [42], and the books [6, 7] by Cartan
himself.

Here we briefly recall the basic definitions and we state and prove the main
equations that are needed in this paper. A moving frame in some open set $U$
of the Riemannian manifold $(M,g)$ is simply a collection of $m$ smooth vector
fields $e_{1},\dots,e_{m}:U\to TU$ defined on $U$ such that $\langle
e_{i},e_{j}\rangle=\delta_{ij}$ at every point of $U$. We will always assume
the moving frame to be positively oriented. The dual coframe
$\theta^{1},\dots\theta^{m}$ will be denoted as $\theta^{i}=e_{i}^{\flat}$. It
is defined by

$\theta^{i}(X)=g(e_{i},X).$

######  Definition A.1.

The _connection forms_ associated with the moving frame $\\{e_{i}\\}$ are the
$1$-forms defined on $U$ by:

$\omega_{ij}(X)=g(e_{i},\nabla_{X}e_{j})=\theta^{i}(\nabla_{X}e_{j}),$

and the _curvature forms_ are the $2$-forms:

$\Omega_{ij}(X,Y)=R(X,Y;e_{i},e_{j}).$

Observe that the connection forms are characterized by the identity

$\nabla_{X}e_{j}=\sum_{i=1}^{m}\omega_{ij}(X)e_{i},$

note also the skew symmetry:

$\omega_{ij}+\omega_{ji}=\Omega_{ij}+\Omega_{ji}=0.$

The next lemma gives us three useful formulas representing the curvature in
the moving frame formalism:

######  Lemma A.2.

Under the previously established notation for the moving coframe, the
curvature forms $\Omega_{ij}$ can ve written as

$\Omega_{ij}=\frac{1}{2}\sum_{k,l=1}^{m}R_{ijkl}\,\theta^{k}\wedge\theta^{l}.$

And the curvature tensor $R$ is expressed as

$\displaystyle R$
$\displaystyle=\frac{1}{4}\sum_{i,j,k,l=1}^{m}R_{ijkl}\,\theta^{i}\wedge\theta^{j}\otimes\theta^{k}\wedge\theta^{l}$
$\displaystyle=\frac{1}{2}\sum_{i,j=1}^{m}\Omega_{ij}\otimes\theta^{i}\wedge\theta^{j},$

where $R_{ijkl}=R(e_{i},e_{j},e_{k},e_{l})$.

###### Proof.

From the symmetries of the curvature tensor we have
$R_{klij}=R_{ijkl}=-R_{ijlk}$, therefore

$\Omega_{ij}=\sum_{k,l=1}^{m}R_{klij}\,\theta^{k}\otimes\theta^{l}=\sum_{k,l=1}^{m}R_{ijkl}\,\theta^{k}\otimes\theta^{l}=\frac{1}{2}\sum_{k,l=1}^{m}R_{ijkl}\,\theta^{k}\wedge\theta^{l}.$

This proves the first equation. The second equation is proved as follows

$\displaystyle R$
$\displaystyle=\sum_{i,j,k,l=1}^{m}R_{ijkl}\left(\theta^{i}\otimes\theta^{j}\right)\otimes\left(\theta^{k}\otimes\theta^{l}\right)=\sum_{i,j,k,l=1}^{m}R_{ijkl}\left(\theta^{k}\otimes\theta^{l}\right)\otimes\left(\theta^{i}\otimes\theta^{j}\right)$
$\displaystyle=\frac{1}{4}\sum_{i,j,k,l=1}^{m}R_{ijkl}\left(\theta^{k}\otimes\theta^{l}-\theta^{l}\otimes\theta^{k}\right)\otimes\left(\theta^{i}\otimes\theta^{j}-\theta^{j}\otimes\theta^{i}\right)$
$\displaystyle=\frac{1}{4}\sum_{i,j,k,l=1}^{m}R_{ijkl}\left(\theta^{k}\wedge\theta^{l}\right)\otimes\left(\theta^{i}\wedge\theta^{j}\right)$
$\displaystyle=\frac{1}{2}\sum_{k,l=1}^{m}\Omega_{ij}\otimes\theta^{i}\wedge\theta^{j}.$

∎

######  Proposition A.3 (É. Cartan’s Structure Equations).

The exterior derivative of the moving coframe and the connection forms are
given by

$d\theta^{i}=-\sum_{j=1}^{m}\omega_{ij}\wedge\theta^{j}\quad\mathrm{and}\quad
d\omega_{ij}=\Omega_{ij}-\sum_{k=1}^{m}\omega_{ik}\wedge\omega_{kj}.$ (A.4)

Proof. We prove the first structure equation as follows:

$\displaystyle d\theta^{i}(X,Y)$
$\displaystyle=X(\theta^{i}(Y))-Y(\theta^{i}(X))-\theta^{i}([X,Y])$
$\displaystyle=Xg(e_{i},Y)-Yg(e_{i},X)-g(e_{i},[X,Y])$
$\displaystyle=g(\nabla_{X}e_{i},Y)+g(e_{i},\nabla_{X}Y)-g(\nabla_{Y}e_{i},X)-g(e_{i},\nabla_{Y}X)-g(e_{i},[X,Y])$
$\displaystyle=g(\nabla_{X}e_{i},Y)-g(\nabla_{Y}e_{i},X)$
$\displaystyle=\sum_{j=1}^{m}\left(g(\nabla_{X}e_{i},e_{j})g(e_{j},Y)-g(\nabla_{Y}e_{i},e_{j})g(e_{j},X)\right)$
$\displaystyle=\sum_{j=1}^{m}\left(\omega_{ji}(X)\theta^{j}(Y)-\omega_{ji}(Y)\theta^{j}(X)\right).$
$\displaystyle=-\sum_{j=1}^{m}\left(\omega_{ij}\wedge\theta^{j}\right)(X,Y)$

(we have used the symmetry of $\nabla$ on the fourth line). To prove the
second structure equation, we compute

$\displaystyle d\omega_{ij}(X,Y)$
$\displaystyle=X(\omega_{ij}(Y))-Y(\omega_{ij}(X))-\omega_{ij}([X,Y])$
$\displaystyle=X(g(e_{i},\nabla_{Y}e_{j}))-Y(g(e_{i},\nabla_{X}e_{j}))-g(e_{i},\nabla_{[X,Y]}e_{j})$
$\displaystyle=(g(\nabla_{X}e_{i},\nabla_{Y}e_{j})+g(e_{i},\nabla_{X}\nabla_{Y}e_{j}))-(g(\nabla_{Y}e_{i},\nabla_{X}e_{j})-g(e_{i},\nabla_{Y}\nabla_{X}e_{j}))-g(e_{i},\nabla_{[X,Y]}e_{j})$
$\displaystyle=g(\nabla_{X}e_{i},\nabla_{Y}e_{j})-g(\nabla_{Y}e_{i},\nabla_{X}e_{j})+g(e_{i},\nabla_{X}\nabla_{Y}e_{j}-\nabla_{Y}\nabla
Xe_{j}-\nabla_{[X,Y]}e_{j})$
$\displaystyle=g(\nabla_{X}e_{i},\nabla_{Y}e_{j})-g(\nabla_{Y}e_{i},\nabla_{X}e_{j})-R(X,Y;e_{j},e_{i}).$

Observe now that on the one hand

$g(\nabla_{X}e_{i},\nabla_{Y}e_{j})=\sum_{k=1}^{m}\theta^{k}(\nabla_{X}e_{i})\theta^{k}(\nabla_{Y}e_{j})=\sum_{k=1}^{m}\omega_{k,i}(X)\omega_{k,j}(Y)=-\sum_{k=1}^{m}\omega_{i,k}(X)\omega_{k,j}(Y),$

therefore

$g(\nabla_{X}e_{i},\nabla_{Y}e_{j})-g(\nabla_{Y}e_{i},\nabla_{X}e_{j})=-\sum_{k=1}^{m}\omega_{ik}\wedge\omega_{kj}(X,Y).$

On the other hand we have

$d\omega_{ij}(X,Y)=\Omega_{ij}(X,Y)-\sum_{k=1}^{m}\omega_{ik}\wedge\omega_{kj}(X,Y),$

Because $-R(X,Y;e_{j},e_{i})=R(X,Y;e_{i},e_{j})=\Omega_{ij}(X,Y)$. ∎

######  Lemma A.4 (The Bianchi Identities).

The curvature forms also satisfy the following identities:

$\sum_{j=1}^{m}\Omega_{ij}\wedge\theta^{j}=0\quad\mathrm{and}\quad
d\Omega_{ij}=\sum_{k=1}^{m}\left(\Omega_{ik}\wedge\omega_{kj}-\omega_{ik}\wedge\Omega_{kj}\right).$
(A.5)

###### Proof.

Since $d^{2}\theta^{i}=0$ and $d^{2}\omega_{ij}=0$, we have for $1\leq i\leq
m$,

$\displaystyle 0$
$\displaystyle=d\left(\sum_{j=1}^{m}\omega_{ij}\wedge\theta^{j}\right)=\sum_{j=1}^{m}d\omega_{ij}\wedge\theta^{j}-\sum_{j=1}^{m}\omega_{ij}\wedge
d\theta^{j}$
$\displaystyle=\sum_{j=1}^{m}\left(\Omega_{ij}-\sum_{k=1}^{m}\omega_{ik}\wedge\omega_{kj}\right)\wedge\theta^{j}+\sum_{j=1}^{m}\omega_{ij}\wedge\left(\sum_{k=1}^{m}\omega_{jk}\wedge\theta^{k}\right)$
$\displaystyle=\sum_{j=1}^{m}\Omega_{ij}\wedge\theta^{j}+\sum_{j,k=1}^{m}\left(-\omega_{ik}\wedge\omega_{kj}\wedge\theta^{j}+\omega_{ij}\wedge\omega_{jk}\wedge\theta^{k}\right)$
$\displaystyle=\sum_{j=1}^{m}\Omega_{ij}\wedge\theta^{j},$

which proves the first equation. To prove the second equation, we compute

$\displaystyle 0$
$\displaystyle=d^{2}\omega_{ij}=d\left(\Omega_{ij}-\sum_{k=1}^{m}\omega_{ik}\wedge\omega_{kj}\right)=d\Omega_{ij}-\sum_{k=1}^{m}d\omega_{ik}\wedge\omega_{kj}+\sum_{k=1}^{m}\omega_{ik}\wedge
d\omega_{kj}$
$\displaystyle=d\Omega_{ij}-\sum_{k=1}^{m}\left(\Omega_{ik}-\sum_{s=1}^{m}\omega_{is}\wedge\omega_{sk}\right)\wedge\omega_{kj}+\sum_{k=1}^{m}\omega_{ik}\wedge\left(\Omega_{kj}-\sum_{s=1}^{m}\omega_{ks}\wedge\omega_{sj}\right)$
$\displaystyle=d\Omega_{ij}-\sum_{k=1}^{m}\left(\Omega_{ik}\wedge\omega_{kj}-\omega_{ik}\wedge\Omega_{kj}\right)+\sum_{k,s=1}^{m}\left(\omega_{is}\wedge\omega_{sk}\wedge\omega_{kj}-\omega_{ik}\wedge\omega_{ks}\wedge\omega_{sj}\right)$
$\displaystyle=d\Omega_{ij}-\sum_{k=1}^{m}\left(\Omega_{ik}\wedge\omega_{kj}-\omega_{ik}\wedge\Omega_{kj}\right).$

∎

We conclude this section by discussing the effect of changing the moving
frame. Suppose that $e_{1}^{\prime},\dots,e_{m}^{\prime}$ is another
orthonormal moving frame defined on $U$; then we have

$e_{j}^{\prime}=\sum_{i=1}^{m}a_{ij}e_{i},$

where $a=(a_{ij})$ is a smooth function defined on $U$ with values in the
orthogonal group $O(m)$. The corresponding moving coframe is

$\theta^{\prime j}=\sum_{i=1}^{m}a_{ij}\theta^{i}$

(indeed we have $\theta^{\prime
j}(X)=g\left(e^{\prime}_{j},X\right)=\sum_{i=1}^{m}a_{ij}g(e_{i},X)=\sum_{i=1}^{m}a_{ij}\theta^{i}(X)$).
About the connection and curvature forms, we have the

######  Lemma A.5.

Under a change of moving frame, the connection forms transform as

$\omega^{\prime}_{ij}=\sum_{r,s=1}^{m}a_{si}a_{rj}\omega_{sr}+\sum_{r=1}^{m}a_{ri}da_{rj},$

and the curvature forms transform as

$\Omega^{\prime}_{ij}=\sum_{r,s=1}^{m}a_{si}a_{rj}\Omega_{sr}.$

Remark. In this lemma we don’t need to assume that our moving frames are
positively oriented. In matrix notation, the transformation formula for the
connection can be written as

$\omega^{\prime}=a^{-1}\omega a+a^{-1}da$

(compare with [38, vol. 2, page 280]). This is the standard gauge
transformation formula for connections in a principal bundle. Note however
that in our case the inverse matrix $a^{-1}$ is simply its transpose.

###### Proof.

Recall first that

$\nabla_{X}e_{j}^{\prime}=\nabla_{X}e_{j}^{\prime}=\nabla_{X}\left(\sum_{r=1}^{m}a_{rj}e_{r}\right)=\sum_{r=1}^{m}a_{rj}\nabla_{X}e_{r}+\sum_{r=1}^{m}da_{rj}(X)e_{r}.$

Therefore

$\displaystyle\omega^{\prime}_{ij}(X)$ $\displaystyle=\theta^{\prime
i}\left(\nabla_{X}e^{\prime}_{j}\right)=\sum_{r,s=1}^{m}a_{si}a_{rj}\left(\theta^{s}\left(\nabla_{X}e_{r}\right)+da_{rj}(X)\theta^{s}(e_{r})\right)$
$\displaystyle=\sum_{r,s=1}^{m}a_{si}a_{rj}\omega_{sr}(X)+\sum_{r=1}^{m}a_{ri}da_{rj}(X).$

This proves the first formula. The proof of the second formula is immediate
from the definition:

$\Omega^{\prime}_{ij}(X,Y)=R(X,Y;e^{\prime}_{i},e^{\prime}_{j})=\sum_{r,s=1}^{m}a_{si}a_{rj}R(X,Y;e_{r},e_{s})=\sum_{r,s=1}^{m}a_{si}a_{rj}\Omega_{sr}(X,Y),$

because of the tensorial nature of $R$. ∎

##### Acknowledgement.

The author extends his gratitude to Niky Kamran for his encouragements and
meticulous review of the manuscript, as well as to Ruth Kellerhals for her
comments on the case of space forms and for pointing to the precise reference
in Ratcliffe’s treatise. We also acknowledge Adrien Marcone for the valuable
discussions we had regarding the Gauss–Bonnet Chern formula; his thesis [30]
has been a helpful resource in preparing the current paper. Special thanks are
also due to Jean-Pierre Bourguignon for his comments and for drawing our
attention to the notable paper [28] by C. Lanczos.

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