# Equivariant Chern Weil Forms and The Families Index Theorem

Richard Wedeen  Department of Mathematics, University of Texas at Austin, 2515
Speedway, Austin TX, 78712, USA<EMAIL_ADDRESS>
###### Abstract.

We apply the equivariance $\rightarrow$ families principle to a geometric
family of Clifford module bundles with an action of a compact Lie group $G$ to
prove an equivariant version of Bismut’s families index theorem on the
differential Borel quotient of the geometric family.

In certain incarnations of the families index theorem, one considers a
geometric family of Clifford module bundles over compact even dimensional
Riemannian manifolds parametrized by a base $S$, i.e. an iterated bundle

(1) $\mathscr{E}\rightarrow X\rightarrow S$

where $X\rightarrow S$ is a fiber bundle with compact even dimensional fibers,
a horizontal distribution, and a metric on the vertical tangent bundle
$T(X/S)$, and $\mathscr{E}\rightarrow X$ is a $\mathbb{C}\ell(T(X/S))$-module
bundle with compatible connection.

This is the setting considered in [Bis86], where Bismut adapted Quillen’s
ideas in [Qui85] to construct a one-parameter family of superconnections

(2) $\mathbb{B}^{t}:\Omega_{S}(\mathscr{H})\rightarrow\Omega_{S}(\mathscr{H})$

on the infinite dimensional vector bundle $\mathscr{H}$ of fiberwise sections
of $\mathscr{E}\rightarrow X$ over $S$ and proved that

(3) $\lim_{t\rightarrow
0}\operatorname{ch}(\mathbb{B}^{t})=\int_{X/S}\hat{A}(\Omega^{X/S})\operatorname{ch}_{\mathbb{\mathscr{E}}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}})$

where $\hat{A}(\Omega^{X/S})$ is the $\hat{A}$-genus of the vertical curvature
and
$\operatorname{ch}_{\mathbb{\mathscr{E}}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}})$
is the relative Chern character of the twisting curvature of $\mathscr{E}$.

In this paper we consider the case when (1) has an action of a Lie group $G$
preserving the geometric data. A similar situation was considered in [Fre16]
which studied the case of a $G$-action on a geometric family $X\rightarrow S$
which induced a $G$-action on the determinant line bundle

(4) $\mathcal{L}\rightarrow S.$

There, the $G$-action was used to construct an iterated map of differential
Borel quotients

(5) $(\mathcal{L}_{G})_{\nabla}\rightarrow(S_{G})_{\nabla}\rightarrow
B_{\nabla}G$

over the simplicial sheaf $B_{\nabla}G$ of principal $G$-bundles with
connection introduced and studied in [FH13]. The $G$-equivariant connection on
$\mathcal{L}\rightarrow S$ from [BF86] was used to construct a connection on
$(\mathcal{L}_{G})_{\nabla}\rightarrow(S_{G})_{\nabla}$ from which the moment
map and curvature were computed. The curvature was identified with the
$G$-equivariant curvature on $\mathcal{L}$ using the identification of
$\Omega_{(X_{G})_{\nabla}}$ with the Weil-model for $X$ [FH13].

We adapt these ideas and consider a geometric family of differential Borel
quotients

(6)
$(\mathscr{E}_{G})_{\nabla}\rightarrow(X_{G})_{\nabla}\rightarrow(S_{G})_{\nabla}$

fibered over $B_{\nabla}G$. From this we construct a family of
superconnections $\mathbb{B}^{t}_{G}$ on a bundle
$(\mathscr{H}_{G})_{\nabla}\rightarrow(S_{G})_{\nabla}$ whose curvature is an
equivariant form in the Weil-model for $S$. We also construct equivariant
curvatures $\Omega_{G}^{X/S}$ and $F_{G}^{\mathscr{E}/\mathbb{S}}$ on
$(X_{G})_{\nabla}$. We then prove our main theorem

(7) $\lim_{t\rightarrow
0}\operatorname{ch}(\mathbb{B}^{t}_{G})=\int_{X/S}\hat{A}(\Omega^{X/S}_{G})\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}}_{G}).$

An outline of the paper is as follows. In §1 we set up the necessary
differential geometry on manifolds and generalized manifolds, i.e. simplicial
sheaves on the site of manifolds. In particular we construct the connection
and curvature of the differential Borel quotient of a vector bundle with
$G$-action and $G$-equivariant connection. In §2 we review the families index
theorem as proved by Bismut following [BGV03]. Finally in §3 we construct the
superconnection $\mathbb{B}^{t}_{G}$ and prove the main result.

I thank Dan Freed, Charlie Reid, and Dan Weser for helpful discussions and the
mathematics department at UT Austin that made this paper possible.

##  1\. Preliminaries

###  1.0.1. Principal Bundles

This section is a summary of the discussion appearing in [Fre16]. Let $G,H$ be
Lie groups and let $\pi:P\rightarrow X$ be a principal $H$-bundle with a
$G$-action. We endow $\pi$ with a $G$-invariant connection, i.e. a horizontal
distribution $W\subset TP$ that is invariant under the $G$-action on the left
and the $H$-action on the right.

Let $r:Q\rightarrow M$ be a principal $G$-bundle.

######  Notation.

Given a $G$-space $Y$ we can form the associated bundle

(8) $Y_{Q}:=Q\times_{G}Y.$

We will denote by $q:Q\times Y\rightarrow Y_{Q}$ the quotient and by
$p:Q\times Y\rightarrow Y$ the projection. Finally, if $E\rightarrow Y$ is a
vector bundle, we will write

(9) $\underline{E}:=p^{\ast}E$

to denote the pullback vector bundle of $E$ over $Q\times Y$.

The map $Q\times P\xrightarrow{id\times\pi}Q\times X$ is $G$-equivariant and
descends to a map $\pi_{Q}:P_{Q}\rightarrow X_{Q}$ which is a principal
$H$-bundle. Suppose given a connection on $r$, i.e. a $G$-invariant horizontal
distribution

(10) $V\subset TQ.$

We use the given data to construct a connection on $\pi_{Q}$.

######  Construction 1.1.

The distribution $V\oplus W\subset TQ\oplus TP$ is $G$-invariant and descends
to an $H$-invariant horizontal distribution $q_{\ast}(V\oplus W)\subset
TP_{Q}$.

Denote by $\mathfrak{g},\mathfrak{h}$ the Lie algebras of $G,H$ and choose a
basis $(e_{a})$ of $\mathfrak{g}$. Let $\Phi\in\Omega^{1}_{Q}(\mathfrak{g})$
be the connection 1-form on $Q$ with $\ker(\Phi)=V$. We will write
$\Phi:=\phi^{a}e_{a}$ where the coefficients $\phi^{a}\in\Omega^{1}_{Q}$ are
$G$-invariant. Let $\Theta\in\Omega^{1}_{P}(\mathfrak{h})$ be the
$G$-invariant connection 1-form on $P$ satisfying $\ker(\Theta)=W$. We set
$\omega:=\omega^{a}e_{a}\in\Omega^{2}_{Q}(\mathfrak{g})$ to be the curvature
of $\Phi$ and $\Omega\in\Omega^{2}_{P}(\mathfrak{h})$ the curvature of
$\Theta$.

Finally, let
$\Theta_{Q}\in\Omega^{1}_{P_{Q}}(\mathfrak{h})\simeq\Omega^{1}_{Q\times
P}(\mathfrak{h})_{\operatorname{bas}}$ be the connection 1-form on $\pi_{Q}$
with $\ker(\Theta_{Q})=q_{\ast}(V\oplus W)$ and let
$\Omega_{Q}\in\Omega^{2}_{P_{Q}}(\mathfrak{h})\simeq\Omega^{2}_{Q\times
P}(\mathfrak{h})_{\operatorname{bas}}$ denote its curvature.

######  Lemma 1.2.

(11) $\displaystyle\Theta_{Q}$ $\displaystyle=\Theta-\phi^{a}\iota_{a}\Theta$
(12) $\displaystyle\Omega_{Q}=\Omega-\phi^{a}\iota_{a}$
$\displaystyle\Omega+\frac{1}{2}\phi^{a}\phi^{b}\iota_{b}\iota_{a}\Omega-\omega^{a}\iota_{a}\Theta$

See [Fre16] for a proof of this calculation.

###  1.2.1. Associated Bundles

We now consider the case where $\pi:P\rightarrow X$ is the principal
$H:=GL_{N}(\mathbf{C})$ bundle of frames for a complex vector bundle
$E\rightarrow X$ of rank $N$ with a $G$-action. This action induces a left
$G$-action on $\pi$ that commutes with the $H$-action on the right.

Pulling back $E$ along $\pi$ gives the trivial bundle of rank $N$ over the
bundle of frames

(13) ${\underline{\mathbf{C}}^{N}}$${E}$${P}$${X}$$\scriptstyle{\pi}$

This implies that sections of $E$ can be expressed as $H$-invariant functions
on $P$ valued in $V$, and more generally,

(14) $\Omega_{X}(E)\simeq\Omega_{P}(\mathbf{C}^{N})_{\operatorname{bas}}$

where the basic forms are those that are horizontal and $H$-invariant under
the action of $H$ on both $P$ and $\mathbf{C}^{N}$.

Suppose given a $G$-equivariant covariant derivative

(15) $\nabla:\Omega^{0}_{X}(E)\rightarrow\Omega^{1}_{X}(E)$

on $E$. Using the identification (14), $\nabla$ becomes a derivation on
$\Omega_{P}(\mathbf{C}^{N})_{\operatorname{bas}}$ and can be expressed as

(16) $\nabla=d+\Theta$

where $\Theta\in\Omega^{1}_{P}(\mathfrak{gl}_{N})$ is a $G$-invariant
connection 1-form on $P$ thought of as a matrix of 1-forms. Applying 1.2, we
can construct a connection 1-form
$\Theta_{Q}\in\Omega^{1}_{P_{Q}}(\mathfrak{gl}_{N})$ on
$\pi_{Q}:P_{Q}\rightarrow X_{Q}$.

We now proceed to construct the covariant derivative on $E_{Q}$ associated to
this connection. First, we note that there are identifications

(17)
$\begin{split}\Omega_{X_{Q}}(E_{Q})&\simeq\Omega_{P_{Q}}(\mathbf{C}^{N})_{\operatorname{bas}_{H}}\\\
&\simeq\Omega_{Q\times P}(\mathbf{C}^{N})_{\operatorname{bas}_{G\times H}}\\\
&\simeq\Omega_{Q\times X}(\underline{E})_{\operatorname{bas}_{G}}.\end{split}$

######  Lemma 1.3.

The connection $\Theta_{Q}$ induces a covariant derivative on
$E_{Q}\rightarrow X_{Q}$, expressed under the identification (17) as a
derivation $\nabla_{Q}:\Omega_{Q\times
X}(\underline{E})_{\operatorname{bas}}\rightarrow\Omega_{Q\times
X}(\underline{E})_{\operatorname{bas}}$, given by

(18) $\nabla_{Q}:=\nabla-\phi^{a}\tau_{a}.$

where $\nabla$ is understood to be the covariant derivative on $\underline{E}$
obtained by pulling back the covariant derivative on $E$ by $p$. Here,
$\tau_{a}\in\operatorname{End}(\underline{E})$ is the endomorphism

(19) $\tau_{a}:=\mathcal{L}_{\xi_{a}}-\nabla_{\xi_{a}}$

where $\xi_{a}$ is the vector field generated by the action of $e_{a}$ on
$Q\times X$and $\mathcal{L}_{\xi_{a}}$ is the Lie derivative on
$\Omega_{Q\times X}(\underline{E})$.

###### Proof.

As a derivation on $\Omega_{Q\times
P}(\mathbf{C}^{N})_{\operatorname{bas}_{G\times H}}$, the covariant derivative
is expressed as

(20) $\begin{split}\nabla_{Q}&=d+\Theta_{Q}\\\
&=d_{Q}+d_{P}+\Theta-\phi^{a}\iota_{a}\Theta.\end{split}$

where $d_{P},d_{Q}$ are the de Rham differentials on $P$ and $Q$. The term
$d_{Q}+d_{P}+\Theta$ on $\Omega_{Q\times P}(\mathbf{C}^{N})$ becomes
identified with $d_{Q}+\nabla=p^{\ast}\nabla$ on $\Omega_{Q\times
X}(\underline{E})$ and the term $\iota_{a}\Theta\in\Omega^{0}_{Q\times
P}(\mathfrak{gl}_{N})$ is identified with
$\tau_{a}\in\operatorname{End}(\underline{E})$. ∎

The following lemma is standard and we state it without proof.

######  Lemma 1.4.

Under the identification
$\Omega_{P_{Q}}(\mathfrak{gl}_{N})_{\operatorname{bas}_{H}}\simeq\Omega_{X_{Q}}(\operatorname{End}(E_{Q}))$
the curvature of $\nabla_{Q}$ is given by (12), i.e.

(21) $\nabla_{Q}^{2}=\Omega_{Q}$

###  1.4.1. Generalized Manifolds

We now enter the category of generalized manifolds with the aim of
constructing a universal connection and covariant derivative. We first review
some background on simplicial sheaves and refer the reader to [FH13] for
proofs and details.

Let $\Omega^{k}:\text{Man}^{op}\rightarrow\Delta\text{-Set}$ be the simplicial
sheaf that assigns a test manifold $M$ the set $\Omega^{k}_{M}$ of $k$-forms
on $M$. The _simplicial de Rham complex_ [FH13] is defined to be the
simplicial sheaf

(22) $\Omega:=\big{(}\bigoplus_{k\geq 0}\Omega^{k},d\big{)}$

which assigns an $n$-dimensional test manifold $M$ the differential graded
algebra of forms $\Omega_{M}$ on $M$.

We will have reason to consider formal series of forms in the completion of
(22) and we review the construction here. For $k\geq 0$ let $\Omega^{\geq
k}\subset\Omega$ be the ideal generated by forms of degree $k$. There is an
inverse system

(23) $\cdots\rightarrow\Omega/\Omega^{\geq k+1}\rightarrow\Omega/\Omega^{\geq
k}\rightarrow\cdots\rightarrow\Omega/\Omega=0$

The completion is defined as the limit

(24)
$\begin{split}\overline{\Omega}:=\lim_{\leftarrow}\left(\cdots\rightarrow\Omega/\Omega^{\geq
k+1}\rightarrow\Omega/\Omega^{\geq k}\rightarrow\cdots\right)\end{split}$

and can be identified with the direct product

(25) $\overline{\Omega}\simeq\prod_{k\geq 0}\Omega^{k}.$

We will later use the completion to consider forms of unbounded degree on
generalized manifolds. For now, we note that for a finite dimensional manifold
$M$, the completion is isomorphic to the ordinary de Rham complex

(26) $\overline{\Omega}_{M}\simeq\Omega_{M}.$

The Koszul complex associated to a vector space $V$ is the differential graded
algebra generated by $\Lambda^{1}V$. It is given by

(27) $\operatorname{Kos}(V):=\Lambda V\otimes\operatorname{Sym}(V)$

with differential

(28) $d_{K}(v)=\tilde{v},\hskip 14.22636ptd_{K}(\tilde{v})=0$

where $v\in\Lambda^{1}V$ and $\tilde{v}\in\operatorname{Sym}^{1}(V)$ is $v$
regarded as an element of the symmetric algebra of total degree 2. As in (24)
we let $\operatorname{Kos}^{\geq k}$ be the ideal generated by elements of
degree $k$ and we define the _completed Koszul algebra_ to be

(29)
$\overline{\operatorname{Kos}}(V):=\lim_{\leftarrow}\big{(}\cdots\rightarrow\operatorname{Kos}(V)/\operatorname{Kos}^{\geq
k+1}(V)\rightarrow\operatorname{Kos}(V)/\operatorname{Kos}^{\geq
k}(V)\rightarrow\cdots\big{)}$

which can be identified with

(30) $\overline{\operatorname{Kos}}(V)\simeq\Lambda
V\otimes\overline{\operatorname{Sym}}(V)$

where $\overline{\operatorname{Sym}}(V)$ is the algebra of formal power series
on $V^{*}$.

We now recall some theorems from [FH13] and their corresponding statements on
completions for future use.

######  Theorem 1.5 ([FH13] Theorem 8.1).

Let $V$ be a finite dimensional real vector space. Then there is an
isomorphism

(31)
$\eta:\operatorname{Kos}(V^{*})\xrightarrow{\sim}\Omega_{\Omega^{1}\otimes V}$

defined on $\ell\in\Lambda^{1}(V^{\ast})$ by

(32) $\begin{split}\eta(\ell):\Omega^{1}\otimes&V\rightarrow\Omega^{1}\\\
\alpha^{i}v_{i}&\mapsto\langle\ell,v_{i}\rangle\alpha^{i}\end{split}$

and extended to $\operatorname{Kos}(V^{\ast})$ by requiring that it be a
morphism of differential graded algebras.

######  Corollary 1.6.

The isomorphism (31) extends to an isomorphism of completions

(33) $\overline{\Omega}_{\Omega^{1}\otimes
V}\simeq\overline{\operatorname{Kos}}(V^{*}).$

Let $E_{\nabla}G$ be the simplicial sheaf of principal $G$ bundles with
connection and trivialization. We remark that pulling down the connection form
along the trivializing section gives a weak equivalence

(34) $E_{\nabla}G\simeq\Omega^{1}\otimes\mathfrak{g}.$

Combining (33) and (34) gives

######  Corollary 1.7.

There is an identification

(35)
$\overline{\Omega}_{E_{\nabla}G}\simeq\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})$

where
$\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast}):=\Lambda\mathfrak{g}^{\ast}\otimes\overline{\operatorname{Sym}}(\mathfrak{g}^{\ast})$
is the completed Weil algebra.

Similarly, for a smooth manifold $Y$ we have

######  Proposition 1.8 ([FH13] Corollary 8.24).

There is an isomorphism

(36) $\Omega_{E_{\nabla}G\times
Y}\simeq\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{Y}$

which extends to an isomorphism of completions

(37) $\overline{\Omega}_{E_{\nabla}G\times
Y}\simeq\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{Y}$

We recall that the groupoid sheaf of principal $G$ bundles with connection is
weakly equivalent to the simplicial sheaf

(38)

where the 1-simplices are determined by the $G$-action on $E_{\nabla}G$ and
the higher simplices by group composition.

For a $G$-space $Y$, one can use (38) to define the differential Borel
quotient as the simplicial sheaf

(39)

constructed using the $G$ action on $E_{\nabla}G\times Y$.

The action of $G$ on $E_{\nabla}G$ in (38) induces two derivations on the de
Rham complex $\Omega_{E_{\nabla}G}$. The _contraction_ ([FH13] Lemma 7.25) in
the direction $\xi\in\mathfrak{g}$ is a degree (-1) derivation on
$\Omega_{E_{\nabla}G}\simeq\operatorname{Kos}(\mathfrak{g}^{*})$ which extends
to a derivation on
$\overline{\Omega}_{E_{\nabla}G}\simeq\overline{\operatorname{Kos}}(\mathfrak{g}^{*})$.
If we choose bases $\theta^{a}\in\Lambda^{1}\mathfrak{g}^{*}$ and
$z^{a}:=d\theta^{a}\in\operatorname{Sym}^{1}(\mathfrak{g}^{*})$ dual to
$e_{a}\in\mathfrak{g}$, we can express this as

(40) $\iota_{e_{b}}\theta^{a}:=\delta^{a}_{b},\hskip
14.22636pt\iota_{e_{b}}z^{a}:=-f^{a}_{bc}\theta^{c}$

where $f^{a}_{bc}:=\langle e^{a},[e_{b},e_{c}]\rangle$.

The _Lie derivative_ in the direction $\xi\in\mathfrak{g}$ is a degree 0
derivation $\mathcal{L}_{\xi}$ on $\operatorname{Kos}(\mathfrak{g}^{*})$ which
extends to $\overline{\operatorname{Kos}}(\mathfrak{g}^{*})$ and is induced
from the coadjoint action of $G$ on $\mathfrak{g}^{\ast}$. It is given by

(41) $\mathcal{L}_{e_{b}}\theta^{a}:=-f^{a}_{bc}\theta^{c},\hskip
14.22636pt\mathcal{L}_{e_{b}}z^{a}:=-f^{a}_{bc}z^{c}.$

These satisfy Cartan’s relation

(42) $\mathcal{L}_{\xi}=d\iota_{\xi}+\iota_{\xi}d.$

Similarly, there are two derivations $\iota,\mathcal{L}$ on the completed de
Rham complex of $\overline{\Omega}_{E_{\nabla}G\times
Y}\simeq\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{Y}$. They act on
$\overline{\operatorname{Kos}}(\mathfrak{g}^{*})$ by the derivations given
above and by contraction and Lie derivative on $\Omega_{Y}$. Finally define
the subcomplexes _basic forms_

(43)
$(\overline{\Omega}_{E_{\nabla}G})_{\operatorname{bas}}\subset\overline{\Omega}_{E_{\nabla}G}$
(44) $(\overline{\Omega}_{E_{\nabla}G\times
Y})_{\operatorname{bas}}\subset\overline{\Omega}_{E_{\nabla}G\times Y}$

to be those elements annilhilated by $\mathcal{L}$ and $\iota$.

######  Theorem 1.9 ([FH13] Theorem 7.20, 7.28).

There are identifications

(45)
$\Omega_{B_{\nabla}G}\simeq(\Omega_{E_{\nabla}G})_{\operatorname{bas}}\simeq(\operatorname{Kos}(\mathfrak{g}^{*}))_{\operatorname{bas}}\simeq\operatorname{Sym}(\mathfrak{g}^{*})^{G}$
(46)
$\Omega_{(Y_{G})_{\nabla}}\simeq(\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{Y})_{\operatorname{bas}}$

and they extend to identifications of their respective completions.

Let $r:Q\rightarrow M$ be a principal $G$-bundle with connection
$\Phi\in\Omega^{1}_{Q}(\mathfrak{g})$ and curvature
$\omega\in\Omega^{2}_{Q}(\mathfrak{g})$. We express $\Phi=\phi^{a}e_{a}$ and
$\omega=\omega^{a}e_{a}$ where $\phi^{a},\omega^{a}$ are $G$-invariant forms
on $Q$. Let

(47) $f:M\rightarrow B_{\nabla}G$

be the classifying map which induces the pullback

(48)
${Q}$${E_{\nabla}G}$${M}$${B_{\nabla}G}$$\scriptstyle{\tilde{f}}$$\scriptstyle{f}$

where $\tilde{f}$ corresponds to the form $\Phi$ under the identification
$E_{\nabla}G(Q)\simeq\Omega^{1}_{Q}(\mathfrak{g})$.

Using the map (32) we see that

(49) $Q\xrightarrow{\tilde{f}}E_{\nabla}G\xrightarrow{\eta(\theta^{b})}\Omega$

is the element of $\Omega_{Q}$ given by
$\langle\theta^{b},e_{a}\rangle\phi^{a}=\phi^{b}$. Pulling back forms along
$\tilde{f}$ gives the Chern Weil map

(50)
$\begin{split}\tilde{f}^{\ast}:\overline{\operatorname{Kos}}(\mathfrak{g}^{*})&\rightarrow\Omega_{Q}\\\
\theta^{a}&\mapsto\phi^{a}.\end{split}$

The map $\tilde{f}^{\ast}$ restricts to a morphism of differential graded
algebras on
$\operatorname{Kos}(\mathfrak{g}^{\ast})\subset\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})$
and this defines the rest of the map. In particular, if we set

(51) $\chi^{a}:=z^{a}+\frac{1}{2}f^{a}_{bc}\theta^{b}\theta^{c}$

in degree 2 of
$\operatorname{Kos}(\mathfrak{g}^{\ast})\subset\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})$,
we see that

(52) $\tilde{f}^{\ast}:\chi^{a}\mapsto\omega^{a}.$

We also note that

(53) $\tilde{f}\times id:Q\times Y\rightarrow E_{\nabla}G\times Y$

is $G$-equivariant and descends to a map

(54) $f_{Y}:Y_{Q}\rightarrow(Y_{G})_{\nabla}$

which induces a map on the level of forms

(55)
$\begin{split}f_{Y}^{\ast}:(\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{Y})_{\operatorname{bas}}&\rightarrow(\Omega_{Q\times
Y})_{\operatorname{bas}}\\\
\theta^{a}\otimes\alpha&\mapsto\phi^{a}\wedge\alpha\\\
\chi^{a}\otimes\alpha&\mapsto\omega^{a}\wedge\alpha.\end{split}$

Finally, let $E\rightarrow Y$ be a vector bundle on $Y$ with a $G$ action.
Then (55) extends to a map on forms valued in sections of $E$

(56)
$f_{Y}^{\ast}:(\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{Y}(E))_{\operatorname{bas}}\rightarrow\Omega_{Y_{Q}}(E_{Q})$

by pulling back sections along $p:Q\times Y\rightarrow Y$.

###  1.9.1. Covariant Derivative on Generalized Manifolds

In this section we construct a covariant derivative on the differential Borel
quotient of a vector bundle. As before, let $E\rightarrow X$ be a complex
vector bundle with a $G$-action and a $G$-equivariant covariant derivative
$\nabla$. The map $E_{\nabla}G\times
E\xrightarrow{id\times\pi}E_{\nabla}G\times X$ is $G$-equivariant and descends
to a map of Borel quotients

(57) $\pi_{G}:(E_{G})_{\nabla}\rightarrow(X_{G})_{\nabla}.$

We will denote the bundle $E_{\nabla}G\times E$ by $\underline{E}$ and we
observe that it can be written as a pullback of simplicial sheaves

(58) ${\underline{E}}$${(E_{G})_{\nabla}}$${E_{\nabla}G\times
X}$${(X_{G})_{\nabla}}$$\scriptstyle{\pi_{G}}$

We define forms on $(X_{G})_{\nabla}$ valued in sections of $(E_{G})_{\nabla}$
by

(59)
$\begin{split}\overline{\Omega}_{(X_{G})_{\nabla}}((E_{G})_{\nabla})&:=\overline{\Omega}_{E_{\nabla}G\times
X}(\underline{E})_{\operatorname{bas}}\\\
&\simeq(\overline{\Omega}_{E_{\nabla}G}\otimes\Omega_{X}(E))_{\operatorname{bas}}\\\
&\simeq(\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(E))_{\operatorname{bas}}.\end{split}$

where we have used 1.8 in the second line and the grading is given by the
total degree.

There is a covariant derivative on
$\Omega_{(X_{G})_{\nabla}}((E_{G})_{\nabla})$ over the algebra
$\Omega_{(X_{G})_{\nabla}}$ given by

(60) $\nabla_{G}:=d_{K}+\nabla-\theta^{a}\tau_{a}.$

Its curvature

(61) $\Omega_{G}:=\nabla_{G}^{2}$

is a basic endomorphism valued 2-form, i.e.
$F_{G}\in(\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(\operatorname{End}(E)))_{\operatorname{bas}}$.

######  Lemma 1.10.

The equivariant curvature satisfies

(62) $f_{X}^{\ast}\Omega_{G}=\Omega_{Q}$

where $\Omega_{Q}$ is (12) and $f_{X}^{*}$ is (56) applied to the bundle
$\operatorname{End}(E)\rightarrow X$.

###### Proof.

Let $\bar{\nabla}_{Q}:\Omega_{Q\times
X}(\underline{E})\rightarrow\Omega_{Q\times X}(\underline{E})$ be the
extension of $\nabla_{Q}:\Omega_{Q\times
X}(\underline{E})_{\operatorname{bas}}\rightarrow\Omega_{Q\times
X}(\underline{E})_{\operatorname{bas}}$ using formula (60). Similarly, let
$\bar{\nabla}_{G}$ be the extension of $\nabla_{G}$ to
$\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{X}(\operatorname{End}(E))$
to all basic and nonbasic forms. Applying definitions shows that

(63)
${\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(E)}$${\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(E)}$${\Omega_{Q\times
X}(\underline{E})}$${\Omega_{Q\times
X}(\underline{E})}$$\scriptstyle{\bar{\nabla}_{G}}$$\scriptstyle{f_{X}^{\ast}}$$\scriptstyle{f_{X}^{\ast}}$$\scriptstyle{\bar{\nabla}_{Q}}$

commutes. which implies that

(64)
${\Omega^{0}_{X}(E)}$${\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(E)}$${\Omega^{0}_{Q\times
X}(\underline{E})}$${\Omega_{Q\times
X}(\underline{E})}$$\scriptstyle{\bar{\nabla}_{G}^{2}}$$\scriptstyle{\tilde{f}_{X}^{\ast}=p^{\ast}}$$\scriptstyle{\tilde{f}_{X}^{*}}$$\scriptstyle{\bar{\nabla}_{Q}^{2}}$

commutes.

Since sections of $\underline{E}\simeq p^{\ast}E$ are generated over the ring
of functions $\Omega^{0}_{Q\times X}$ by $p^{\ast}\sigma$ for
$\sigma\in\Omega^{0}_{X}(E)$, we see that
$\bar{\nabla}_{Q}^{2}=\tilde{f}_{X}^{\ast}\bar{\nabla}_{G}^{2}$ which implies
$\nabla_{Q}^{2}=f_{X}^{\ast}\nabla_{G}^{2}$. Applying Lemma 1.4 finishes the
proof. ∎

##  2\. Review of The Families Index Theorem

We now recall the statement of the families index theorem [Bis86]. This
section is based on [BGV03] to which we refer the reader for a more thorough
exposition.

###  2.0.1. The Bismut Superconnection

######  Definition 2.1.

A _geometric family_ is

  * •

A smooth fiber bundle $X\xrightarrow{\pi}S$ with compact even dimensional
fibers

  * •

A horizontal distribution $H\subset TX$; i.e. $\pi$ induces an isomorphism on
the fibers $\pi_{\ast}:H_{x}\xrightarrow{\sim}TS_{\pi(x)}$

  * •

A metric $g^{X/S}$ on $T(X/S):=\ker(\pi_{\ast})$.

Let $(X\xrightarrow{\pi}S,H,g^{X/S})$ be a geometric family. We construct a
connection $\nabla^{X/S}$ on $T(X/S)$ as follows.

######  Construction 2.2.

Choose a metric $g^{S}$ on $TS$ and pull it back to a metric $\pi^{\ast}g^{S}$
on $H$. The metric $g^{X}:=g^{X/S}\oplus\pi^{\ast}g^{S}$ induces a Levi-Civita
connection $\nabla^{g^{X}}$ on $TX$. The horizontal distribution induces a
split short exact sequence of bundles

(65) $0\rightarrow H\hookrightarrow
TX\mathrel{{}\mathop{\rightleftarrows}\limits_{\iota}^{P}}T(X/S)\rightarrow
0.$

Define $\nabla^{X/S}:=P\circ\nabla^{g^{X}}\circ\iota$, i.e. the compression of
the Levi-Civita connection to the vertical tangent space.

######  Remark 2.3.

The proof that this construction is independent of the choice of metric
$g^{S}$ is straightforward and can be found for example in [BGV03].

######  Remark 2.4.

The curvature of $\nabla^{X/S}$

(66) $\Omega^{X/S}:=(\nabla^{X/S})^{2}$

is a 2-form in $\Omega_{X}(\mathfrak{so}(T(X/S)))$.

######  Definition 2.5.

A _geometric Clifford family_ is

  * •

A geometric family $(X\xrightarrow{\pi}S,H,g^{X/S})$

  * •

A $\mathbf{Z}/2$-graded $\mathbb{C}\ell(T(X/S),g^{X/S})$-module bundle
$\mathscr{E}\rightarrow X$

  * •

A _Clifford connection_ $\nabla^{\mathscr{E}}$ on $\mathscr{E}$ satisfying the
Leibniz rule for Clifford multiplication: for $v$ a section of
$T(X/S)\subset\mathbb{C}\ell(T(X/S))$ and $\sigma\in C^{\infty}(\mathscr{E})$

(67)
$\nabla^{\mathscr{E}}(v\cdot\sigma)=\nabla^{X/S}v\cdot\sigma+v\cdot\nabla^{\mathscr{E}}\sigma.$

######  Remark 2.6.

The $\mathbf{Z}/2$-grading
$\mathscr{E}\simeq\mathscr{E}^{+}\oplus\mathscr{E}^{-}$ is an eigen-
decomposition for the action of the _chirality operator_ , $\Gamma\in
C^{\infty}(\mathbb{C}\ell(TX/S))$, defined locally as

(68) $\Gamma:=i^{n/2}e_{1}\cdots e_{n}$

where $(e_{i})$ is a local orthonormal basis of $T(X/S)$. It satisfies
$\Gamma^{2}=1$ and its $\pm 1$ eigenbundle is $\mathscr{E}^{\pm}$.

Let $(\mathscr{E}\rightarrow
X\xrightarrow{\pi}S,H,g^{X/S},\nabla^{\mathscr{E}})$ be a geometric Clifford
family and let $\mathscr{H}\rightarrow S$ be the $\mathbf{Z}/2$-graded
infinite dimensional vector bundle whose fiber over $z\in S$ is the space of
$C^{\infty}$ sections of $\mathscr{E}$ over $X_{z}:=\pi^{-1}(z)$. We can
identify its space of sections as
$\Gamma(S;\mathscr{H}):=\mathbf{C}^{\infty}(X;\mathscr{E})$ and we define
forms valued in $\mathscr{H}$ to be

(69) $\Omega_{S}(\mathscr{H}):=C^{\infty}(X;\Lambda H^{*}\otimes\mathscr{E})$

where we are using that $H$ is horizontal to identify $\pi^{\ast}\Lambda
T^{\ast}S\simeq\Lambda H^{*}$. There is a $\mathbf{Z}/2$-grading on
$\mathscr{E}$ and a $\mathbf{Z}$-grading on $\Lambda H^{*}$ and we use the
total (mod 2) to endow $\Omega_{S}(\mathscr{H})$ with a
$\mathbf{Z}/2$-grading.

A _superconnection_ on $\mathscr{H}$ is a differential operator $\mathbb{A}$
on $\Omega_{S}(\mathscr{H})$ of odd degree such that

(70)
$\mathbb{A}(\nu.\sigma)=d_{S}\nu\cdot\sigma+(-1)^{|\nu|}\nu\cdot\mathbb{A}\sigma$

where $\nu\in\Omega_{S}$ and $\sigma\in\Gamma(S;\mathscr{H})$. There is a
decomposition $\mathbb{A}=\sum_{i}\mathbb{A}_{[i]}$ where

(71)
$\mathbb{A}_{[i]}:\Omega^{\bullet}_{S}(\mathscr{H})\rightarrow\Omega^{\bullet+i}_{S}(\mathscr{H})$

raises the total degree by $i$.

It follows from the definition that $\mathbb{A}_{[1]}$ is a connection on
$\mathscr{H}$. The other components $\mathbb{A}_{[i\neq 1]}$ supercommute with
the action of the algebra of forms $\Omega_{S}$ which shows that
$\mathbb{A}_{[i\neq 1]}$ acts by exterior multiplication by forms on $S$
valued in vertical differential operators. More precisely, let

(72) $\mathcal{D}(\mathscr{E})\rightarrow S$

be the bundle whose fiber over $z\in S$ is the space of differential operators
on $\Gamma(X_{z};\mathscr{E}_{z})$ where $\mathscr{E}_{z}$ is the restriction
of $\mathscr{E}$ to $X_{z}$; a section of $\mathcal{D}(\mathscr{E})$ is a
smooth family of vertical differential operators on $\mathscr{E}$. Then

(73) $\mathbb{A}_{[i\neq 1]}\in\Omega^{i}_{S}(\mathcal{D}(\mathscr{E})).$

There is a specific superconnection constructed from the data of a geometric
Clifford family, the _Bismut superconnection_ , which has components up to
degree 2 and is defined as

(74)
$\begin{split}\mathbb{B}&=\mathbb{B}_{[0]}+\mathbb{B}_{[1]}+\mathbb{B}_{[2]}\\\
&:=\not{D}+\left(\tilde{\nabla}^{\mathscr{E}}+\frac{1}{2}k\right)-\frac{1}{4}c(T).\end{split}$

The terms in the formula are as follows:

  * •

$\not{D}\in\Omega^{0}_{S}(\mathcal{D}(\mathscr{E}))$ is the vertical Dirac
operator given locally by

(75)
$\not{D}=\sum_{i}c(e_{i})\nabla_{e_{i}}:C^{\infty}(\mathscr{E})\rightarrow
C^{\infty}(\mathscr{E})$

where $(e_{i})$ is a local orthonormal basis of $T(X/S)$.

  * •

For $\xi\in\Gamma(TS)$ and $\sigma\in\Omega^{0}_{S}(\mathscr{H})$, we define

(76)
$\tilde{\nabla}^{\mathscr{E}}_{\xi}\sigma:=\nabla^{\mathscr{E}}_{\tilde{\xi}}\sigma$

where $\tilde{\xi}\in\Gamma(H)$ is the horizontal lift of $\xi$.

  * •

$T\in C^{\infty}(\Lambda^{2}H^{*}\otimes T(X/S))$ is the Frobenius tensor
defined by

(77) $T(\eta,\zeta)=-P[\eta,\zeta]$

for $\eta,\zeta\in\Gamma(H)$, where $P$ is the projection onto the vertical
defined in (65).

  * •

$c(T)\in
C^{\infty}(\Lambda^{2}H^{*}\otimes\operatorname{End}(\mathscr{E}))\subset\Omega^{2}_{S}(\mathcal{D}(\mathscr{E}))$
is obtained by applying the Clifford map

(78) $c:\mathbb{C}\ell(T(X/S))\rightarrow\operatorname{End}(\mathscr{E})$

to the Frobenius tensor $T$.

  * •

$k\in C^{\infty}(H^{*})\subset
C^{\infty}(H^{*}\otimes\operatorname{End}(\mathscr{E}))\subset\Omega^{1}_{S}(\mathcal{D}(\mathscr{E}))$
is multiplication by the mean curvature tensor of the fibers. Using the
horizontal distribution, both $k$ and the vertical volume form $\nu^{X/S}\in
C^{\infty}(\Lambda^{n}T^{\ast}(X/S))$ can be identified with forms of degree 1
and $n$ in $\Omega_{X}$. Then the mean curvature form is characterized by the
equation

(79) $d\nu^{X/S}=k\wedge\nu^{X/S}+\sum_{i=1}^{n}\langle
T,e_{i}\rangle\iota_{e_{i}}\nu^{X/S}.$

######  Remark 2.7.

$\not{D}$ is valued in vertical differential operators of order 1 on
$\mathscr{E}$ while $k,c(T)$ are valued in vertical differential operators of
order 0.

The curvature of the Bismut superconnection

(80) $\mathcal{F}:=\mathbb{B}^{2}$

is an element of $\Omega_{S}(\mathcal{D}(\mathscr{E}))$. We express it as

(81) $\mathcal{F}=\not{D}^{2}+\mathcal{F}^{+}$

where $\not{D}^{2}\in\Omega^{0}_{S}(\mathscr{D}(\mathscr{E}))$ is the Dirac
Laplacian and $\mathcal{F}^{+}\in\Omega^{i\geq
1}_{S}(\mathcal{D}(\mathscr{E}))$ is a finite sum of forms valued in vertical
differential operators of order 1.

######  Remark 2.8.

The heat operator $e^{-t\not{D}^{2}}$ of the Dirac Laplacian has a smoothing
kernel and is therefore trace-class due to compactness of the fibers. We refer
the reader to [BGV03] for a construction.

###  2.8.1. Heat Operators and Smoothing Kernels

We now recount from [BGV03] the definition of the heat operator for the Bismut
superconnection and the argument to show it has a smoothing kernel with an eye
toward generalization in the next section. To do so, we first recall some
facts about smoothing operators.

Let

(82) $\mathcal{K}(\mathscr{E})\rightarrow S$

be the bundle whose fiber over $z\in S$ is the space of smoothing operators on
$\Gamma(X_{z};E_{z})$. A section $K\in\mathcal{K}(\mathscr{E})$ is a smooth
family of vertical smoothing operators on $\mathscr{E}$ and has a smooth
family of smooth integral kernels $k\in
C^{\infty}(X\times_{\pi}X;\mathscr{E}\boxtimes\mathscr{E}^{\ast})$. The smooth
family $K$ acts on sections $\varphi\in\Gamma(\mathscr{E})$ by

(83) $(K\varphi)(x):=\int_{X/S}k(x,y)\varphi(y)d\text{Vol}_{X/S}(y)$

where $d\text{Vol}_{X/S}$ is the vertical Riemannian volume measure for the
vertical metric $g^{X/S}$.

The restriction of $K$ to the diagonal $X\subset X\times_{\pi}X$ is a section
of $\operatorname{End}(\mathscr{E})$. Applying the supertrace on
$\operatorname{End}(\mathscr{E})$ to this section gives a function on $X$
which can be integrated along the compact vertical fibers using the vertical
volume form to give a smooth function on $S$. This is the _supertrace_ on
sections of $\mathcal{K}(\mathscr{E})$:

(84) $\begin{split}sTr:\Gamma(&\mathcal{K}(\mathscr{E}))\rightarrow
C^{\infty}(S)\\\
K&\mapsto\int_{X/S}sTr\left(k(x,x)\right)d\text{Vol}_{X/S}(x).\end{split}$

For $t>0$ the heat operator is defined using the _Volterra series_

(85) $e^{-t\mathcal{F}}:=e^{-t\not{D}^{2}}+\sum_{k\geq 1}(-t)^{k}I^{(k)}$

where $e^{-t\not{D}^{2}}\in\Omega^{0}_{S}(\mathcal{K}(\mathscr{E}))$ is the
heat operator for the Dirac Laplacian $\not{D}^{2}$ (see Remark 2.8) and

(86)
$I^{(k)}:=\int_{\Delta_{k}}e^{-t\sigma_{0}\not{D}^{2}}\mathcal{F}^{+}e^{-t\sigma_{1}\not{D}^{2}}\cdots\mathcal{F}^{+}e^{-t\sigma_{k}\not{D}^{2}}d\sigma$

is a section of vertical operators obtained by integrating a continuous family
of vertical operators over the $k$-simplex
$\Delta_{k}:=\\{\sigma\in\mathbf{R}_{+}^{k+1}|\sigma_{0}+\cdots+\sigma_{k}=1\\}$
with the standard Lebesgue measure. Note that (85) is a finite sum since
$I^{(k)}$ is a form of degree $\geq k$ and therefore vanishes for large $k$
due to the finite dimensionality of the exterior algebra $\Lambda(T^{\ast}S)$.

The proof of the following lemma can be found in the appendix to Chapter 9 of
[BGV03].

######  Lemma 2.9.

Let $D\in\Gamma(\mathcal{D}(\mathscr{E}))$ be a smooth family of vertical
differential operators. Then the vertical operator

(87)
$\int_{\Delta_{k}}e^{-t\sigma_{0}\not{D}^{2}}De^{-t\sigma_{1}\not{D}^{2}}\cdots
De^{-t\sigma_{k}\not{D}^{2}}d\sigma$

is a family of vertical smoothing operators, i.e. it is a section of
$\mathcal{K}(\mathscr{E})$.

######  Proposition 2.10.

The section of vertical operators $I^{(k)}$ is a form on $S$ valued in
vertical smoothing operators, i.e.
$I^{(k)}\in\Omega{}_{S}(\mathcal{K}(\mathscr{E}))$

###### Proof.

Since $\mathcal{F}^{+}$ is valued in sections of $\mathcal{D}(\mathscr{E})$,
applying Lemma 2.9 to each term gives the desired result. ∎

######  Corollary 2.11.

For $t>0$, the heat operator associated to the Bismut superconnection is a
form valued in vertical smoothing operators, i.e.
$e^{-t\mathcal{F}}\in\Omega_{S}(\mathcal{K}(\mathscr{E}))$

###  2.11.1. Bismut’s Theorem

We now recall the ingredients necessary to state Bismut’s index theorem for
geometric families.

There is a one parameter family of geometric Clifford families obtained by
scaling the vertical metric to $\frac{1}{t}g^{X/S}$ for each $t>0$. This gives
rise to a one parameter family of Bismut superconnections

(88)
$\mathbb{B}^{t}:=t^{1/2}\not{D}+\left(\nabla^{\mathscr{\mathscr{E}}}+\frac{1}{2}k\right)-t^{1/2}\frac{1}{4}c(T)$

which can be expressed as

(89) $\mathbb{B}^{t}=t^{1/2}\delta_{t}\mathbb{A}\delta_{t}^{-1}$

if we set
$\delta_{t}:\Omega_{S}(\mathscr{H})\rightarrow\Omega_{S}(\mathscr{H})$ to be
the operator that multiplies $\Omega^{i}_{S}(\mathscr{H})$ by $t^{-i/2}$.

The curvature of $\mathbb{B}^{t}$ is given by

(90)
$\mathcal{F}^{t}=(\mathbb{B}^{t})^{2}=\delta_{t}(t\mathcal{F})\delta_{t}^{-1}.$

Using Corollary 2.11 the _Chern character_ of $\mathbb{B}^{t}$ is defined as
the supertrace

(91)
$\begin{split}\operatorname{ch}(\mathbb{B}^{t})&:=sTr(e^{-\mathcal{F}^{t}})\\\
&=sTr(e^{-\delta_{t}t\mathcal{F}\delta_{t}^{-1}})\\\
&=\delta_{t}sTr(e^{-t\mathcal{F}})\delta_{t}^{-1}\end{split}$

where we have extended the supertrace (84) to a map
$sTr:\Omega_{S}(\mathcal{K}(\mathscr{E}))\rightarrow\Omega_{S}$ on the level
of forms.

Recall there is a map

$\displaystyle\alpha:\hskip 7.11317pt$ $\displaystyle\mathfrak{so}_{n}$
$\displaystyle\rightarrow\hskip 7.11317pt$
$\displaystyle\mathfrak{spin}_{n}\subset\mathbb{C}\ell_{n}$ (92)
$\displaystyle A_{ij}$ $\displaystyle\mapsto$
$\displaystyle\frac{1}{4}\sum_{ij}A_{ij}c(e_{i})c(e_{j})$

The _twisting curvature_
$F^{\mathscr{E}/\mathbb{S}}\in\Omega^{2}_{X}(\operatorname{End}_{\mathbb{C}\ell(T(X/S))}(\mathscr{E}))$
is

(93)
$F^{\mathscr{E}/\mathbb{S}}:=(\nabla^{\mathscr{E}})^{2}-c(\alpha(\Omega^{X/S}))$

Here $c:\mathbb{C}\ell(T(X/S))\rightarrow\operatorname{End}(\mathscr{E})$ is
(78) and $\Omega^{X/S}$ is (66). We refer the reader to Proposition 3.43 in
[BGV03] for a justification of the statement that $F^{\mathscr{E}/\mathbb{S}}$
is valued in endomorphisms that commute with the vertical Clifford action. In
particular, $F^{\mathscr{E}/\mathbb{S}}$ commutes with $\Gamma$ defined in
(68) and therefore preserves the $\mathbf{Z}/2$-grading on $\mathscr{E}$.

The _relative Chern character_ of $F^{\mathscr{E}/\mathbb{S}}$ is then defined
to be

(94)
$\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}}):=2^{-n/2}sTr(\exp(-\Gamma\cdot
F^{\mathscr{E}/\mathbb{S}})$

where $n$ is the dimension of the even dimensional fiber of $\pi$.

We can now state the families index theorem:

######  Theorem 2.12 ([Bis86]).

(95) $\lim_{t\rightarrow
0}\operatorname{ch}(\mathbb{B}^{t})=\int_{X/S}\hat{A}(\Omega^{X/S})\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}})$

##  3\. The Equivariant Families Index Theorem

We now embark on proving an equivariant version of Theorem 2.12 using the
tools laid out in §1.

Let $(\mathscr{E}\rightarrow
X\xrightarrow{\pi}S,g^{X/S},H,\nabla^{\mathscr{E}})$ be a geometric Clifford
family with a $G$-action that preserves the geometric data. There is an
iterated bundle of Borel quotients

(96)
$(\mathscr{E}_{G})_{\nabla}\rightarrow(X_{G})_{\nabla}\rightarrow(S_{G})_{\nabla}\rightarrow
B_{\nabla}G$

and pulling back (96) along the map (47) gives an iterated map of fiber
bundles

(97) $\mathscr{E}_{Q}\rightarrow X_{Q}\xrightarrow{\pi_{Q}}S_{Q}\rightarrow M$

associated to the principal $G$-bundle $Q\rightarrow M$ with connection
$\Phi$.

We recall that the total space of the bundle $\mathscr{H}\rightarrow S$ is
infinite dimensional and we regard it as a sheaf of sets on the site of
manifolds. In particular, a map from the test manifold $M$ to $\mathscr{H}$
consists of a map $\varphi:M\rightarrow S$ and a section of the pullback
bundle $\varphi^{\ast}\mathscr{H}\rightarrow M$, which can be identified as a
section of $\varphi^{\ast}\mathscr{E}\rightarrow\varphi^{\ast}X$. Thus the set
$\mathscr{H}(M)$ fibers over $S(M)$ where the fiber over $\varphi\in S(M)$ is
$\Gamma(\varphi^{\ast}X;\varphi^{\ast}\mathscr{E})$.

The action on sections of $\mathscr{E}$ induces an action on the bundle
$\mathscr{H}$. We use this action to construct the simplicial sheaf
$(\mathscr{H}_{G})_{\nabla}$ whose value on $M$ is the simplicial set

(98)
$(\mathscr{H}_{G})_{\nabla}(M):=E_{\nabla}G(M)\times\mathscr{H}(M)\leftleftarrows
E_{\nabla}G(M)\times\mathscr{H}(M)\times
G(M)\mathrel{\begin{subarray}{c}\textstyle\leftarrow\\\\[-2.58334pt]
\textstyle\leftarrow\\\\[-2.58334pt]
\textstyle\leftarrow\end{subarray}}\cdots$

This fits into a bundle of simplicial Borel quotients

(99) $(\mathscr{H}_{G})_{\nabla}\rightarrow(S_{G})_{\nabla}\rightarrow
B_{\nabla}G$

that pulls back along $f$ to

(100) $\mathscr{H}_{Q}\rightarrow S_{Q}\rightarrow M.$

There is a horizontal distribution $H_{Q}:=q_{\ast}(V\oplus H)$ on $\pi_{Q}$
obtained by applying Construction 1.1 to $\pi$, where $V$ is (10) . By
$G$-invariance, the metric $p^{\ast}g^{X/S}$ descends to a metric
$g^{X/S}_{Q}$ on $T(X_{Q}/S_{Q})$. Thus we have shown

######  Lemma 3.1.

$(X_{Q}\xrightarrow{\pi_{Q}}S_{Q},H_{Q},g^{X/S}_{Q})$ is a geometric family.

Applying Lemma 1.3 to $\nabla^{\mathscr{E}}$ implies there is a covariant
derivative $\nabla^{\mathscr{E}}_{Q}$ on $\mathscr{E}_{Q}\rightarrow X_{Q}$.
Similarly there is a covariant derivative $\nabla^{X/S}_{Q}$ on
$T(X_{Q}/S_{Q})$. In fact, there is yet another covariant derivative
$\nabla^{X_{Q}/S_{Q}}$ on $T(X_{Q}/S_{Q})$ obtained by applying Construction
2.2 to the geometric family and they are equal:

(101) $\nabla^{X_{Q}/S_{Q}}=\nabla^{X/S}_{Q}.$

We refer the reader to Lemma 16 of [Fre16] for a proof of this statement.

######  Remark 3.2.

The curvature

(102) $\Omega_{Q}^{X/S}:=(\nabla^{X/S}_{Q})^{2}$

is a form in
$\Omega_{X_{Q}}(\mathfrak{so}(TX/S))\simeq\Omega_{\mathcal{B}_{SO}(T(X_{Q}/S_{Q}))}(\mathfrak{so}(n))$.
It is identified with (12) when $P=\mathcal{B}_{SO}(T(X/S))$ is the bundle of
frames on $T(X/S)$.

######  Proposition 3.3.

$(\mathscr{E}_{Q}\rightarrow X_{Q}\rightarrow
S_{Q},g^{X/S}_{Q},H_{Q},\nabla^{\mathscr{E}}_{Q})$ is a geometric Clifford
family.

###### Proof.

In light of Lemma 3.1, we are left with showing that
$\mathscr{E}_{Q}\rightarrow X_{Q}$ is a vertical Clifford module bundle with
Clifford connection $\nabla^{\mathscr{E}}_{Q}$. Clifford multiplication on
$\mathscr{E}_{Q}$ by $\mathbb{C}\ell(T(X_{Q}/S_{Q}),g^{X/S}_{Q})$ is defined
fiberwise as the action of $\mathbb{C}\ell(T(X/S),g^{X/S})$ on $\mathscr{E}$,
i.e. sections $v\in\Gamma(X_{Q};T(X_{Q}/S_{Q}))$,
$\sigma\in\Gamma(X_{Q},\mathscr{E}_{Q})$ lift to $G$-equivariant sections
$\tilde{v}:=q^{\ast}v\in\Gamma(Q\times X;\underline{T(X/S)})^{G}$,
$\tilde{\sigma}:=q^{\ast}\sigma\in\Gamma(Q\times
X;\underline{\mathscr{E}})^{G}$ and we set

(103) $v\cdot\sigma:=q_{\ast}(\tilde{v}\cdot\tilde{\sigma}).$

Recall that by Lemma 1.3

(104)
$q^{\ast}\nabla^{\mathscr{E}}_{Q}=\nabla^{\mathscr{E}}-\phi^{a}\tau^{\mathscr{E}}_{a}$

where $\tau^{\mathscr{E}}_{a}:=\mathcal{L}_{a}-\nabla^{\mathscr{E}}_{a}$ in
particular has the property

(105)
$\begin{split}\tau^{\mathscr{E}}_{a}(\tilde{v}\cdot\tilde{\sigma})&=(\mathcal{L}_{a}-\nabla^{\mathscr{E}}_{a})(\tilde{v}\cdot\tilde{\sigma})\\\
&=-\nabla^{\mathscr{E}}_{a}(\tilde{v}\cdot\tilde{\sigma})\\\
&=-\nabla^{X/S}_{a}\tilde{v}\cdot\tilde{\sigma}-\tilde{v}\cdot\nabla^{\mathscr{E}}_{a}\tilde{\sigma}\\\
&=\tau^{X/S}_{a}(\tilde{v})\cdot\tilde{\sigma}+\tilde{v}\cdot\tau^{\mathscr{E}}_{a}(\tilde{\sigma})\end{split}$

where we used the $G$-equivariance of $\tilde{v},\tilde{\sigma}$ in the second
and fourth equalities.

We now check

(106)
$\begin{split}\nabla^{\mathscr{E}}_{Q}(v\cdot\sigma)&=q_{*}\left(\nabla^{\mathscr{E}}(\tilde{v}\cdot\tilde{\sigma})-\phi^{a}\tau^{\mathscr{E}}_{a}(\tilde{v}\cdot\tilde{\sigma})\right)\\\
&=q_{*}\left(\nabla^{X/S}\tilde{v}\cdot\tilde{\sigma}+\tilde{v}\cdot\nabla^{\mathscr{E}}\tilde{\sigma}-\phi^{a}(\tau^{X/S}_{a}\tilde{v})\cdot\tilde{\sigma}-\phi^{a}\tilde{v}\cdot\tau^{\mathscr{E}}_{a}\tilde{\sigma}\right)\\\
&=q_{*}\left(\left(\nabla^{X/S}\tilde{v}-\phi^{a}\tau^{X/S}_{a}\tilde{v}\right)\cdot\tilde{\sigma}+\tilde{v}\cdot\left(\nabla^{\mathscr{E}}\tilde{\sigma}-\phi^{a}\tau^{\mathscr{E}}_{a}\tilde{\sigma}\right)\right)\\\
&=(\nabla_{Q}^{X/S}v)\cdot\sigma+v\cdot\nabla^{\mathscr{E}}_{Q}\sigma\end{split}$

∎

As before, there is a 1-parameter family of Bismut superconnections

(107)
$\mathbb{B}^{t}_{Q}:\Omega_{S_{Q}}(\mathscr{H}_{Q})\rightarrow\Omega_{S_{Q}}(\mathscr{H}_{Q})$

associated to the 1-parameter family of geometric Clifford families
$(\mathscr{E}_{Q}\rightarrow X_{Q}\rightarrow
S_{Q},\frac{1}{t}g^{X/S}_{Q},H_{Q},\nabla^{\mathscr{E}}_{Q})$. For $t>0$,
$\mathbb{B}^{t}_{Q}$ is an odd derivation over the algebra of forms
$\Omega_{S_{Q}}$defined on $\Omega_{S_{Q}}(\mathscr{H}_{Q})$ by

(108)
$\mathbb{B}^{t}_{Q}:=t^{1/2}\not{D}_{Q}+\left(\tilde{\nabla}^{\mathscr{E}}_{Q}+\frac{1}{2}k_{Q}\right)-t^{-1/2}\frac{1}{4}c(T_{Q}).$

Here $\not{D}_{Q}\in\Omega^{0}_{S_{Q}}(\mathcal{D}(\mathscr{E}_{Q}))$ is the
vertical Dirac operator, $\tilde{\nabla}^{\mathscr{E}}_{Q}$ is the covariant
derivative on $\mathscr{H}_{Q}$ constructed from $\nabla^{\mathscr{E}}_{Q}$ as
in (76), $k_{Q}\in\Omega^{1}_{S_{Q}}(\mathcal{D}(\mathscr{E}_{Q}))$ is
multiplication by the mean curvature tensor (79), and
$c(T_{Q})\in\Omega^{2}_{S_{Q}}(\mathcal{D}(\mathscr{E}_{Q}))$ is Clifford
multiplication by the Frobenius tensor (77). The curvature

(109) $\mathcal{F}_{Q}^{t}:=(\mathbb{B}^{t}_{Q})^{2}$

is a form in $\Omega_{S_{Q}}(\mathcal{D}(\mathscr{E}_{Q}))$.

######  Lemma 3.4.

The vertical Dirac operator satisfies

(110) $\not{D}_{Q}=\not{D}$

in $\Omega^{0}_{Q\times
X}(\mathcal{D}(\underline{\mathscr{E}}))_{\operatorname{bas}}$.

###### Proof.

The vertical Dirac operator
$\not{D}\in\Omega^{0}_{X}(\mathcal{D}(\mathscr{E}))$ is $G$-invariant because
the action preserves the vertical metric $g^{X/S}$. This implies
$p^{\ast}\not{D}\in\Omega^{0}_{Q\times
X}(\mathcal{D}(\underline{\mathscr{E}}))$ is also $G$-invariant and it
descends to $\not{D}_{Q}$. ∎

There is a projection [MQ86], which we call the _Mathai-Quillen map_ ,

(111) $\exp(-\phi^{a}\iota_{a}):\Omega_{Q\times X}\rightarrow(\Omega_{Q\times
X})^{hor}$

onto horizontal forms on $Q\times X$, i.e. those that are annihilated by
$\iota_{a}$ for all $e_{a}\in\mathfrak{g}$. It restricts to a map on
$G$-invariants

(112) $\Omega_{X}^{G}\xhookrightarrow{p^{\ast}}\Omega_{Q\times
X}^{G}\xrightarrow{\exp(-\phi^{a}\iota_{a})}(\Omega_{Q\times
X})_{\operatorname{bas}}.$

We note that the vertical volume form for the geometric family
$(\pi,g^{X/S},H)$ is $G$-invariant and we identify it with a form
$\nu^{X/S}\in(\Omega^{n}_{X})^{G}$ using the horizontal distribution $H$.

######  Lemma 3.5.

The image of $\nu^{X/S}$ under the Mathai-Quillen map is the vertical volume
form $\nu^{X_{Q}/S_{Q}}\in(\Omega^{n}_{Q\times X})_{\operatorname{bas}}$ on
the geometric family $(\pi_{Q},g^{X/S}_{Q},H_{Q})$, i.e.

(113) $\begin{split}\nu^{X_{Q}/S_{Q}}&=\exp(-\phi^{a}\iota_{a})\nu^{X/S}\\\
&=\nu^{X/S}-\phi^{a}\iota_{a}\nu^{X/S}+\frac{1}{2}\phi^{a}\phi^{b}\iota_{b}\iota_{a}\nu^{X/S}-\cdots\end{split}$

###### Proof.

The restriction to a fiber of the form $\exp(-\phi^{a}\iota_{a})\nu^{X/S}$
agrees with the vertical Riemannian volume form. Moreover, for any $\xi\in
V\oplus H$, $\iota_{\xi}\exp(-\phi^{a}\iota_{a})\nu^{X/S}=0$. Since these two
properties uniquely characterize $\nu^{X_{Q}/S_{Q}}$, this concludes the
proof. ∎

######  Proposition 3.6.

The mean curvature satisfies

(114) $k_{Q}=k-\phi^{a}\iota_{a}k$

in $\Omega^{1}_{Q\times
S}(\mathcal{D}(\underline{\mathscr{E}}))_{\operatorname{bas}}$.

###### Proof.

For the purposes of this proof only, we will call a form
$\omega\in\Omega_{Q\times X}$ _irrelevant_ if

(115) $\iota_{e_{1}}\cdots\iota_{e_{n}}\omega=0$

for $(e_{i})$ an orthonormal frame of the vertical tangent bundle, i.e.
$\omega$ is irrelevant if it does not contain a factor of $\nu^{X/S}$.

We now apply the previous lemma. The equation characterizing $k_{Q}$ is

(116)
$\begin{split}d\nu^{X_{Q}/S_{Q}}&=k_{Q}\wedge\nu^{X_{Q}/S_{Q}}+\sum_{i=1}^{n}\langle
T_{Q},e_{i}\rangle\iota_{e_{i}}\nu^{X_{Q}/S_{Q}}\\\
&=k_{Q}\wedge\nu^{X/S}+\text{irrelevant terms}\end{split}$

The left hand side of (116) is

(117)
$\begin{split}d\nu^{X_{Q}/S_{Q}}&=d\nu^{X/S}+\phi^{a}d\iota_{a}\nu^{X/S}+\text{irrelevant
terms}\\\ &=k\wedge\nu^{X/S}-\phi^{a}\iota_{a}d\nu^{X/S}+\text{irrelevant
terms}\\\ &=(k-\phi^{a}\iota_{a}k)\wedge\nu^{X/S}+\text{irrelevant
terms}\end{split}$

where we used $G$-invariance of $\nu^{X/S}$ to say that
$\mathcal{L}_{a}\nu^{X/S}=[d,\iota_{a}]\nu^{X/S}=0$ in the second equality.
Matching the relevant terms gives

(118) $k_{Q}=k-\phi^{a}\iota_{a}k$

as forms in $(\Omega_{Q\times X})_{\operatorname{bas}}$. Let
$(\Omega^{H_{Q}}_{Q\times X})_{\operatorname{bas}}$ denote those forms
annihilated by $\iota_{\zeta}$ where $\zeta$ is a section of $T(X/S)$ and let
$\pi_{\ast}\underline{\mathbf{R}}$ denote the bundle on $Q\times S$ whose
preimage over a point in $Q\times S$ is the space of smooth functions on the
fiber. Under the identification

(119) $(\Omega^{H_{Q}}_{Q\times X})_{\operatorname{bas}}\simeq\Omega_{Q\times
S}(\pi_{\ast}\underline{\mathbf{R}})_{\operatorname{bas}}\subset\Omega_{Q\times
S}(\mathcal{D}(\underline{\mathscr{E}}))_{\operatorname{bas}}$

(118) becomes (114). ∎

######  Proposition 3.7.

The Frobenius tensor satisfies

(120)
$T_{Q}=T-\phi^{a}\iota_{a}T+\frac{1}{2}\phi^{a}\phi^{b}\iota_{b}\iota_{a}T-\omega^{a}P\xi_{a}$

where $P$ is (65) and $\xi_{a}$ is the vector field on $X$ generated by the
action of $e_{a}\in\mathfrak{g}$.

###### Proof.

We first set some notation. Given a split short exact sequence of vector
bundles

(121) $0\rightarrow W\hookrightarrow
E\mathrel{{}\mathop{\rightleftarrows}\limits_{\iota}^{j}}V\rightarrow 0$

we denote by $P^{V}_{W}:=\iota\circ j\in\operatorname{End}(E)$ the projection
operator onto $\iota(V)$ with kernel $W$.

By definition,

(122)
$T_{Q}:=P^{(T(X_{Q}/S_{Q})}_{H_{Q}}[P^{H_{Q}}_{T(X_{Q}/S_{Q})}-,P^{H_{Q}}_{T(X_{Q}/S_{Q})}-]$

which is a tensor, i.e. an element of $\Omega^{2}_{X_{Q}}(T(X_{Q}/S_{Q}))$. To
show (120), we will evaluate $q^{\ast}T_{Q}\in\Omega^{2}_{Q\times
X}(\underline{T(X/S)})_{\operatorname{bas}}$ pointwise on vectors
$\eta,\zeta\in T(Q\times X)$. By tensoriality, this value does not change
regardless of how we extend $\eta,\zeta$ to global vector fields and we will
make use of this freedom when doing the calculation.

Now we observe that pointwise

(123) $P^{H_{Q}}_{T(X_{Q}/S_{Q})}\circ q_{\ast}=q_{\ast}\circ P^{V\oplus
H}_{\dot{\rho}(\mathfrak{g})\oplus T(X/S)}$

and

(124) $P^{T(X_{Q}/S_{Q})}_{H_{Q}}\circ q_{\ast}=q_{\ast}\circ
P_{\dot{\rho}(\mathfrak{g})\oplus V\oplus H}^{T(X/S)}.$

Here $\dot{\rho}(\mathfrak{g})\subset T(Q\times X)$ is the subbundle generated
the action of $\mathfrak{g}$ on $Q\times X$. We will use
$\dot{\rho}_{X}:\mathfrak{g}\rightarrow C^{\infty}(TX)$ and
$\dot{\rho}_{Q}:\mathfrak{g}\rightarrow C^{\infty}(TQ)$ to denote the action
of $\mathfrak{g}$ on $X$ and $Q$.

We now use our freedom and extend $\eta,\zeta$ to $G$-invariant vector fields
on $Q\times X$. Then

(125)
$\begin{split}(q^{*}T_{Q})(\eta,\zeta)&=q^{*}P^{T(X_{Q}/S_{Q})}_{H_{Q}}[P^{H_{Q}}_{T(X_{Q}/S_{Q})}q_{\ast}\eta,P^{H_{Q}}_{T(X_{Q}/S_{Q})}q_{\ast}\zeta]\\\
&=q^{*}P^{T(X_{Q}/S_{Q})}_{H_{Q}}q_{*}[P^{V\oplus
H}_{\dot{\rho}(\mathfrak{g})\oplus T(X/S)}\eta,P^{V\oplus
H}_{\dot{\rho}(\mathfrak{g})\oplus T(X/S)}\zeta]\\\
&=P_{\dot{\rho}(\mathfrak{g})\oplus V\oplus H}^{T(X/S)}[P^{V\oplus
H}_{\dot{\rho}(\mathfrak{g})\oplus T(X/S)}\eta,P^{V\oplus
H}_{\dot{\rho}(\mathfrak{g})\oplus T(X/S)}\zeta]\end{split}$

where (123) and (124) were used in the second and third equalities.

We note that $\eta$ has a decomposition

(126) $\eta=\eta_{Q}+\eta_{X}$

into components in $TQ\oplus TX$, which has a further decomposition

(127)
$\eta=\underbrace{\left(\dot{\rho}_{Q}\Phi(\eta)+\dot{\rho}_{X}\Phi(\eta)\right)}_{\in\dot{\rho}(\mathfrak{g})}+\underbrace{\left(\eta_{Q}-\dot{\rho}_{Q}\Phi(\eta)\right)}_{\eta_{V}\in
V}+\underbrace{P^{H}_{T(X/S)}\left(\eta_{X}-\dot{\rho}_{X}\Phi(\eta)\right)}_{\eta_{H}\in
H}+\underbrace{P^{T(X/S)}_{H}\left(\eta_{X}-\dot{\rho}_{X}\Phi(\eta)\right)}_{\in
T(X/S)}.$

There is a similar decomposition for $\zeta$.

Continuing (125),

(128)
$\begin{split}(q^{*}T_{Q})(\eta,\zeta)&=P_{\dot{\rho}(\mathfrak{g})\oplus
V\oplus H}^{T(X/S)}[\eta_{H}+\eta_{V},\zeta_{H}+\zeta_{V}]\\\
&=P_{\dot{\rho}(\mathfrak{g})\oplus V\oplus
H}^{T(X/S)}\left([\eta_{H},\zeta_{H}]+\underbrace{[\eta_{H},\zeta_{V}]+[\eta_{V},\zeta_{H}]}_{\in
V\oplus H}+[\eta_{V},\zeta_{V}]\right)\\\
&=\underbrace{P^{T(X/S)}_{H}[P^{H}_{T(X/S)}\eta_{X},P^{H}_{T(X/S)}\zeta_{X}]}_{\iota_{\zeta}\iota_{\eta}T}\\\
&\hskip
11.38092pt-\underbrace{\left(P^{T(X/S)}_{H}[P^{H}_{T(X/S)}\dot{\rho}_{X}\Phi(\eta),P^{H}_{T(X/S)}\zeta_{X}]+P^{T(X/S)}_{H}[P^{H}_{T(X/S)}\eta_{X},P^{H}_{T(X/S)}\dot{\rho}_{X}\Phi(\zeta)]\right)}_{\iota_{\zeta}\iota_{\eta}\phi^{a}\iota_{a}T}\\\
&\hskip
11.38092pt+\underbrace{P^{T(X/S)}_{H}[P^{H}_{T(X/S)}\dot{\rho}_{X}\Phi(\eta),P^{H}_{T(X/S)}\dot{\rho}_{X}\Phi(\zeta)]}_{\frac{1}{2}\iota_{\xi}\iota_{\eta}\phi^{a}\phi^{b}\iota_{b}\iota_{a}T}\\\
&\hskip
11.38092pt-\underbrace{P^{T(X/S)}_{H}\dot{\rho}_{X}\omega(\eta,\zeta)}_{\iota_{\zeta}\iota_{\eta}\omega^{a}P\xi_{a}}\\\
\end{split}$

∎

Applying the Clifford map (78) and the identification (119), we get

######  Corollary 3.8.

There is an equality

(129)
$c(T_{Q})=T-\phi^{a}\iota_{a}c(T)+\frac{1}{2}\phi^{a}\phi^{b}\iota_{b}\iota_{a}c(T)-\omega^{a}c(P\xi_{a})$

of forms in $\Omega_{Q\times
S}(\mathcal{D}(\underline{\mathscr{E}}))_{\operatorname{bas}}$

###  3.8.1. The Equivariant Bismut Superconnection

We define the equivariant Bismut superconnection to be the derivation

(130)
$\mathbb{B}_{G}:\Omega_{(S_{G})_{\nabla}}((\mathscr{H}_{G})_{\nabla})\rightarrow\Omega_{(S_{G})_{\nabla}}((\mathscr{H}_{G})_{\nabla})$

over the algebra
$\Omega_{(S_{G})_{\nabla}}\simeq\left(\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{S}\right)_{\operatorname{bas}}$
given by

(131)
$\mathbb{B}_{G}:=\not{D}+\left(\tilde{\nabla}^{\mathscr{E}}_{G}+\frac{1}{2}k_{G}\right)-\frac{1}{4}c(T)_{G}$

where the terms are given as follows:

  * •

$\not{D}\in\Omega^{0}_{S}(\mathcal{D}(\mathscr{E}))$ is the vertical Dirac
operator. It is $G$-equivariant and defines a degree 0 map on
$\Omega_{(S_{G})_{\nabla}}((\mathscr{H}_{G})_{\nabla}):=(\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathscr{H}))_{\operatorname{bas}}$.

  * •

$\tilde{\nabla}^{\mathscr{E}}_{G}$ is defined as

(132)
$\tilde{\nabla}^{\mathscr{E}}_{G}:=d_{K}+\tilde{\nabla}^{\mathscr{E}}-\theta^{a}\tau_{a}$

where $\tilde{\nabla}^{\mathscr{E}}$ is (76) and $\tau_{a}$ is (19)

  * •

$k_{G}\in(\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathcal{D}(\mathscr{E})))^{\bullet=1}_{\operatorname{bas}}$
is

(133) $k_{G}:=k-\theta^{a}\iota_{a}k$

  * •

$c(T)_{G}\in(\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathcal{D}(\mathscr{E})))_{\operatorname{bas}}^{\bullet=2}$
is

(134)
$c(T)_{G}:=c(T)-\theta^{a}\iota_{a}c(T)+\frac{1}{2}\theta^{a}\theta^{b}\iota_{b}\iota_{a}c(T)-\chi^{a}c(P(\xi_{a}))$

where $\chi^{a}$ is (51).

######  Proposition 3.9.

The curvature

(135) $\mathcal{F}_{G}:=(\mathbb{B}_{G})^{2}$

is a form in
$(\operatorname{Kos}(\mathfrak{g}^{\ast})\otimes\Omega_{S}(\mathcal{D}(\mathscr{E})))_{\operatorname{bas}}$
and it satisfies

(136) $f_{S}^{\ast}\mathcal{F}_{G}=\mathcal{F}_{Q}$

###### Proof.

The map

(137)
$\mathbb{B}_{G}^{2}:(\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathscr{H}))_{\operatorname{bas}}\rightarrow(\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathscr{H}))_{\operatorname{bas}}$

supercommutes with exterior multiplication by forms in
$\Omega_{(S_{G})_{\nabla}}$, and therefore defines an element of
$(\operatorname{Kos}(\mathfrak{g}^{\ast})\otimes\Omega_{S}(\mathcal{D}(\mathscr{E})))_{\operatorname{bas}}$.

Let $\overline{\mathbb{B}}_{Q}:\Omega_{Q\times
S}(\underline{\mathscr{H}})\rightarrow\Omega_{Q\times
S}(\underline{\mathscr{H}})$ be the extension of $\mathbb{B}_{Q}$ using
formula (107) with $t=1$. Similarly, let $\overline{\mathbb{B}}_{G}$ be the
extension of $\mathbb{B}_{G}$ to
$\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathscr{H})$.
Then
$\overline{\mathbb{B}}_{G}^{2}:\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathscr{H})\rightarrow\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathscr{H})$.
One checks that

(138)
${\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{S}(\mathscr{H})}$${\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{S}(\mathscr{H})}$${\Omega_{Q\times
S}(\underline{\mathscr{H}})}$${\Omega_{Q\times
S}(\underline{\mathscr{H}})}$$\scriptstyle{\overline{\mathbb{B}}_{G}}$$\scriptstyle{f_{S}^{\ast}}$$\scriptstyle{f_{S}^{\ast}}$$\scriptstyle{\overline{\mathbb{B}}_{Q}}$

commutes, which implies that

(139)
${\Omega^{0}_{S}(\mathscr{H})}$${\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(E)}$${\Omega^{0}_{Q\times
S}(\underline{\mathscr{H}})}$${\Omega_{Q\times
S}(\underline{\mathscr{H}})}$$\scriptstyle{\overline{\mathbb{B}}_{G}^{2}}$$\scriptstyle{f_{S}^{\ast}}$$\scriptstyle{f_{S}^{*}}$$\scriptstyle{\overline{\mathbb{B}}_{Q}^{2}}$

commutes.

Since sections of $\underline{\mathscr{H}}\simeq p^{\ast}\mathscr{H}$ are
generated over the ring of functions on $Q\times S$ by $p^{\ast}\sigma$ for
$\sigma\in\Omega^{0}_{S}(\mathscr{H})$, we see that
$\overline{\mathbb{B}}_{Q}^{2}=f_{S}^{\ast}\overline{\mathbb{B}}_{G}^{2}$
which implies $\mathcal{F}_{Q}=f_{S}^{\ast}\mathcal{F}_{G}$. ∎

We can write

(140) $\mathcal{F}_{G}=\not{D}^{2}+\mathcal{F}^{+}_{G}$

where
$\mathcal{F}^{+}_{G}\in(\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\mathcal{D}(\mathscr{E}))_{\operatorname{bas}}$
is a form with terms of degree between 1 and 4, valued in vertical
differential operators of order 1.

As in (85) we define the heat operator of the equivariant Bismut
superconnection as the Volterra series

(141) $e^{-t\mathcal{F}_{G}}:=e^{-t\not{D}^{2}}+\sum_{k\geq
1}(-t)^{k}I^{(k)}_{G}$

where

(142)
$I^{(k)}_{G}:=\int_{\Delta_{k}}e^{-t\sigma_{0}\not{D}^{2}}\mathcal{F}^{+}_{G}e^{-t\sigma_{1}\not{D}^{2}}\cdots\mathcal{F}^{+}_{G}e^{-t\sigma_{k}\not{D}^{2}}d\sigma.$

A notable difference to (85) is that (141) defines a form of unbounded degree
on $(S_{G})_{\nabla}$ with coefficients in vertical operators on
$\mathscr{E}$. However, we have the following analogous statements

######  Proposition 3.10.

The operators $I_{G}^{(k)}$ are forms of degree $\geq k$ on $(S_{G})_{\nabla}$
valued in vertical smoothing operators, i.e.
$I^{(k)}_{G}\in(\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathcal{K}(\mathscr{E})))_{\operatorname{bas}}$.

###### Proof.

The form $\mathcal{F}^{+}_{G}$ is valued in sections of
$\mathcal{D}(\mathscr{E})$ so applying Lemma (2.9) to each term shows that
$I^{(k)}_{G}$ is valued in sections of $\mathcal{K}(\mathscr{E})$. It is clear
that $I^{(k)}_{G}$ is horizontal and $G$-invariant. ∎

######  Corollary 3.11.

For $t>0$, the heat operator is an unbounded form valued in vertical smoothing
operators, i.e.
$e^{-t\mathcal{F}_{G}}\in(\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathcal{K}(\mathscr{E})))_{\operatorname{bas}}$.

######  Lemma 3.12.

The supertrace

(143)
$sTr:\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathcal{K}(\mathscr{E}))\rightarrow\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}$

is $G$-equivariant.

###### Proof.

For $K\in\mathcal{K}(\mathscr{E})$, $\varphi\in C^{\infty}(\mathscr{E})$, and
$g\in G$ we have

(144) $\begin{split}(g.K)(\varphi)&=gK(g^{-1}.\varphi)\\\
&=g\int_{X/S}k(x,y)\varphi(g.y)\\\
&=\int_{X/S}k(g^{-1}x,y)\varphi(g.y)d\text{Vol}_{X/S}(y)\\\
&=\int_{X/S}k(g^{-1}x,g^{-1}y)\varphi(y)d\text{Vol}_{X/S}(y)\end{split}$

where we have used the $G$-invariance of $g^{X/S}$ in the fourth equality.
Thus

(145)
$\begin{split}sTr(g.K)(z)&=\int_{X_{z}/S_{z}}sTr\left(k(g^{-1}x,g^{-1}x)\right)d\text{Vol}_{X/S}(x)\\\
&=sTr(K)(g^{-1}.z)=g.sTr(K)(z)\end{split}$

∎

Lemma 3.12 implies that the map

(146) $1\otimes
sTr:\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathcal{K}(\mathscr{E}))\rightarrow\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}$

is $G$-equivariant and preserves horizontal forms so it restricts to a map on
basic forms

(147)
$sTr:(\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{S}(\mathcal{K}(\mathscr{E})))_{\operatorname{bas}}\rightarrow(\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{S})_{\operatorname{bas}}.$

Analogous to (88), for $t>0$ the scaled equivariant superconnection satisfies

(148)
$\begin{split}\mathbb{B}^{t}_{G}&:=t^{1/2}\not{D}+\left(\tilde{\nabla}^{\mathscr{E}}_{G}+\frac{1}{2}k_{G}\right)-t^{-1/2}\frac{1}{4}c(T)_{G}\\\
&=t^{1/2}\delta_{t}\circ\mathbb{B}_{G}\circ\delta_{t}^{-1}.\end{split}$

We now define the Chern character of the scaled equivariant superconnection to
be

(149)
$\begin{split}\operatorname{ch}(\mathbb{B}^{t}_{G})&:=sTr(e^{-\mathcal{F}^{t}_{G}})\\\
&=sTr(e^{-\delta_{t}t\mathcal{F}_{G}\delta_{t}^{-1}})\\\
&=\delta_{t}sTr(e^{-t\mathcal{F}_{G}})\delta_{t}^{-1}\end{split}$

which is a form in $\Omega_{(S_{G})_{\nabla}}$.

######  Proposition 3.13.

The Chern Weil map (55) applied to the Chern character is a form in
$\Omega_{S_{Q}}$ that satisfies

(150)
$f_{S}^{\ast}\operatorname{ch}(\mathbb{B}^{t}_{G})=\operatorname{ch}(\mathbb{B}^{t}_{Q})$

###### Proof.

It follows from 3.9 that

(151) $f_{S}^{*}e^{-t\mathcal{F}_{G}}=e^{-t\mathcal{F}_{Q}}$

and a check of the definitions shows that

(152)
${(\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S}(\mathcal{K}(\mathscr{E})))_{\operatorname{bas}}}$${(\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{S})_{\operatorname{bas}}}$${\Omega_{S_{Q}}(\mathcal{K}(\mathscr{E}_{Q}))}$${\Omega_{S_{Q}}}$$\scriptstyle{sTr}$$\scriptstyle{f_{S}^{\ast}}$$\scriptstyle{f_{S}^{\ast}}$$\scriptstyle{sTr}$

commutes.

∎

Let $\nabla^{X/S}_{G}$ be the covariant derivative on
$(\overline{\operatorname{Kos}}(\mathfrak{g}^{*})\otimes\Omega_{X}(T(X/S)))_{\operatorname{bas}}$
given by

(153) $\nabla^{X/S}_{G}:=d_{K}+\nabla^{X/S}-\theta^{a}\tau_{a}.$

Its curvature

(154) $\Omega^{X/S}_{G}:=(\nabla^{X/S}_{G})^{2}$

is a form in
$(\operatorname{Kos}(\mathfrak{g}^{*})\otimes\Omega_{X}(\mathfrak{so}(T(X/S))))_{\operatorname{bas}}$.
By Lemma 1.10, it satisfies

(155) $f_{X}^{\ast}\Omega^{X/S}_{G}=\Omega^{X/S}_{Q}$

where $\Omega^{X/S}_{Q}$ is (102). The $\hat{A}$ polynomial applied to the
curvature $\Omega^{X/S}_{G}$ is the unbounded form

(156)
$\hat{A}(\Omega^{X/S}_{G}):=\text{det}^{1/2}\left(\frac{\Omega^{X/S}_{G}/2}{\sinh(\Omega^{X/S}_{G}/2)}\right)$

lying in
$(\overline{\operatorname{Kos}}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(\mathfrak{so}(T(X/S))))_{\operatorname{bas}}$.
Applying (155) gives

(157) $\hat{A}(\Omega_{Q}^{X/S})=f_{X}^{\ast}\hat{A}(\Omega_{G}^{X/S}).$

Finally, let
$F^{\mathscr{E}}_{G}\in(\operatorname{Kos}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(\operatorname{End}(\mathscr{E})))_{\operatorname{bas}}$
be the curvature of $\nabla^{\mathscr{E}}_{G}$ and set

(158)
$F^{\mathscr{E}/\mathbb{S}}_{G}:=F_{G}^{\mathscr{E}}-c(\alpha(\Omega_{G}^{X/S}))$

in
$(\operatorname{Kos}(\mathfrak{g}^{\ast})\otimes\Omega_{X}(\operatorname{End}(\mathscr{E})))_{\operatorname{bas}}$,
where
$\alpha:\mathfrak{so}(n)\rightarrow\mathfrak{spin}_{n}\subset\mathbb{C}\ell_{n}$
is (92) and
$c:\mathbb{C}\ell(T(X/S))\rightarrow\operatorname{End}(\mathscr{E})$ is the
Clifford map. Again, Lemma 1.10 implies that

(159)
$F^{\mathscr{E}/\mathbb{S}}_{Q}=f_{X}^{\ast}F^{\mathscr{E}/\mathbb{S}}_{G}$

and applying the relative chern character (94) gives the relation

(160)
$\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}}_{Q})=f_{X}^{\ast}\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}}_{G}).$

We now prove the main theorem.

######  Theorem 3.14.

There is an equality

(161) $\lim_{t\rightarrow
0}\operatorname{ch}(\mathbb{B}^{t}_{G})=\int_{X/S}\hat{A}(\Omega^{X/S}_{G})\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}}_{G}).$

of unbounded forms in $\overline{\Omega}_{(S_{G})_{\nabla}}$.

###### Proof.

The statement is proved if we show that the equation is satisfied on pullbacks
along maps $f:M\rightarrow B_{\nabla}G$:

(162) $\begin{split}\lim_{t\rightarrow
0}f_{S}^{\ast}\operatorname{ch}(\mathbb{B}^{t}_{G})&=\lim_{t\rightarrow
0}\operatorname{ch}(\mathbb{B}^{t}_{Q})\\\
&=\int_{X_{Q}/S_{Q}}\hat{A}(\Omega_{Q}^{X/S})\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F_{Q}^{\mathscr{E}/\mathbb{S}})\\\
&=\int_{X_{Q}/S_{Q}}f_{X}^{\ast}\hat{A}(\Omega_{G}^{X/S})f_{X}^{\ast}\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}}_{G})\\\
&=f_{S}^{\ast}\int_{X/S}\hat{A}(\Omega^{X/S}_{G})\operatorname{ch}_{\mathscr{E}/\mathbb{S}}(F^{\mathscr{E}/\mathbb{S}}_{G})\end{split}$

where we used Proposition 3.13 in the first equality and Theorem 2.12 in the
second. The fourth equality follows from the equality

(163) $\int_{(Q\times X)/(Q\times S)}p^{\ast}\omega=p^{\ast}\int_{X/S}\omega$

for all $\omega\in\Omega_{X}$. ∎

## References

  * [BF86] Jean-Michel Bismut and Daniel S. Freed.  The analysis of elliptic families. i. metrics and connections on determinant bundles.  Communications in Mathematical Physics, 106(1):159–176, 1986.
  * [BGV03] Nicole Berline, Ezra Getzler, and Michele Vergne.  Heat kernels and Dirac operators.  Springer Science & Business Media, 2003.
  * [Bis86] Jean-Michel Bismut.  The atiyah-singer index theorem for families of dirac operators: Two heat equation proofs.  Inventiones mathematicae, 83(1):91–151, 1986.
  * [FH13] Daniel S. Freed and Michael J. Hopkins.  Chern-weil forms and abstract homotopy theory, 2013.
  * [Fre16] Daniel S. Freed.  On equivariant chern-weil forms and determinant lines, 2016.
  * [MQ86] Varghese Mathai and Daniel Quillen.  Superconnections, thom classes, and equivariant differential forms.  Topology, 25(1):85–110, 1986.
  * [Qui85] Daniel Quillen.  Superconnections and the chern character.  Topology, 24(1):89–95, 1985.


# Minimal immersions of Riemannian manifolds in products of space forms

Fernando Manfio and Feliciano Vitório

###### Abstract

In this note, we give natural extensions to cylinders and tori of a classical
result due to T. Takahashi [9] about minimal immersions into spheres. More
precisely, we deal with Euclidean isometric immersions whose projections in
$\mathbb{R}^{N}$ satisfy a spectral condition of their Laplacian.

MSC 2000: 53C42, 53A10.

Key words: Minimal immersions, isometric immersions, Riemannian product of
space forms.

##  1 Introduction

An isometric immersion $f:M^{m}\to N^{n}$ of a Riemannian manifold $M$ in
another Riemannian manifold $N$ is said to be minimal if its mean curvature
vector field $H$ vanishes. The study of minimal surfaces is one of the oldest
subjects in differential geometry, having its origin with the work of Euler
and Lagrange. In the last century, a series of works have been developed in
the study of properties of minimal immersions, whose ambient space has
constant sectional curvature. In particular, minimal immersions in the sphere
$\mathbb{S}^{n}$ play a important role in the theory, as for example the
famous paper of J. Simons [8].

Let $f:M^{m}\to\mathbb{R}^{n}$ be an isometric immersion of a $m$-dimensional
manifold $M$ into the Euclidean space $\mathbb{R}^{n}$. Associated with the
induced metric on $M$, it is defined the Laplace operator $\Delta$ acting on
$\mathrm{C}^{\infty}(M)$. This Laplacian can be extended in a natural way to
the immersion $f$. A well-known result by J. Eells and J. H. Sampson [4]
asserts that the immersion $f$ is minimal if and only if $\Delta f=0$. The
following result, due to T. Takahashi [9], states that the immersion $f$
realizes a minimal immersion in a sphere if and only if its coordinate
functions are eigenfunctions of the Laplace operator with the same nonzero
eigenvalue.

######  Theorem 1.

Let $F:M^{m}\to\mathbb{R}^{n+1}$ be an isometric immersion such that

$\Delta F=-mcF$

for some constant $c\neq 0$. Then $c>0$ and there exists a minimal isometric
immersion $f:M^{m}\to\mathbb{S}^{n}_{c}$ such that $F=i\circ f$.

O. Garay generalized the Theorem 1 for the hypersurfaces
$f:M^{n}\to\mathbb{R}^{n+1}$ satisfying $\Delta f=Af$, where $A$ is a constant
$(n+1)\times(n+1)$ diagonal matrix. He proved in [5] that such a hypersurface
is either minimal or an open subset of a sphere or of a cylinder. In this
direction, J. Park [7] classified the hypersurfaces in a space form or in
Lorentzian space whose immersion $f$ satisfies $\Delta f=Af+B$, where $A$ is a
constant square matrix and $B$ is a constant vector. Similar results were
obtained in [1], where the authors study and classify pseudo-Riemannian
hypersurfaces in pseudo-Riemannian space forms which satisfy the condition
$\Delta f=Af+B$, where $A$ is an endomorphism and $B$ is a constant vector. In
a somewhat different direction, B. Chen in a series of papers discuss the
problem of determining the geometrical structure of a submanifold knowing some
simple analytic information (see for example [2], [3]).

In this work we shall deal with an isometric immersion
$f:M^{m}\to\mathbb{R}^{N}$ of a Riemannian manifold $M^{m}$ into the Euclidean
space $R^{N}$. If the submanifold $f(M)$ is contained in a cylinder
$\mathbb{S}^{n}\times\mathbb{R}^{k}\subset\mathbb{R}^{N}$ or in a torus
$\mathbb{S}^{n}\times\mathbb{S}^{k}\subset\mathbb{R}^{N}$, we shall call that
the immersion $f$ realizes an immersion in a cylinder or in a torus,
respectively. Motivated by recent works on the submanifold theory in the
product of space forms [6], we obtain theorems that give us necessary and
sufficient conditions for an isometric immersion $f:M^{m}\to\mathbb{R}^{N}$
realize a minimal immersion in a cylinder or in a torus (cf. Theorems 4 and
8).

##  2 Preliminaries

Let $M^{m}$ be a Riemannian manifold and $h\in\text{C}^{\infty}(M)$. The
hessian of $h$ is the symmetric section of $\text{Lin}(TM\times TM)$ defined
by

$\mathrm{Hess}\,h(X,Y)=XY(h)-\nabla_{X}Y(h)$

for all $X,Y\in TM$. Equivalently,

$\mathrm{Hess}\,h(X,Y)=\langle\nabla_{X}\mathrm{grad}\,h,Y\rangle$

where $X,Y\in TM$ and $\mathrm{grad}\,h$ is the gradient of $h$. The Laplacian
$\Delta h$ of a function $h\in\text{C}^{\infty}(M)$ at the point $p\in M$ is
defined as

$\Delta
h(p)=\mathrm{trace}\,\mathrm{Hess}\,h(p)=\mathrm{div}\,\mathrm{grad}\,h(p).$

Consider now an isometric immersion $f:M^{m}\to\mathbb{R}^{n}$. For a fixed
$v\in\mathbb{R}^{n}$, let $h\in\text{C}^{\infty}(M)$ be the height function
with respect to the hyperplane normal to $v$, given by $h(p)=\langle
f(p),v\rangle$. Then

$\displaystyle\mathrm{Hess}\,h(X,Y)=\langle\alpha_{f}(X,Y),v\rangle$ (1)

for any $X,Y\in TM$. For an isometric immersion $f:M^{n}\to\mathbb{R}^{n}$, by
$\Delta f(p)$ at the point $p\in M$ we mean the vector

$\Delta f(p)=(\Delta f_{1}(p),\ldots,\Delta f_{n}(p)),$

where $f=(f_{1},\ldots,f_{n})$. Taking traces in (1) we obtain

$\displaystyle\Delta f(p)=mH(p),$ (2)

where $H(p)$ is the mean curvature vector of $f$ at $p\in M$.

##  3 Minimal submanifolds in $\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}$

Let $\mathbb{S}^{n}_{c}$ denote the sphere with constant sectional curvature
$c>0$ and dimension $n$. We use the fact that $\mathbb{S}^{n}_{c}$ admits a
canonical isometric embedding in $\mathbb{R}^{n+1}$ as

$\mathbb{S}^{n}_{c}=\left\\{X\in\mathbb{R}^{n+1}:\langle
X,X\rangle=1/c\right\\}.$

Thus, $\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}$ admits a canonical isometric
embedding

$i:\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}\to\mathbb{R}^{n+k+1}.$

Denote by $\pi:\mathbb{R}^{n+k+1}\to\mathbb{R}^{n+1}$ the canonical
projection. Then, the normal space of $i$ at each point
$z\in\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}$ is spanned by $N(z)=c(\pi\circ
i)(z))$, and the second fundamental form of $i$ is given by

$\alpha_{i}(X,Y)=-c\langle\pi X,Y\rangle\pi\circ i.$

If we consider a parallel orthonormal frame $E_{1},\ldots,E_{n+k+1}$ of
$\mathbb{R}^{n+k+1}$ such that

$\displaystyle\mathbb{R}^{k}=\mathrm{span}\\{E_{n+2},\ldots,E_{n+k+1}\\},$ (3)

we can express the second fundamental form $\alpha_{i}$ as

$\displaystyle\alpha_{i}(X,Y)=-c\left(\langle
X,Y\rangle-\sum_{i=n+2}^{n+k+1}\langle X,E_{i}\rangle\langle
Y,E_{i}\rangle\right)\pi\circ i.$ (4)

The following result shows that minimal immersions of a $m$-dimensional
Riemannian manifold into the cylinder $\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}$
are precisely those immersions whose $n+1$ first coordinate functions in
$\mathbb{R}^{n+k+1}$ are eigenfunctions of the Laplace operator in the induced
metric.

######  Proposition 2.

Let $f:M^{m}\to\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}$ be an isometric
immersion and set $F=i\circ f$, where
$i:\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}\to\mathbb{R}^{n+k+1}$ is the
canonical inclusion. Let $E_{1},\ldots,E_{n+k+1}$ be a parallel orthonormal
frame of $\mathbb{R}^{n+k+1}$ as in (3). Then $f$ is a minimal immersion if
and only if

$\displaystyle\Delta
F=-c\left(m-\sum_{j=n+2}^{n+k+1}\|T_{j}\|^{2}\right)\pi\circ F,$ (5)

where $T_{j}$ denotes the orthogonal projection of $E_{j}$ onto $TM$.

###### Proof.

The second fundamental forms of $f$ and $F$ are related by

$\alpha_{F}(X,Y)=i_{\ast}\alpha_{f}(X,Y)+\alpha_{i}(f_{\ast}X,f_{\ast}Y)$

for all $X,Y\in TM$. From (4) we get that

$\alpha_{F}(X,Y)=i_{\ast}\alpha_{f}(X,Y)-c\left(\langle
X,Y\rangle-\sum_{j=n+2}^{n+k+1}\langle X,T_{j}\rangle\langle
Y,T_{j}\rangle\right)\pi\circ F,$

where $T_{j}$ denotes the orthogonal projection of $E_{i}$ onto $TM$. Taking
traces and using (2) yields

$\Delta
F=mi_{\ast}H^{f}-c\left(m-\sum_{j=n+2}^{n+k+1}\|T_{j}\|^{2}\right)\pi\circ F,$

and the conclusion follows. ∎

######  Remark 3.

In case $f:M^{m}\to\mathbb{S}^{n}_{c}\times\mathbb{R}$, a tangent vector field
$T$ on $M$ and a normal vector field $\eta$ along $f$ are defined by

$\frac{\partial}{\partial t}=f_{\ast}T+\eta,$

where $\frac{\partial}{\partial t}$ is an unit vector field tangent to
$\mathbb{R}$. In this case, $f$ is a minimal immersion if and only if

$\Delta F=-c(m-\|T\|^{2})\pi\circ F.$

The next result states that any isometric immersion of a Riemannian manifold
$M^{m}$ into Euclidean space $\mathbb{R}^{n+k+1}$, whose Laplacian satisfies a
condition as in (5), arises for a minimal isometric immersion of $M$ into some
cylinder $\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}$.

######  Theorem 4.

Let $F:M^{m}\to\mathbb{R}^{n+k+1}$ be an isometric immersion and let
$E_{1},\ldots,E_{n+k+1}$ be a parallel orthonormal frame in
$\mathbb{R}^{n+k+1}$ such that

$\Delta
F=-c\left(m-\sum_{j=n+2}^{n+k+1}\|T_{j}\|^{2}\right)\left(F-\sum_{j=n+2}^{n+k+1}\langle
F,E_{j}\rangle E_{j}\right),$

for some constant $c\neq 0$, where $T_{j}$ denotes the orthogonal projection
of $E_{j}$ onto the tangent bundle $TM$. Then $c>0$ and there exists a minimal
isometric immersion $f:M^{m}\to\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}$ such
that $F=i\circ f$.

###### Proof.

Since $\Delta F=mH$ by (2), the assumption implies that the vector field

$N=F-\sum_{j=n+2}^{n+k+1}\langle F,E_{j}\rangle E_{j}$

is normal to $F$. On the other hand,

$\displaystyle\langle N,E_{j}\rangle$ $\displaystyle=$
$\displaystyle\left\langle F-\sum_{l=n+2}^{n+k+1}\langle F,E_{l}\rangle
E_{l},E_{j}\right\rangle$ $\displaystyle=$ $\displaystyle\langle
F,E_{j}\rangle-\langle F,E_{j}\rangle=0$

for all $n+2\leq j\leq n+k+1$. Hence, for any $X\in TM$ we have

$X\langle N,N\rangle=2\left\langle F_{\ast}X-\sum_{j=n+2}^{n+k+1}\langle
F_{\ast}X,E_{j}\rangle E_{j},N\right\rangle=0,$

and it follows that $\langle N,N\rangle=r^{2}$ for some constant $r$. Now we
claim that

$\displaystyle\Delta\|F\|^{2}=2\sum_{j=n+2}^{n+k+1}\|T_{j}\|^{2}.$ (6)

To see this, fix a point $p\in M$ and consider a local geodesic frame
$\\{X_{1},\ldots,X_{m}\\}$ in $p$. Then

$\mathrm{grad}\,\|F\|^{2}=\sum_{\alpha=1}^{m}X_{\alpha}(\|F\|^{2})X_{\alpha}=2\sum_{\alpha=1}^{m}\langle
F_{\ast}X_{\alpha},F\rangle X_{\alpha}=2F^{T}.$

Since $N$ is normal to $F$, we have

$F^{T}=\sum_{j=n+2}^{n+k+1}\langle F,E_{j}\rangle
T_{j}=\sum_{j=n+2}^{n+k+1}\sum_{\alpha=1}^{m}\langle F,E_{j}\rangle\langle
E_{j},X_{\alpha}\rangle X_{\alpha},$

and it follows that

$\mathrm{grad}\,\|F\|^{2}=2\sum_{j=n+2}^{n+k+1}\sum_{\alpha=1}^{m}\langle
F,E_{j}\rangle\langle E_{j},X_{\alpha}\rangle X_{\alpha}.$

Therefore,

$\displaystyle\Delta\|F\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{\beta=1}^{m}\left\langle\nabla_{X_{\beta}}\mathrm{grad}\,\|F\|^{2},X_{\beta}\right\rangle$
$\displaystyle=$ $\displaystyle
2\sum_{\alpha,\beta=1}^{m}\sum_{j=n+2}^{n+k+1}\left\langle\nabla_{X_{\beta}}\langle
F,E_{j}\rangle\langle E_{j},X_{\alpha}\rangle
X_{\alpha},X_{\beta}\right\rangle$ $\displaystyle=$ $\displaystyle
2\sum_{\alpha,\beta=1}^{m}\sum_{j=n+2}^{n+k+1}\langle
F_{\ast}X_{\beta},E_{j}\rangle\langle E_{j},X_{\alpha}\rangle\langle
X_{\alpha},X_{\beta}\rangle$ $\displaystyle=$ $\displaystyle
2\sum_{\alpha=1}^{m}\sum_{j=n+2}^{n+k+1}\langle
X_{\alpha},T_{j}\rangle^{2}=2\sum_{j=n+2}^{n+k+1}\|T_{j}\|^{2},$

and this proves our claim. Finally, using the fact that

$\Delta\|F\|^{2}=2(\langle\Delta F,F\rangle+m),$

we get that

$\displaystyle\sum_{j=n+2}^{n+k+1}\|T_{j}\|^{2}$ $\displaystyle=$
$\displaystyle\langle\Delta
F,F\rangle+m=\left\langle-c\left(m-\sum_{j=n+2}^{n+k+1}\|T_{j}\|^{2}\right)N,N\right\rangle+m$
$\displaystyle=$
$\displaystyle-\left(m-\sum_{j=n+2}^{n+k+1}\|T_{j}\|^{2}\right)cr^{2}+m,$

and the equality above implies that $c=1/r^{2}$. We conclude that there exists
an isometric immersion $f:M^{m}\to\mathbb{S}^{n}_{c}\times\mathbb{R}^{k}$ such
that $F=i\circ f$, and minimality of $f$ follows from Proposition 2. ∎

As an application of Theorem 4 we will construct an example into
$\mathbb{S}^{2n-1}\times\mathbb{R}$.

######  Example 5.

For each real number $a$, with $\sqrt{n-1}<a\leq\sqrt{n}$, we claim that there
exists a real number $b$ such that the immersion
$f:\mathbb{R}^{n}\to\mathbb{R}^{2n+1}$, given by

$f(x_{1},\ldots,x_{n})=\frac{1}{\sqrt{n}}\left(e^{iax_{1}},\ldots,e^{iax_{n}},b\sum_{j=1}^{n}x_{j}\right),$

is a minimal immersion into $\mathbb{S}^{2n-1}\times\mathbb{R}$. In fact, we
need check the hypothesis of the Theorem 4 for a suitable choice of
$(a,b)\in\mathbb{R}^{2}$. First observe that

$\Delta f=-a^{2}\left(f-\left\langle f,\frac{\partial}{\partial
t}\right\rangle\frac{\partial}{\partial t}\right),$

where $\frac{\partial}{\partial t}$ denotes a unit vector field tangent to the
factor $\mathbb{R}$. Now, if we denote by $T$ the orthogonal projection of
$\frac{\partial}{\partial t}$ onto $f$, a direct computation give us

$\displaystyle\|T\|^{2}=\frac{nb^{2}}{a^{2}+nb^{2}}.$ (7)

On the other hand, follows from (6) that

$\displaystyle\|T\|^{2}=\langle\Delta f,f\rangle+n=n-a^{2}.$ (8)

It follows from (7) and (8) that

$b^{2}=\frac{a^{2}(n-a^{2})}{n(a^{2}-n+1)}.$

##  4 Minimal submanifolds in the product $\mathbb{S}^{n}\times\mathbb{S}^{k}$

Let $\mathbb{S}^{n}$ and $\mathbb{S}^{k}$ denote the spheres of dimension $n$
and $k$, respectively. Using the fact that the spheres admit a canonical
isometric embedding $\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$ and
$\mathbb{S}^{k}\subset\mathbb{R}^{k+1}$, the product
$\mathbb{S}^{n}\times\mathbb{S}^{k}$ admits a canonical isometric embedding

$\displaystyle i:\mathbb{S}^{n}\times\mathbb{S}^{k}\to\mathbb{R}^{n+k+2}.$ (9)

Denote by $\pi_{1}:\mathbb{R}^{n+k+2}\to\mathbb{R}^{n+1}$ and
$\pi_{2}:\mathbb{R}^{n+k+2}\to\mathbb{R}^{k+1}$ the canonical projections.
Then, the normal space of $i$ at each point
$z\in\mathbb{S}^{n}\times\mathbb{S}^{k}$ is spanned by
$N_{1}(z)=\pi_{1}(i(z))$ and $N_{2}(z)=\pi_{2}(i(z))$, and the second
fundamental form of $i$ is given by

$\alpha_{i}(X,Y)=-\langle\pi_{1}X,Y\rangle N_{1}-\langle\pi_{2}X,Y\rangle
N_{2}$

for all $X,Y\in T_{z}\mathbb{S}^{n}\times\mathbb{S}^{k}$.

Now, let $f:M^{m}\to\mathbb{S}^{n}\times\mathbb{S}^{k}$ be an isometric
immersion of a Riemannian manifold. Then, writing $F=i\circ f$, the unit
vector fields $N_{1}=\pi_{1}\circ F$ and $N_{2}=\pi_{2}\circ F$ are normal to
$F$. Consider a parallel orthonormal frame $E_{1},\ldots,E_{n+k+2}$ of
$\mathbb{R}^{n+k+2}$ such that

$\displaystyle\mathbb{R}^{n+1}=\mathrm{span}\\{E_{1},\ldots,E_{n+1}\\}\quad\text{and}\quad\mathbb{R}^{k+1}=\mathrm{span}\\{E_{n+2},\ldots,E_{n+k+2}\\}.$
(10)

In terms of this frame, we can express the vector fields $N_{1}$ and $N_{2}$
as

$\displaystyle N_{1}=F-\sum_{j=n+2}^{n+k+2}\langle F,E_{j}\rangle
E_{j}\quad\text{and}\quad N_{2}=F-\sum_{l=1}^{n+1}\langle F,E_{l}\rangle
E_{l}.$ (11)

######  Proposition 6.

Let $f:M^{m}\to\mathbb{S}^{n}\times\mathbb{S}^{k}$ be an isometric immersion
and set $F=i\circ f$, where
$i:\mathbb{S}^{n}\times\mathbb{S}^{k}\to\mathbb{R}^{n+k+2}$ is the canonical
inclusion. Let $E_{1},\ldots,E_{n+k+2}$ be a parallel orthonormal frame of
$\mathbb{R}^{n+k+2}$ as in (10). Then $f$ is a minimal immersion if and only
if

$\displaystyle\Delta
F=-\left(m-\sum_{j=n+2}^{n+k+2}\|T_{j}\|^{2}\right)N_{1}-\left(m-\sum_{l=1}^{n+1}\|T_{l}\|^{2}\right)N_{2},$
(12)

where $T_{j}$ denotes the orthogonal projection of $E_{j}$ onto $TM$.

###### Proof.

The second fundamental forms of $f$ and $F$ are related by

$\alpha_{F}(X,Y)=i_{\ast}\alpha_{f}(X,Y)+\alpha_{i}(f_{\ast}X,f_{\ast}Y)$

for all $X,Y\in TM$. Let $E_{1},\ldots,E_{n+k+2}$ be a parallel orthonormal
frame of $\mathbb{R}^{n+k+2}$ as in (10). Given $X\in TM$, we can write

$\pi_{1}X=X-\sum_{j=n+1}^{n+k+2}\langle X,T_{j}\rangle
E_{j}\quad\text{and}\quad\pi_{2}X=X-\sum_{l=1}^{n+1}\langle X,T_{l}\rangle
E_{l},$

and so, we have

$\langle\pi_{1}X,Y\rangle=\langle X,Y\rangle-\sum_{j=n+1}^{n+k+2}\langle
X,T_{j}\rangle\langle Y,T_{j}\rangle$

and

$\langle\pi_{2}X,Y\rangle=\langle X,Y\rangle-\sum_{l=1}^{n+1}\langle
X,T_{l}\rangle\langle Y,T_{l}\rangle.$

Then the second fundamental form of $F$ can be expressed by

$\displaystyle\alpha_{F}(X,Y)$ $\displaystyle=$ $\displaystyle
i_{\ast}\alpha_{f}(X,Y)-\left(\langle X,Y\rangle-\sum_{j=n+1}^{n+k+2}\langle
X,T_{j}\rangle\langle Y,T_{j}\rangle\right)N_{1}$ $\displaystyle-\left(\langle
X,Y\rangle-\sum_{l=1}^{n+1}\langle X,T_{l}\rangle\langle
Y,T_{l}\rangle\right)N_{2}.$

Taking traces and using (2) yields

$\Delta
F=mH^{F}=mi_{\ast}H^{f}-\left(m-\sum_{j=n+1}^{n+k+2}\|T_{j}\|^{2}\right)N_{1}-\left(m-\sum_{l=1}^{n+1}\|T_{l}\|^{2}\right)N_{2}$

and the conclusion follows. ∎

######  Remark 7.

Observe that an isometric immersion
$f:M^{m}\to\mathbb{S}^{n}\times\mathbb{S}^{k}$ can be seen as an isometric
immersion $\widetilde{f}=\iota\circ f:M^{m}\to\mathbb{S}_{\kappa}^{n+k+1}$
into the sphere with constant sectional curvature $\kappa=1/2$, where
$\iota:\mathbb{S}^{n}\times\mathbb{S}^{k}\to\mathbb{S}_{\kappa}^{n+k+1}$
denotes the canonical inclusion.

The next result states that any isometric immersion of a Riemannian manifold
$M^{m}$ into the sphere $\mathbb{S}_{\kappa}^{N-1}$ with constant sectional
curvature $\kappa=1/2$, whose Laplacian of coordinate functions satisfies a
condition as in (12), arises for a minimal isometric immersion of $M^{m}$ into
a product of spheres
$\mathbb{S}^{n}\times\mathbb{S}^{k}\subset\mathbb{R}^{N}$.

######  Theorem 8.

Let $F:M^{m}\to\mathbb{S}_{\kappa}^{N-1}$ be an isometric immersion. Fixed a
choice of two integers $n$ and $k$, with $N=n+k+2$, let $E_{1},\ldots,E_{N}$
be a parallel orthonormal frame in $\mathbb{R}^{N}$ as in (10) such that

$\Delta\widetilde{F}=-\left(m-\sum_{j=n+2}^{n+k+2}\|T_{j}\|^{2}\right)N_{1}-\left(m-\sum_{l=1}^{n+1}\|T_{l}\|^{2}\right)N_{2},$

where $\widetilde{F}=h\circ F$, $h:\mathbb{S}_{\kappa}^{N-1}\to\mathbb{R}^{N}$
is the umbilical inclusion, $T_{i}$ denotes the orthogonal projection of
$E_{i}$ onto $TM$ and $N_{1}$ and $N_{2}$ as in (11). Then there exists a
minimal isometric immersion $f:M^{m}\to\mathbb{S}^{n}\times\mathbb{S}^{k}$
such that $F=i\circ f$.

###### Proof.

We first prove that $N_{1}$ and $N_{2}$ are normal to $F$. In fact, in terms
of an orthonormal frame $\\{X_{1},\ldots,X_{m}\\}$ of $TM$, we have

$\displaystyle\sum_{i=1}^{N}\|T_{i}\|^{2}=\sum_{l=1}^{n+1}\|T_{l}\|^{2}+\sum_{j=n+2}^{N}\|T_{j}\|^{2}=m.$
(13)

Then, as $\widetilde{F}=N_{1}+N_{2}$, we can write:

$\displaystyle\begin{aligned}
\Delta\widetilde{F}=&-\left(\sum_{l=1}^{n+1}\|T_{l}\|^{2}\right)N_{1}-\left(\sum_{j=n+2}^{N}\|T_{j}\|^{2}\right)N_{2}\\\
=&-\left(\sum_{l=1}^{n+1}\|T_{l}\|^{2}\right)\widetilde{F}+\left(\sum_{l=1}^{n+1}\|T_{l}\|^{2}-\sum_{j=n+2}^{N}\|T_{j}\|^{2}\right)N_{2}.\end{aligned}$
(14)

If

$\sum_{l=1}^{n+1}\|T_{l}\|^{2}=\sum_{j=n+2}^{N}\|T_{j}\|^{2},$

we have, by using (13), that

$\Delta\widetilde{F}=-\frac{1}{2}m\widetilde{F}.$

Thus, it follows from Theorem 1 that $F:M^{m}\to\mathbb{S}_{\kappa}^{N-1}$ is
a minimal isometric immersion. Suppose from now on that

$\displaystyle\sum_{l=1}^{n+1}\|T_{l}\|^{2}\neq\sum_{j=n+2}^{N}\|T_{j}\|^{2}.$
(15)

As $\Delta\widetilde{F}=mH$ and $\widetilde{F}$ is normal to $M$, we conclude
from (14) that $N_{2}$ is normal to $M$. Similary we obtain that $N_{1}$ is
normal to $M$. Now, for $n+2\leq j\leq n+k+2$, we have

$\displaystyle\langle N_{1},E_{j}\rangle$ $\displaystyle=$
$\displaystyle\left\langle\widetilde{F}-\sum_{i=n+2}^{N}\langle\widetilde{F},E_{i}\rangle
E_{i},E_{j}\right\rangle$ $\displaystyle=$
$\displaystyle\langle\widetilde{F},E_{j}\rangle-\langle\widetilde{F},E_{j}\rangle=0.$

Hence, for any $X\in TM$ we have

$X\langle
N_{1},N_{1}\rangle=2\left\langle\widetilde{F}_{\ast}X-\sum_{j=n+2}^{N}\langle\widetilde{F}_{\ast}X,E_{j}\rangle
E_{j},N_{1}\right\rangle=0,$

and it follows that $\langle N_{1},N_{1}\rangle=r^{2}$ for some constant $r$.
The same argument gives $\langle N_{2},N_{2}\rangle=s^{2}$ for some constant
$s$. Since $\widetilde{F}=N_{1}+N_{2}$ and
$\Delta\|\widetilde{F}\|^{2}=2(\langle\Delta\widetilde{F},\widetilde{F}\rangle+m)$,
we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Delta\|\widetilde{F}\|^{2}=\langle\Delta\widetilde{F},\widetilde{F}\rangle+m$
$\displaystyle=$
$\displaystyle-\left(m-\sum_{j=n+2}^{N}\|T_{j}\|^{2}\right)r^{2}-\left(m-\sum_{l=1}^{n+1}\|T_{l}\|^{2}\right)s^{2}+m.$
$\displaystyle=$
$\displaystyle-\left(\sum_{l=1}^{n+1}\|T_{l}\|^{2}\right)r^{2}-\left(\sum_{j=n+2}^{N}\|T_{j}\|^{2}\right)s^{2}+m.$

Since $r^{2}+s^{2}=2$, we can rewrite the above equation as

$\displaystyle\left(\sum_{l=1}^{n+1}\|T_{l}\|^{2}-\sum_{j=n+2}^{N}\|T_{j}\|^{2}\right)s^{2}$
$\displaystyle=$ $\displaystyle 2\sum_{l=1}^{n+1}\|T_{l}\|^{2}-m$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{n+1}\|T_{l}\|^{2}-\sum_{j=n+2}^{n+k+2}\|T_{j}\|^{2}.$

As we are assuming (15), we obtain $s^{2}=1$, and therefore $r^{2}=1$. We
conclude that there exists an isometric immersion
$f:M^{m}\to\mathbb{S}^{n}\times\mathbb{S}^{k}$ such that $F=i\circ f$, and
minimality of $f$ follows from Proposition 6.

∎

## References

  * [1] L. J. Alías, A. Ferrández, P. Lucas, Hypersurfaces in space forms satisfying the condition $\Delta x=Ax+B$, Trans. Amer. Math. Soc., 347 (1995), 1793–1801.
  * [2] B.-Y. Chen., A report on submanifolds of finite type, Soochow J. Math. 22 (1996), 117–337.
  * [3] B.-Y. Chen, O. J. Garay, $\delta(2)$-ideal null $2$-type hypersurfaces of Euclidean space are spherical cylinders, Kodai Math. J. 35 (2012), 382–391.
  * [4] J. Eells, J. H. Sampson, Harmonic mapping of Riemannian manifold, Amer. J. Math., 186 (1964), 109–160.
  * [5] O. J. Garay, An extension of Takahashi’s theorem, Geom. Dedicata 34 (1990), 105–112.
  * [6] J. H. Lira, R. Tojeiro, F. Vitório, A Bonnet theorem for isometric immersions into products of space forms, Archiv der Math. 95 (2010), 469–479.
  * [7] J. Park, Hypersurfaces satisfying the equation $\Delta x=Rx+b$, Proc. Amer. Math. Soc. 120 (1994), 317–328.
  * [8] J. Simons, Minimal Varietes in Riemannian Manifolds, Ann. of Math. 88 (1968) 62–105.
  * [9] T. Takahashi, Minimal Immersions of Riemannian Manifolds, J. Math. Soc. Japan. 18 (1966), 380–385.

| Fernando Manfio
---
ICMC, Universidade de São Paulo
São Carlos-SP, 13561-060
Brazil
<EMAIL_ADDRESS>
|  | Feliciano Vitório
---
IM, Universidade Federal de Alagoas
Maceió-AL, 57072-900
Brazil
<EMAIL_ADDRESS>