# Lecture Notes on the Fundamental Structures of Differential Geometry

Dmitrii Pedchenko 111UCLA Mathematics Department, Los Angeles, CA 90095-1555,
Email addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
(April 3, 2022)

###### Contents

  1. 1 Introduction
  2. 2 Differential Operators
    1. 2.1 Differential operators over a ring
      1. 2.1.1 Differential operators over the polynomial ring
      2. 2.1.2 Differential operators over the ring of smooth functions
    2. 2.2 Algebra of symbols of differential operators over a ring
    3. 2.3 Algebra of symbols of differential operators over the ring of smooth functions on a manifold

##  Chapter 1 Introduction

This is the first part of the lecture notes that grew out of the special
course given by the author to the audience of exceptional undergraduate and
first and second year graduate students at UCLA interested in geometry during
the 2021/2022 academic year. In these lecture notes we present an approach to
the fundamental structures in differential geometry that uses the vernacular
of sheaves, differential operators and horizontal subbundles.

This language is now standard in other disciplines, notably algebraic and
complex geometry, but has been slow to enter the textbooks for students
aspiring to work in differential geometry or use differential geometry in
other geometric disciplines.

A notable exception is the recent textbook [Wed16] that uses the language of
sheaves and cohomology to introduce the basics of smooth manifolds. This
textbook, however, does not tap into the differential geometry territory per
se, as it does not discuss metrics, connections, curvature, parallel transport
or holonomy groups.

Another feature of this book is the treatment of the notion of connection from
various different viewpoints: as a Koszul connection, as a covariant
derivative, as an Ehresmann connection, as a parallel transport, etc. A
similar extensive exposition of the notion of connection can be found in
[Spi99] and in [Pos88], but those expositions do not use the modern language
that we try to develop in these notes.

A byproduct of the chosen approach is a relatively simple and conceptually
clear construction of canonical connections (Levi-Civita, Chern, Bismut, etc.)
without a heavy reliance on explicit local expressions as a starting point.

Finally, these lecture notes were heavily inspired by the course [Ver13] in
differential geometry given by Misha Verbitsky in 2013.

##### Prerequisites

This book assumes familiarity with the language of sheaves and the basics of
smooth manifolds on the level of first year graduate courses.

Both of these topics are covered in the already mentioned [Wed16]. For a more
extensive modern treatment of the basics of smooth manifolds see the wonderful
textbook [Lee09]. Other references include [Lee13], [Pos88] or the classic
[KN96].

Familiarity with scheme theory and complex geometry will also be very helpful.

##### Acknowledgments

First, we would like to thank Misha Verbitsky for a wonderful and broad
education in geometry. The author took $5$ geometric courses and sat on
numerous seminars organized by Misha while being an undergraduate student in
the National Research University Higher School of Economics.

Second, I want to thank my scientific advisors Jack Huizenga and Alexei
Pirkovskii for the ongoing support and interest in our work.

Last but not least, these lecture notes were prepared during the ongoing
Russian invasion of Ukraine. Our heart is with those who had to bear the
unnecessary and intolerable suffering one encounters during the war time. We
also thank UCLA for the accommodations provided to the author during this
rough period of life.

##  Chapter 2 Differential Operators

###  2.1 Differential operators over a ring

######  Definition 2.1.

A commutator of two linear operators $A,B:V\to V$ is a linear operator defined
as

$[A,B]:=A\circ B-B\circ A.$

The next key definition is due to A. Grothendieck.

######  Definition 2.2.

Let $R$ be a (commutative) $k$-algebra. A differential operator of order
$\mathbf{0}$ is a morphism of $R$-modules $D^{0}:R\to R$. Denote differential
operators of order $0$ by $Diff^{0}(R)$.

More generally, a $k$-linear morphism $D^{i}:R\to R$ is a differential
operator of order $\mathbf{i}$, $D^{i}\in Diff^{i}(R)$, if for any $D^{0}\in
Diff^{0}(R)$ the commutator $[D^{i},D^{0}]$ lies in $Diff^{i-1}(R)$.

We have

$Diff^{0}(R)\subset Diff^{1}(R)\subset Diff^{2}(R)\subset...,$

and the union $Diff^{\bullet}(R):=\cup_{i\geq 0}Diff^{i}(R)$ is called the
algebra of differential operators on $\mathbf{R}$.

######  Example 2.3 (Differential operators of order $0$).

It follows from the above definition that

$Diff^{0}(R)\simeq R$

via $D^{0}\mapsto D^{0}(1)$. The inverse morphism is given by
$a\mapsto(m_{a}:b\mapsto ab).$ We will consistently use this identification
throughout this chapter.

######  Example 2.4 (Differential operators of order $1$).

Let us show that

$Diff^{1}(R)\simeq Der(R)\oplus R$

as $k$-vector spaces. First note that for any derivation $X\in Der(R)$, any
$m_{a}\in Diff^{0}(R)$ and any $b\in R$ we have

$[X,m_{a}](b)=X(ab)-aX(b)=aX(b)+bX(a)-aX(b)=X(a)b,$

so we see that $[X,m_{a}]$ is just $m_{X(a)}\in Diff^{0}(R)$. Thus
$Der(R)\subset Diff^{1}(R).$

Now consider $D^{1}\in Diff^{1}(R)$. Define

$X(a):=D^{1}(a)-D^{1}(1)a,\quad\forall a\in R.$

######  Exercise 2.5.

Check that $X$ defined above is a derivation.

This exercise shows that any $D^{1}\in Diff^{1}(R)$ can be written as

$D_{1}=X+m_{D^{1}(1)}.$

Note that if $D^{i}\in Diff^{i}(R)$ and $D^{j}\in Diff^{j}(R)$, then for any
$m_{a}\in Diff^{0}(R)$:

$\displaystyle Diff^{i+j-1}$ $\displaystyle\ni[m_{a},D^{i}]\circ
D^{j}+D^{i}\circ[m_{a},D^{j}]=m_{a}\circ D^{i}\circ D^{j}-D^{i}\circ
m_{a}\circ D^{j}$ $\displaystyle+D^{i}\circ m_{a}\circ D^{j}-D^{i}\circ
D^{j}\circ m_{a}=m_{a}\circ D^{i}\circ D^{j}-D^{i}\circ D^{j}\circ
m_{a}=[m_{a},D^{i}\circ D^{j}]\ $

which shows that $D^{i}\circ D^{j}\in Diff^{i+j}(R)$. This way, we get a
filtered algebra $\mathbf{Diff^{\bullet}(R)}$ of differential operators.

######  Exercise 2.6.

Let $D\in Diff^{k}(R)$.

  1. (a)

Prove that
$D(a_{1}a_{2}...a_{k})=D^{\prime}(a_{2}a_{3}...a_{k})+a_{1}D(a_{2}a_{3}...a_{k}),$
where $D^{\prime}=[D,m_{a_{1}}]\in Diff^{k-1}(R)$.

  2. (b)

Let $I\subset R$ be an ideal. Use induction to prove $D(I^{k+1})\subset I.$

######  Exercise 2.7.

Use induction and the Jacobi identity

$[A,[B,C]]=[[A,B],C]+[B,[A,C]],\quad A,B,C\ -\text{linear operators},$

to prove that if $D^{i}\in Diff^{i}(R),D^{j}\in Diff^{j}(R)$, then the
commutator $[D^{i},D^{j}]$ lies in $Diff^{i+j-1}(R)$.

####  2.1.1 Differential operators over the polynomial ring

For this subsection, let $R:=\boldsymbol{\mathbb{R}}[t_{1},...,t_{n}]$.

######  Lemma 2.8.

If $D\in Diff^{k}(R)$ is a differential operator that vanishes on all
polynomials of degree $\leq k$, then $D=0$.

###### Proof.

Note that if $t_{1}^{i_{1}}\cdot...\cdot t_{n}^{i_{n}}$ is a monomial of
degree $|I|=i_{1}+...+i_{n}=k+1$, then by Exercise 2.6 (a)

$D(t_{1}^{i_{1}}\cdot...\cdot
t_{n}^{i_{n}})=D^{\prime}(t^{|I|-1})+t_{i}D(t^{|I|-1}),\quad
D^{\prime}=[D,m_{t_{i}}]\in Diff^{k-1}(R),\quad|I|-1=k.$

The first term is zero by induction on the order of a differential operator
and the fact that $D^{\prime}$ vanishes on polynomials of degree $\leq k-1$.
The second term is zero by the assumption of the lemma.

The inductive step for monomials $t_{1}^{i_{1}}\cdot...\cdot t_{n}^{i_{n}}$ of
degree $|I|=i_{1}+...+i_{n}>k+1$ is handled using the same formula. ∎

######  Lemma 2.9.

Let $R_{\leq k}=:\boldsymbol{\mathbb{R}}[t_{1},...,t_{n}]_{\leq k}$ be the
space of polynomials of degree $\leq k$. Let $A:R_{\leq k}\to R$ be a linear
operator. Then there exists $D\in Diff^{k}(R)$ such that $D|_{R_{\leq
k}}:R_{\leq k}\to R$ coincides with $A$.

###### Proof.

Consider a differential operator
$D_{f,i_{1}...i_{n}}=\frac{f(t_{1},...,t_{n})}{i_{1}!\cdot...\cdot
i_{n}!}\left(\frac{d}{dt_{1}}\right)^{\circ
i_{1}}\circ...\circ\left(\frac{d}{dt_{n}}\right)^{\circ i_{n}}$ for $f\in R$.
Note that $D_{f,i_{1}...i_{n}}$ is zero on all monomials of degree
$<|I|=i_{1}+...+i_{n}$,

$D_{f,i_{1}...i_{n}}(t_{1}^{i_{1}}...t_{n}^{i_{n}})=f(t_{1},...,t_{n}),$

and is zero on all other monomials of degree $|I|$.

Thus

$D=\sum_{i_{1}+...+i_{n}\leq
k}D_{A(t_{1}^{i_{n}}...t_{n}^{i_{n}}),i_{1}...i_{n}}$

is the desired differential operator as monomials
$A(t_{1}^{i_{n}}...t_{n}^{i_{n}})$ with $i_{1}+...+i_{n}\leq k$ form the basis
of $R_{\leq k}$. ∎

######  Proposition 2.10.

The algebra of differential operators
$Diff^{\bullet}\left(\boldsymbol{\mathbb{R}}[t_{1},...,t_{n}]\right)$ is
generated by $t_{i}$ and $\frac{d}{dt_{j}}$ for $i,j=1,...,n$.

###### Proof.

Let $D$ be a differential operator of order $k$. Consider $D|_{R_{\leq
k}}:R_{\leq k}\to R$. By Lemma 2.9, there is $\tilde{D}\in Diff^{k}(R)$ that
lies in the subalgebra generated by $t_{i}$ and $\frac{d}{dt_{j}}$ and that
satisfies $D|_{R_{\leq k}}=\tilde{D}|_{R_{\leq k}}$. The differential operator
$D-\tilde{D}\in Diff^{k}(R)$ vanishes on all polynomials of degree $\leq k$ by
construction and therefore, by Lemma 2.8, $D=\tilde{D}$. ∎

######  Exercise 2.11.

Prove that $Diff^{\bullet}(R)$ is an $\boldsymbol{\mathbb{R}}$-algebra
generated by $t_{1},...,t_{n},\frac{d}{dt_{1}},...,\frac{d}{dt_{n}}$ with
relations

$[t_{i},t_{j}]=0,\quad\left[\frac{d}{dt_{i}},\frac{d}{dt_{j}}\right]=0,\quad\forall
i,j,$ $\left[t_{i},\frac{d}{dt_{i}}\right]=1,$
$\left[\frac{d}{dt_{i}},t_{j}\right]=0,\quad i\neq j.$

####  2.1.2 Differential operators over the ring of smooth functions

In this subsection, we consider differential operators over the ring
$C^{\infty}(\boldsymbol{\mathbb{R}}^{n})$ of smooth function on
$\boldsymbol{\mathbb{R}}^{n}$. We denote by $x_{i}$ the standard coordinate
functions $x_{i}:(p_{1},...p_{n})\mapsto p_{i}.$

Let $m_{x}=\\{f\in R\ |f(x)=0\\}$ be the maximal ideal of smooth functions
vanishing at point $x$. By Exercise 2.6 (b), for any differential operator of
order $i$ and $f\in m_{x}^{i+1}$ we have $D(f)=0$. This observation allows us
to prove the following key lemma.

######  Lemma 2.12.

Any differential operator $D\in
Diff^{i}(C^{\infty}(\boldsymbol{\mathbb{R}}^{n}))$ that vanishes on
polynomials of degree $\leq i$ is trivial.

###### Proof.

Consider $D\in Diff^{i}(C^{\infty}(\boldsymbol{\mathbb{R}}^{n}))$ that
vanishes on $\boldsymbol{\mathbb{R}}[x_{1},...,x_{n}]_{\leq i}$. Take any
$f\in C^{\infty}(\boldsymbol{\mathbb{R}}^{n})$ and any point
$x\in\boldsymbol{\mathbb{R}}^{n}$. Take the Taylor polynomial $P$ for function
$f$ of degree $i$ at point $x$. We have

$f-P\in m_{x}^{i+1}.$

By the observation above and the assumption that $D$ vanishes on
$\boldsymbol{\mathbb{R}}[x_{1},...,x_{n}]_{\leq i}$ we conclude

$D(f-P)\in m_{x}\quad\implies\quad D(f-P)=0\quad\implies\quad
D(f)=D(P)+D(f-P)=0.$

∎

Next we prove an analogue of Proposition 2.10 for
$Diff^{\bullet}(C^{\infty}(\boldsymbol{\mathbb{R}}^{n})$.

######  Proposition 2.13.

The algebra $Diff^{\bullet}(C^{\infty}(\boldsymbol{\mathbb{R}}^{n}))$ is
generated by $\frac{d}{dx_{i}}$ as a
$C^{\infty}(\boldsymbol{\mathbb{R}}^{n})$-algebra.

###### Proof.

Similar to the proof of Lemma 2.9, given an $\boldsymbol{\mathbb{R}}$-linear
map

$A:\boldsymbol{\mathbb{R}}[x_{1},...,x_{n}]_{\leq k}\to
C^{\infty}(\boldsymbol{\mathbb{R}}^{n})$

we can explicitly construct a differential operator

$D\in C^{\infty}(\boldsymbol{\mathbb{R}}^{n})\left<\frac{\partial}{\partial
x_{1}},...,\frac{\partial}{\partial x_{n}}\right>\subset
Diff^{k}(C^{\infty}(\boldsymbol{\mathbb{R}}^{n}))$

whose restriction to $\boldsymbol{\mathbb{R}}[x_{1},...,x_{n}]_{\leq k}$
coincides with $A$ as a sum of differential operators

$D=\sum_{i_{1}+...+i_{n}\leq
k}D_{A(x_{1}^{i_{n}}...x_{n}^{i_{n}}),i_{1}...i_{n}}\ ,$
$D_{f,i_{1}...i_{n}}=\frac{f(x_{1},...,x_{n})}{i_{1}!\cdot...\cdot
i_{n}!}\left(\frac{\partial}{\partial x_{1}}\right)^{\circ
i_{1}}\circ...\circ\left(\frac{\partial}{\partial x_{n}}\right)^{\circ
i_{n}},\quad f\in C^{\infty}(\boldsymbol{\mathbb{R}}^{n}).$

Now consider an arbitrary differential operator $D$ of order $k$. Consider the
restriction

$D|_{\boldsymbol{\mathbb{R}}[x_{1},...,x_{n}]_{\leq
k}}:\boldsymbol{\mathbb{R}}[x_{1},...,x_{n}]_{\leq k}\to
C^{\infty}(\boldsymbol{\mathbb{R}}^{n}).$

As above, construct a differential operator $\tilde{D}$ of order $i$ as a sum
of $D_{f,i_{1},...,i_{n}}$ with the property $D=\tilde{D}$ on
$\boldsymbol{\mathbb{R}}[x_{1},...,x_{n}]_{\leq k}$.

Consider $\overline{D}=D-\tilde{D}\in
Diff^{k}(C^{\infty}(\boldsymbol{\mathbb{R}}^{n})).$ By construction
$\overline{D}$ vanishes on all polynomials of order $\leq k$. By Lemma 2.12
$\overline{D}=0$ or, equivalently

$D=\tilde{D}\in
C^{\infty}(\boldsymbol{\mathbb{R}}^{n})\left<\frac{\partial}{\partial
x_{1}},...,\frac{\partial}{\partial x_{n}}\right>\subset
Diff^{k}(C^{\infty}(\boldsymbol{\mathbb{R}}^{n})).$

∎

This way, in the case $R=C^{\infty}(\boldsymbol{\mathbb{R}}^{n})$ Proposition
2.13 allows to relate Definition 2.2 to the historic analytical definition of
a differential operator $D\in
Diff^{k}(C^{\infty}(\boldsymbol{\mathbb{R}}^{n}))$ as an expression

$D=\sum_{i_{1}+...+i_{n}\leq
k}f_{i_{1},...,i_{n}}(x_{1},...,x_{n})\frac{\partial^{i_{1}+...+i_{n}}}{\partial
x_{1}^{i_{1}}...x_{n}^{i_{n}}},\quad f_{i_{1},...,i_{n}}(x_{1},...,x_{n})\in
C^{\infty}(\boldsymbol{\mathbb{R}}^{n}).$

The key Definition 2.2 can be sheafified if one replaces a ring $R$ with a
_sheaf_ of algebras $\mathcal{R}$ on a topological space $X$.

######  Definition 2.14.

Let $\mathcal{R}$ be a sheaf of (commutative) $k$-algebras on a topological
space $X$. A sheaf of differential operators on $\bm{\mathcal{R}}$ is a
functor

$\\{U\subset X\ \text{open}\\}^{op}\to\text{Filtered $k$-algebras},$ $U\mapsto
Diff^{\bullet}(\mathcal{R}(U)),$

where $Diff^{\bullet}(\mathcal{R}(U))$ is the filtered algebra of differential
operators over $\mathcal{R}(U)$ in the sense of Definition 2.2. One readily
checks that this assignment satisfies the sheaf axioms.

We will denote this sheaf by $Diff^{\bullet}_{\mathcal{R}}$ or simply by
$Diff^{\bullet}$ if the underlying sheaf $\mathcal{R}$ is clear from the
context. Observe that after forgetting the filtration the sheaf of
differential operators is a subsheaf of the sheaf of internal homomorphisms

$Diff^{\bullet}_{\mathcal{R}}\subset\mathscr{H}\text{\kern-3.0pt{\calligra\large
om}}\,_{k}(\mathcal{R},\mathcal{R}),$

where homomorphisms are taken with respect to the $k$-vector space structure
of $\mathcal{R}$.

Finally, if $D\in Diff^{i}_{\mathcal{R}}(U)=Diff^{i}(\mathcal{R}(U))$, then we
say that $D$ is a differential operator of order $\bm{i}$ on $\bm{U}$.

###  2.2 Algebra of symbols of differential operators over a ring

Given a $k$-algebra $R$ or a sheaf of $k$-algebras $\mathcal{R}$, consider the
filtered algebra of differential operators

$Diff^{0}(R)\subset Diff^{1}(R)\subset Diff^{2}(R)\subset...\subset
Diff^{\bullet}(R)$

and the filtered sheaf of algebras of differential operators

$Diff^{0}_{\mathcal{R}}\subset Diff^{1}_{\mathcal{R}}\subset
Diff^{2}_{\mathcal{R}}\subset...\subset Diff^{\bullet}_{\mathcal{R}}.$

######  Proposition 2.15.

The associated graded algebra $\oplus_{i\geq 0}Diff^{i}(R)/Diff^{i-1}(R)$ and
the associated graded sheaf of algebras $\oplus_{i\geq
0}Diff^{i}_{\mathcal{R}}/Diff^{i-1}_{\mathcal{R}}$ are commutative.

###### Proof.

By Exercise 2.7, for $D^{i}\in Diff^{i}(R),D^{j}\in Diff^{j}(R)$, we have

$[D^{i},D^{j}]=D^{i}\circ D^{j}-D^{j}\circ D^{i}\in Diff^{i+j-1}(R).$

Hence as elements of $Diff^{i+j}(R)$

$D^{i}\circ D^{j}\equiv D^{j}\circ D^{i}\quad\mod Diff^{i+j-1}(R).$

∎

######  Definition 2.16.

The associated graded algebra

$Symb^{*}(Diff^{\bullet}(R)):=\oplus_{i\geq
0}Symb^{i}:=gr(Diff^{\bullet}(R))=\oplus_{i\geq 0}Diff^{i}(R)/Diff^{i-1}(R)$

is called the algebra of symbols of differential operators over $\bm{R}$. Note
that $S^{0}=Diff^{0}(R)$ is canonically isomorphic to $R$, so
$Symb(Diff^{\bullet}(R))$ is a (commutative) $R$-algebra.

We can similarly define a sheaf of (commutative) $\bm{\mathcal{R}}$-algebras
of symbols of differential operators over $\bm{\mathcal{R}}$:

$Symb^{*}(Diff^{\bullet}_{\mathcal{R}}):=\oplus_{i\geq
0}Symb^{i}:=gr(Diff^{\bullet}_{\mathcal{R}})=\oplus_{i\geq
0}Diff^{i}_{\mathcal{R}}/Diff^{i-1}_{\mathcal{R}}.$

Here

######  Example 2.17.

Let us show that $Symb^{1}=Diff^{1}(R)/Diff^{0}(R)$ is isomorphic to the
$R$-module $Der(R)$ of derivations of $R$.

###### Proof.

In Example 2.4 we showed that every derivation is also a differential operator
of order $1$. This way we get a morphism

$Der(R)\xhookrightarrow{}Diff^{1}(R)\twoheadrightarrow
Diff^{1}(R)/Diff^{0}(R).$

The inverse morphism is defined by

$Diff^{1}(R)\ni D\to D-m_{D(1)}\in Der(R).$

One easily checks that $D-m_{D(1)}$ is indeed a derivation. Note that for
$D=m_{a}\in Diff^{0}(R)$ the difference $D-m_{D(1)}=m_{a}-m_{a\cdot 1}=0$, so
the inverse morphism is in fact defined as a map

$Diff^{1}(R)/Diff^{0}(R)\to Der(R).$

∎

######  Proposition 2.18.

There is a natural homomorphism of graded commutative $R$-algebras

$Sym^{*}_{R}(Der(R))=\oplus_{i\geq 0}Sym^{i}_{R}(Der(R))\to\oplus_{i\geq
0}Symb^{i}=Symb^{*}(Diff^{\bullet}(R))$

that extends the isomorphism
$Der(R)\stackrel{{\scriptstyle\simeq}}{{\to}}Symb^{1}$, and an analogous
natural homomorphism of sheaves of graded commutative $\mathcal{R}$-algebras

$Sym^{*}(Der_{\mathcal{R}})\to Symb^{*}(Diff^{\bullet}_{\mathcal{R}}).$

###### Proof.

Consider the following diagram of solid arrows:

${Sym^{*}(Der(R))}$${Symb^{*}(Diff^{\bullet}(R))}$${Der(R)}$${Symb^{1}.}$$\scriptstyle{\simeq}$

By Proposition 2.15 the target algebra $Symb^{*}(Diff^{\bullet}(R))$ is
commutative. The upper horizontal dashed arrow is then just the universal
property of the symmetric algebra applied to the bottom isomorphism of Example
2.17 followed by the inclusion
${Symb^{1}\xhookrightarrow{}Symb^{*}(Diff^{\bullet}(R))}$ on the right of the
commutative diagram. ∎

###  2.3 Algebra of symbols of differential operators over the ring of smooth
functions on a manifold

In this section we will specialize the results of §2.2 on the symbols of
differential operators to the setting of differential operators on a smooth
manifold $M$.

Consider the following objects:

  * •

the ring $R=C^{\infty}(M)$ of smooth functions on a smooth manifold $M$,

  * •

the sheaf $\mathcal{R}=C^{\infty}_{M}$ of smooth functions on $M$ (so that
$R=C^{\infty}(M)=\Gamma(M,C^{\infty}_{M})=\Gamma(M,\mathcal{R})$),

  * •

the algebra $Diff^{\bullet}(M):=Diff^{\bullet}(C^{\infty}(M))$ of global
differential operators on $M$,

  * •

the sheaf of algebras $Diff^{\bullet}_{M}:=Diff^{\bullet}_{C^{\infty}_{M}}$ of
differential operators on $M$ in the sense of Definition 2.14 (so that
${Diff^{\bullet}(M)=\Gamma(M,Diff^{\bullet}_{M}))}$.

While the similarity of notation for the global and local objects might appear
slightly confusing at first, the reader will soon find that working with
sheafified objects is almost identical to working with global objects. This
justifies our choice of notation.

The well known equivalence between vector fields and derivations of the sheaf
of smooth functions $C^{\infty}_{M}$ yields an isomorphism of sheaves

$Der(C^{\infty}_{M})\simeq TM,$

where we identify the tangent bundle $TM$ with the locally free sheaf of its
sections (vector fields). In this setting, the morphism of Proposition 2.18
becomes

$Sym^{*}(TM)\to Symb^{*}(Diff^{\bullet}_{M}).$

The first goal of this section will be to prove the following key theorem.

######  Theorem 2.19.

Let $M$ be a smooth manifold. The natural homomorphism of sheaves

$Sym^{*}(TM)\to Symb^{*}(Diff^{\bullet}_{M})$

is an isomorphism.

An immediate corollary of this theorem is that the sheaf of symbols of
differential operators on a manifold is _locally free_. We will prove the
above theorem as a series of lemmas.

## Bibliography

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