[Let $E$,$F$ be vector bundles of rank $m$ and $n$ respectively over $X$ a manifold of dimension $k$. A **a rank**$\mathbf {k}$**differential operator** is a map $D:\Gamma (E)\rightarrow \Gamma (F)$ satisfying the following: About each point choose an open set such that $E$ and $F$ are trivial. $s\in \Gamma (E)$ and $Ds\in \Gamma (F)$ may be expressed locally as $s:U\rightarrow \mathbb {R} ^{m}$, $Ds:U\rightarrow \mathbb {R} ^{n}$. We say that $D$ is a differential operator of order $k$ if locally $Ds=As$ where A is a matrix with coefficients of the form:

$\sum f_{i_{1},...,i_{j}}{\frac {\partial ^{j}}{\partial x_{i_{1}}...x_{i_{j}}}}$

and

$j\leq k$.

**Exercise 13.8**

- The Laplace operator. $\Delta =\sum {\frac {\partial ^{2}}{\partial x_{i}^{2}}}$ acting on complex functions is a differential operator of degree $2$.
- The exterior derivative. $d:\Omega ^{p}\rightarrow \Omega ^{p+1}$ is a differential operator of degree $1$ (exercise).]

Suppose that D is a differential operator of degree $k$. Now for each 1-form $\omega \neq 0$ we define $\sigma (D)(\omega )\in Hom(E,F)$ called the **symbol** of $D$. This measures the top-order behaviour of the operator and may be described locally:

We form a new matrix $A^{P}$ from $A$ above by forgetting all differentials of order $<p$, now write the 1-form $\omega$ in local coordinates $\omega (p)=(v_{1},...,v_{k})$. Replace each differential ${\frac {\partial }{\partial x_{i}}}$ in the matrix $A^{p}$ with $v_{i}\in \mathbb {R}$. This matrix gives a linear map $E_{p}\rightarrow V_{p}$ and globally we have a bundle homomorphism $\sigma (D)(\omega )\in Hom(E,F)$ which is called the symbol.

**Exercise 13.9**

Consider the cotangent bundle $\pi :T_{X}^{\star }\setminus 0\rightarrow X$, we may pull back the bundles to obtain $\pi ^{\star }(E),\pi ^{\star }(F)$. Rearrange the above to show that the symbol can be expressed as a bundle morphism $\sigma (D):\pi ^{\star }(E)\rightarrow \pi ^{\star }(F)$.

**Definition 13.4**

$D$ is called an Elliptic operator if $\sigma (D)(\omega )$ is a bundle isomorphism for all $\omega \in \Omega _{X}^{1}\setminus 0$.

**Exercise 13.10**

For the Exterior derivative we may check that $\sigma (d)(\alpha ):\Omega ^{k}\rightarrow \Omega ^{k+1}=\cdot \mapsto \alpha \wedge \cdot$.

By computing the rank of $\Omega ^{k}$ we see that this can't be an isomorphism in all but the most trivial cases, therefore $d$ is not an elliptic operator. Define $d^{\star }:\Omega ^{k+1}\rightarrow \Omega ^{k}$ by

$fdx_{1}...dx_{p}\mapsto \sum _{i}{\frac {\partial f}{\partial x_{i}}}dx_{1}...{\hat {dx_{i}}}...dx_{p}$

. Then

$d^{\star }$ is the formal adjoint of d.

$Ker(d)=coker(d^{\star })$ and

$coker(d)=ker(d^{\star })$, and

$d+d^{\star }:\Omega ^{\star }\rightarrow \Omega ^{\star }$ is an elliptic operator (this is seen by computing the symbol:

$\sigma (d+d^{\star })(a)(\cdot )=a\wedge \cdot +a\lrcorner \cdot$). Note that

$(d+d^{\star })(\Omega ^{2\star })\subset \Omega ^{2*+1}$, restricting the operator in this way

$Ker(d+d^{\star })=\oplus H^{2\star }(X)$,

$Coker(d+d^{\star })=\oplus H^{2\star +1}(X)$.

It turns out that an elliptic complex is always a Fredholm operator ($\Gamma (E),\Gamma (F)$ considered as trivial bundles over $X$) and we see in this case $Index(D)=\sum dim(H^{2i})-\sum dim(H^{2i+1})=e(X)$ (exercise).

Elliptic operators are invertible up to lower order operators. Using the compact Rellich lemma (which states that $L_{k}^{2}\rightarrow L_{k-1}^{2}$ is a compact embedding) shows that

$L^{2}(E){\xrightarrow {D}}L^{2}(F)$

is Fredholm. Index(D) depends only on the homotopy class of the map

$T_{X}^{*}\setminus 0\rightarrow Iso(E,F)$ given

$\omega \mapsto \sigma (D)(\omega )$ and can provide information about the topology of

$X$.

Consider $\pi ^{\star }(E)-\pi ^{\star }(F)\in K(T^{\star }(X))$. Subtracting $\rho ^{\star }(F)$ to get a class ${\tilde {\sigma }}\in {\tilde {K}}(X)$.

We may embed $X\hookrightarrow N_{X}\hookrightarrow \mathbb {R} ^{n}$. Suppose that $N_{X}$ was trivial $N_{X}=X\times \mathbb {R} ^{m}$ ($m=codim(X)$).

Now there exists an operator on the trivial bundle $\mathbb {R} \times \mathbb {C}$ over $\mathbb {R}$. Whose symbol over $T^{\star }\mathbb {R} \cong \mathbb {R} ^{2}$, gives the Bott class $b\in {\tilde {K}}(\mathbb {R} ^{2})$. We want to form $D\boxplus B^{\boxplus m}\in {\tilde {K}}(T^{\star }X\times R^{2m})={\tilde {K}}(T^{\star }(N_{X}))$.