Global analysis, analysis on manifolds Definition

Elliptic operator on a compact manifold

Builds on Differential operator between vector bundles Principal symbol of an operator on a manifold Elliptic operator Vector bundle Smooth manifold Laplace–Beltrami operator Exterior derivative

Definition

Let $M$ be a compact smooth manifold without boundary, and let $E$ and $F$ be smooth vector bundles over $M$ . Let $P \colon \Gamma(E) \to \Gamma(F)$ be a differential operator of order $m$ with principal symbol $\sigma_m(P)$ . The operator $P$ is elliptic if the linear map $\sigma_m(P)(x, \xi) \colon E_x \to F_x$ is invertible for every $x \in M$ and every nonzero $\xi \in T^*_x M$ .

Remarks

When $\dim M \ge 1$ , the condition forces $E$ and $F$ to have the same rank, since an invertible linear map exists only between spaces of equal dimension. On an open subset of $\R^n$ with $E$ and $F$ trivial of rank $1$ , it is the usual condition for an elliptic operator.

Example: let $(M, g)$ be a closed Riemannian manifold and let $\Delta = -\operatorname{div} \operatorname{grad}$ be the Laplace–Beltrami operator on functions. Its principal symbol is $\sigma_2(\Delta)(x, \xi) = -\abs{\xi}_g^2$ , which is a nonzero number for $\xi \ne 0$ , so $\Delta$ is elliptic.

Non-example: on a closed manifold $M$ of dimension $n \ge 2$ , the exterior derivative $d$ from functions to $1$ -forms is not elliptic. Its symbol at $\xi$ is the map $\R \to T^*_x M$ , $c \mapsto c \, \xi$ , which is injective but not surjective, since $T^*_x M$ has dimension $n > 1$ .

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Differential form concept
Differential operator between vector bundles concept
Elliptic operator concept
Laplace–Beltrami operator concept
Principal symbol of an operator on a manifold concept
Smooth manifold concept
Vector bundle concept
Exterior derivative concept
Elliptic operator on a compact manifold definition

Tag	`W4B84CGEG1`
URI	`m2://W4B84CGEG1`
Kind	definition
Writing	W0 stub
MSC	58J05, 47F10, 35J46
Defines	Elliptic operator on a compact manifold
Uses	Differential operator between vector bundles, Principal symbol of an operator on a manifold, Elliptic operator, Vector bundle, Smooth manifold, Laplace–Beltrami operator, Exterior derivative

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