Elliptic complex

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Definition

Symbol-free definition

A differential complex is said to be elliptic if its sequence of symbols is exact.

Definition with symbols

Let $M$ be a differential manifold and $E_{i}$ be smooth vector bundles over $M$. Let $P_{i}:\Gamma (E_{i})\to \Gamma (E_{i+1})$ form a differential complex (viz $P_{i}\circ P_{i-1}=0$. Then this differential complex is said to be elliptic if the following sequence of symbols is exact:

$\ldots \pi ^{*}(E_{i-1})\to ^{\sigma (P_{i-1})}\pi ^{*}(E_{i})\to ^{\sigma (P_{i})}\pi ^{*}{E_{i+1}}\ldots$