Elliptic operator on a compact manifold

Summary

An elliptic operator on a compact manifold is a differential operator between vector bundles whose principal symbol is invertible at every nonzero covector. It extends the notion of an elliptic operator on a domain of $\R^n$ to sections of bundles on a closed manifold. On a compact manifold such an operator is Fredholm between suitable Sobolev spaces, so it has a finite index. The Hodge Laplacian, the operator $d + d^*$ on forms, Dirac operators and the operator $\bar\partial + \bar\partial^*$ built from the Dolbeault operator are the main examples. The Atiyah–Singer index theorem computes their indices.