Principal symbol of an operator on a manifold

Summary

The principal symbol of a differential operator between vector bundles records the top-order part of the operator. It is a bundle map that depends on a covector. It is a function on the cotangent bundle with values in linear maps between the fibres. It is homogeneous in the covector, of degree the order of the operator. Unlike the full local expression of the operator, it does not depend on coordinates or trivialisations. Ellipticity, characteristic directions and the index theorem are all phrased in terms of the principal symbol. It is the manifold version of the principal symbol on $\R^n$.