Invariant Differential Operator - Examples

Examples

  1. The usual gradient operator acting on real valued functions on Euclidean space is invariant with respect to all Euclidean transformations.
  2. The differential acting on functions on a manifold with values in 1-forms (its expression is

    in any local coordinates) is invariant with respect to all smooth transformations of the manifold (the action of the transformation on differential forms is just the pullback).
  3. More generally, the exterior derivative

    that acts on n-forms of any smooth manifold M is invariant with respect to all smooth transformations. It can be shown that the exterior derivative is the only linear invariant differential operator between those bundles.
  4. The Dirac operator in physics is invariant with respect to the Poincaré group (if we choose the proper action of the Poincaré group on spinor valued functions. This is, however, a subtle question and if we want to make this mathematically rigorous, we should say that it is invariant with respect to a group which is a double cover of the Poincaré group)
  5. The conformal Killing equation

    is a conformally invariant linear differential operator between vector fields and symmetric trace-free tensors.

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