Invariant Differential Operator - Invariance in Terms of Abstract Indices

Invariance in Terms of Abstract Indices

Given two connections and and a one form, we have

for some tensor . Given an equivalence class of connections, we say that an operator is invariant if the form of the operator does not change when we change from one connection in the equivalence class to another. For example, if we consider the equivalence class of all torsion free connections, then the tensor Q is symmetric in its lower indices, i.e. . Therefore we can compute

where brackets denote skew symmetrization. This shows the invariance of the exterior derivative when acting on one forms. Equivalence classes of connections arise naturally in differential geometry, for example:

• in conformal geometry an equivalence class of connections is given by the Levi Civita connections of all metrics in the conformal class;
• in projective geometry an equivalence class of connection is given by all connections that have the same geodesics;
• in CR geometry an equivalence class of connections is given by the Tanaka-Webster connections for each choice of pseudohermitian structure