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
Read more about this topic: Invariant Differential Operator
Famous quotes containing the words terms and/or abstract:
“I am a patient manalways willing to forgive on the Christian terms of repentance; and also to give ample time for repentance.”
—Abraham Lincoln (18091865)
“What is important, then, is not that the critic should possess a correct abstract definition of beauty for the intellect, but a certain kind of temperament, the power of being deeply moved by the presence of beautiful objects.”
—Walter Pater (18391894)