Invariance of the Discrete Wedge Product¶
Consider a manifold and its polyhedrization in the context of discrete exterior calculus, and let denote the simplices and the corresponding basis functions, such that the pairing given by integration is natural
Given the projection and reconstruction operators ,
the discrete wedge product is defined as a three tensor
Consider now a second manifold and a diffeomorphism
which induces a new polyhedrization with the corresponding simplices and basis functions .
Given a simplex and a form , the following identities hold for push-forwards and pull-backs:
The discrete wedge product on this new manifold will be elementwise equal to the wedge product on the original manifold. To see why this is the case:
So we have shown that the discrete wedge product is indeed invariant under diffeomorphisms.
This result is consistent with theory in differential geometry, where the wedge product is thought of as a pureley topological operator that is independent of the metric.