Ricci Curvature - Generalization To Affine Connections

Generalization To Affine Connections

The Ricci tensor can also be generalized to arbitrary affine connections, where it is an invariant that plays an especially important role in the study of the projective geometry (geometry associated to unparameterized geodesics) (Nomizu & Sasaki 1994). If denotes an affine connection, then the curvature tensor is the tensor defined by

for any vector fields . The Ricci tensor is defined to be the trace:

In this more general situation, the Ricci tensor is symmetric if and only if there exist locally a parallel volume form for the connection.