Metric Equivalence Problem

The metric equivalence problem concerns two related questions for Riemannian metrics or pseudo-Riemannian metrics:

1. Find a complete system of invariants, or

2. Decide when two metric tensors differ locally only by a coordinate transformation.

The second formulation asks whether suitable neighborhoods are related by a local isometry. Christoffel (1869) and Lipschitz (1869) studied this question for Riemannian metrics.

The moving frame approach compares components of the Riemann tensor and its covariant derivatives. For a fully regular -dimensional pseudo-Riemannian manifold, Cartan's bound is , where counts covariant derivatives of the Riemann tensor (Cartan 1946; Milson and Wylleman 2013, p. 2). Here full regularity means that, at every derivative order, the ranks of the Jacobian matrices of the maps collecting the components of the Riemann tensor and its successive covariant derivatives, as functions of position and frame, are locally constant.

In four dimensions, Cartan's bound is 10. For four-dimensional Lorentzian manifolds, Karlhede's frame-normalization method reduces the bound to 7 (Karlhede 1980; Milson and Wylleman 2013, p. 2). These are upper bounds on the derivative order needed for local classification, and lower orders can suffice. The bound of 7 is attained (Milson and Pelavas 2008).

This problem concerns local preservation of a metric tensor. It is different from equivalent metrics in the topological sense, which need only induce the same topology.