Strong Pseudo-Riemannian Metric

A strong pseudo-Riemannian metric on a smooth manifold modeled on a real Banach space is a smooth tensor field of continuous symmetric bilinear forms for which, at each , the map

is a bounded linear isomorphism of onto its continuous dual normed space , with continuous inverse function (Fischer et al. 1980, p. 182). This condition is called strong non-degeneracy.

In finite dimensions, strong and weak non-degeneracy coincide, since an injective linear transformation between vector spaces of the same finite dimension is an isomorphism. The distinction matters in infinite dimensions.

Strong pseudo-Riemannian metrics which are also positive definite are called strong Riemannian metrics. This agrees with the convention that ordinary semi-Riemannian metrics include the positive definite case. Every strong pseudo-Riemannian (respectively, strong Riemannian) metric is weak pseudo-Riemannian (respectively, weak Riemannian) due to the fact that strong non-degeneracy implies weak non-degeneracy.