Weak Pseudo-Riemannian Metric

A weak pseudo-Riemannian metric on a smooth manifold M is a (0,2) tensor field g which is symmetric and for which, at each m in M,


for all w_m in T_mM implies that v_m=0. This latter condition is most commonly referred to as non-degeneracy though, in the presence of so-called strong non-degeneracy, is more accurately described as weak non-degeneracy.

Weak pseudo-Riemannian metrics which are also positive definite are called weak Riemannian metrics. This use of terminology is in stark contrast to the case of pseudo-Riemannian (and hence, semi-Riemannian) metrics which fundamentally cannot be Riemannian due to the existence of negative-squared terms in their metric signatures. In most literature, weak Riemannian metrics are simply called Riemannian.

See also

Lorentzian Manifold, Metric Signature, Metric Tensor, Metric Tensor Index, Positive Definite Tensor, Pseudo-Euclidean Space, Pseudo-Riemannian Manifold, Semi-Riemannian Manifold, Semi-Riemannian Metric, Smooth Manifold, Strong Pseudo-Riemannian Metric, Strong Riemannian Metric, Weak Riemannian Metric

This entry contributed by Christopher Stover

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Marsden, J. E.; Ratiu, T.; and Abraham, R. Manifolds, Tensor Analysis, and Applications, 3rd ed. Springer-Verlag Publishing Company, 2002.

Cite this as:

Stover, Christopher. "Weak Pseudo-Riemannian Metric." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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