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Strong Pseudo-Riemannian Metric

A strong 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, the map

v_m|->g_m(v_m,·)

is an isomorphism of T_mM onto T_m^*M. This latter condition is called strong non-degeneracy.

Strong pseudo-Riemannian metrics which are also positive definite are called strong 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. 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.

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 Riemannian Metric, Weak Pseudo-Riemannian Metric, Weak Riemannian Metric

This entry contributed by Christopher Stover

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References

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

Cite this as:

Stover, Christopher. "Strong Pseudo-Riemannian Metric." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/StrongPseudo-RiemannianMetric.html

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