Weak Riemannian Metric

A weak Riemannian metric on a smooth manifold M is a (0,2) tensor field g which is both a weak pseudo-Riemannian metric and positive definite.

In a very precise way, the condition of being a weak Riemannian metric is considerably less stringent than the condition of being a strong Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, any strong Riemannian metric provides an isomorphism between the tangent and cotangent spaces T_mM and T_m^*M, respectively, for all m in M; conversely, weak Riemannian metrics are merely injective linear maps from T_mM to T_m^*M (Marsden et al. 2002).

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 Pseudo-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.Stacey, A. "How to Construct a Dirac Operator in Infinite Dimensions." 2008.

Cite this as:

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

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