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Weak Riemannian Metric

A weak Riemannian metric g on a smooth manifold M modeled on a real Banach space, possibly infinite dimensional, is a smooth assignment of continuous inner products g_m to its tangent spaces. The induced linear transformation

g_m^♭(v)=g_m(v,·),

from T_mM to its dual normed space T_m^*M is injective, but need not be surjective.

If each induced linear transformation is an isomorphism of Banach spaces, then g is a strong Riemannian metric. The two notions coincide in finite dimensions, since an injective linear transformation between vector spaces of the same finite dimension is an isomorphism. Thus "weak" does not exclude strong metrics (Stacey 2008).

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

Portions of 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. New York: Springer-Verlag, 2002.Stacey, A. "How to Construct a Dirac Operator in Infinite Dimensions." 18 Sep 2008. https://arxiv.org/abs/0809.3104.

Referenced on Wolfram|Alpha

Weak Riemannian Metric

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Weak Riemannian Metric." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeakRiemannianMetric.html

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