TOPICS

# Strong Riemannian Metric

A strong Riemannian metric on a smooth manifold is a tensor field which is both a strong pseudo-Riemannian metric and positive definite.

In a very precise way, the condition of being a strong Riemannian metric is considerably more stringent than the condition of being a weak Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, strong Riemannian metrics provide an isomorphism between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injective linear maps from to .

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, Weak Pseudo-Riemannian Metric, Weak Riemannian Metric

This entry contributed by Christopher Stover

## Explore with Wolfram|Alpha

More things to try:

## References

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. http://www.math.ntnu.no/~stacey/documents/constructdirac.2up.pdf.

## Cite this as:

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