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Semi-Riemannian Manifold

A smooth manifold M=(M,g) is said to be semi-Riemannian if the indexMetric Tensor Index of g is nonzero. Alternatively, a smooth manifold is semi-Riemannian provided that it comes equipped with a semi-Riemannian metric.

In nearly all literature, the term semi-Riemannian is used synonymously with the term pseudo-Riemannian and is used to describe manifolds whose metric tensor g fails to be positive definite. Alternatively, a manifold is semi-Riemannian (or pseudo-Riemannian) if its infinitesimal distance (ds)^2 is equivalent to that of a pseudo-Euclidean space of signature (p,q) for q!=0, i.e., if

(ds)^2=sum_(j=1)^p(dx^j)^2-sum_(j=p+1)^n(dx^j)^2

with the rightmost summand nonzero (Snygg 2012).

Lorentzian manifolds are examples of semi-Riemannian manifolds.

See also

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

This entry contributed by Christopher Stover

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References

Sachs, R. K. and Wu, H. General Relativity for Mathematicians. New York: Springer-Verlag, 1977.Snygg, J. A New Approach to Differential Geometry using Clifford's Geometric Algebra. New York: Springer Science+Business Media, 2012.

Cite this as:

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

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