Metric Tensor Index

The index associated to a metric tensor g on a smooth manifold M is a nonnegative integer I for which


for all x in M. Here, the notation index(gx) denotes the quadratic form index associated with gx.

The index I=I_g of a metric tensor g provides an alternative tool by which to define a number of various notions typically associated to the signature (p,q) of g. For example, a Lorentzian manifold can be defined as a pair (M,g) for which dimM>=2 and for which I=1, a definition equivalent to its more typical definition as a manifold M of dimension no less than two equipped with a tensor g of metric signature (1,dim(M)-1) (or, equivalently, (dim(M)-1,1)).

See also

Lorentzian Manifold, Metric Signature, Metric Tensor, Quadratic Form Index, Smooth Manifold

This entry contributed by Christopher Stover

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Sachs, R. K. and Wu, H. General Relativity for Mathematicians. New York: Springer-Verlag, 1977.

Cite this as:

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

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