Associated to any metric tensor is the quadratic form defined in each tangent space by

If is positive for all non-zero , then the metric is positive-definite at . If the metric is positive-definite at every , then is called a Riemannian metric. More generally, if the quadratic forms have constant signature independent of , then the signature of is this signature, and is called a pseudo-Riemannian metric. If is connected, then the signature of does not depend on .

By Sylvester's law of inertia, a basis of tangent vectors can be chosen locally so that the quadratic form diagonalizes in the following manner

for some between 1 and . Any two such expressions of (at the same point of ) will have the same number of positive signs. The signature of is the pair of integers , signifying that there are positive signs and negative signs in any such expression. Equivalently, the metric has signature if the matrix of the metric has positive and negative eigenvalues.

Certain metric signatures which arise frequently in applications are:

স্থানাঙ্ক রূপান্তরের অধীনে পরিধির চাপের ধ্রুবক
Signature of a metric
Components of the metric
The round metric on a sphere