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

for all . Here, the notation denotes the quadratic form index associated with .

The index of a metric tensor provides an alternative tool by which to define a number of various notions typically associated to the signature of . For example, a Lorentzian manifold can be defined as a pair for which and for which , a definition equivalent to its more typical definition as a manifold of dimension no less than two equipped with a tensor of metric signature (or, equivalently, ).