The metric tensor index at a point 
of a smooth manifold 
is the quadratic
form index of the metric tensor 
at 
, denoted
 |
Here, 
is the bilinear form on the tangent
space 
.
When 
is a constant function on 
, its value is denoted 
or 
(Sachs and Wu 1977, pp. 2 and 4).
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 metric tensor 
of metric signature 
when negative squares are listed first, or 
when positive squares
are listed first.
