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, 
).
