The index 
associated to a symmetric, non-degenerate, and bilinear 
over a finite-dimensional vector
space 
is a nonnegative integer defined by
 |
where the set 
is defined to be
 |
As a concrete example, a pair 
consisting of a smooth
manifold 
with a symmetric 
tensor field 
is said to be a Lorentzian manifold if and
only if 
and the index 
associated to the quadratic form 
satisfies 
for all 
(Sachs and Wu 1977). This particular definition succinctly
conveys the fact that Lorentzian manifolds have indefinite metric
tensors of signature 
(or equivalently 
) without having to make precise any definitions related
to metric signatures, quadratic form signatures,
etc.
The above example also illustrates the deep connection between the index of a quadratic form and the notion of the index of a metric
tensor 
defined on a smooth manifold 
. In particular, the index of a metric tensor 
is defined to be the quadratic form index associated to 
for any element 
. Because of this connection, indices are especially significant
to various fields: In some literature on differential and Riemannian geometry, for
example, the notion of an index is used as a main tool by which to define
a metric tensor (Sachs and Wu 1977).
