Chapter 19 Semi‐Riemannian Manifolds
Looking back at the material presented on smooth manifolds and smooth manifolds with a connection, we see that many of the concepts and results motivated by our study of regular surfaces and regular surfaces in 
have been recovered—but not all. Missing from our recent efforts has been any discussion of “length” and “area”. To address this shortfall, it is necessary to endow smooth manifolds with additional structure, which entails a consideration of semi‐Riemannian manifolds.
19.1 Semi‐Riemannian Manifolds
The following material builds on portions of the discussion in Section 12.2. Let
M
be a smooth
m
‐manifold, and let 
be a symmetric tensor field in 
, so that 
is a symmetric tensor in 
for all
p
in
M
. We say that 
is a metric on
M
if:
[G1] is nondegenerate on T p (M) for all p in M .
[G2] ind is independent of p
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