Riemannian vs. semi-Riemannian manifolds

Question

Some results on Riemannian manifolds are valid on semi-Riemannian manifolds and the others are not. Sometimes it takes too long to recognize between them.

My question is: Is there a reference gathering the (in)valid such results?

If no, are there any hints to help? I know the answer is not easy so I want to be clear here. What I am looking for is an alarm. So, comments or answers such that:

1. Results concerning(depending on,....) .... are usually valid.
2. You have to check results concerning(depending on,....) ....

are helpful. Thanks in advance.

Answer 1

Obviously the semi-riemannian manifolds constitute a larger more general class of manifolds; and I don't know of a reference that concerns itself specifically and extensively with the differences. Of course this doesn't mean such a reference doesn't exist. (Even though my former advisor wrote a GTM riemannian geometry, I failed to become an expert on the subject and didn't complete my phd ). But I'd say riemannian manifolds seem to get the lion share of the attention, and it is often just noted in passing that certain results that hold for one also hold for the other . .. I did learn that riemannian geometry is a difficult subtopic of differential geometry, about the latter of which gauss described the theorems as being "few but ripe". He was apparently indicating that there aren't that many results "out there ", but that the ones there are are good ones... I would imagine there would be even fewer results in the semi-riemannian category , since it's larger (but I could be wrong ) . That is, if $(M,g)$ is riemannian, then it is semi-riemannian. But not vice versa. So $R\subsetneq R'$, where $R $ and $R'$ are the riemannian and semi-riemannian categories respectively...

Answer 2

Barrett O'Neill's Semi-Riemannian Geometry is a good reference, in the (unlikely?) event you don't already know the book.

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To me, three of the "most surprising" properties of a semi-Riemannian manifold $(M, g)$ with indefinite metric are:

1. There exist non-zero tangent vectors $v$ such that $g(v, v) = 0$. Consequently, although every subspace $W$ of a tangent space $T_{p}M$ has an orthogonal complement $W^{\perp}$, it's not true that $T_{p} M = W \oplus W^{\perp}$.

2. There exist tangent vectors $v$ such that $g(v, v) < 0$. Consequently, arc length as measured by $g$ does not define a topological metric (i.e., a distance function) on $M$.

3. The unit sphere in $T_{p}M$, i.e., the set of $v$ in $T_{p}M$ such that $g(v, v) = 1$, is non-compact. This is where analysis on an indefinite manifold really breaks down, e.g., a sequence of unit vectors at a point need not have a convergence subsequence, and a smooth semi-Riemannian metric on a compact manifold need not be geodesically complete.

At risk of over-generalizing, results using only linear algebra of the metric have a chance of being true, while those involving analysis (including geodesic equations) need more careful examination.

Answer 3

As a rule of thumb, no analogy between Riemannian and semi-Riemannian should be taken for granted beyond definitions of affine connections and the curvature tensors (full, Ricci and scalar). Already sectional curvature is not well defined, length of curves are not defined (but energy is).