Properties that only Lorentzian manifolds have

Question

I wonder what are some statements that although they can be formulated for pseudoriemannian manifolds of arbitrary signature they turn out to be true only in the lorentzian case.

I admit things like: "no analog to cauchy hypersuface can be defined in arbitrary pseudoriemanniann manifolds, just in lorentzian ones". I don't know if this last statement is true, it's just to exemplify.

Another one could be: the equivalent topological condition for a given compact manifold to admit a pseudoriemannian metric of a given signature is to have 0 euler characteristic. I think this is true only for lorentzian manifolds.

Thanks in advance!

Answer 1

A pseudo-Riemannian manifold of signature $(p,q)$ with $p, q\geq 2$ certainly does not admit Cauchy hypersurfaces, since spacelike submanifolds have dimension at most $p$. However, you could define a notion of "Cauchy $p$-surface" as a spacelike submanifold of dimension $p$ intersecting any inextensible timelike $q$-disc at one point. Though it is not explicitly stated, recent progresses in the study of discrete group actions on pseudo-hyperbolic spaces tend to construct higher signature analogs of globally hyperbolic Cauchy compact anti-de Sitter spacetimes.

Concerning the Euler characteristic: A manifold $M$ admits a pseudo-Riemannian metric of signature $(p,q)$ if and only if its tangent bundle $TM$ admits a splitting as $E^{(p)} \oplus F^{(q)}$. This implies the vanishing of the (rational) Euler characteristic when $p$ or $q$ is odd.

More generally, the specificities of Lorentzian geometry should really come from the fact that the "timelike dimension" is $1$. For instance, there cannot be a distinction between "past" and "future", hence no notion of causality in higher signature.

Answer 2

The reverse Cauchy-Schwarz inequality: for any two causal vectors (orientation in this `squared' formulation is not relevant) $v, w$ we have $$ \vert g(v,w)\vert^2\ge g(v,v) g(w,w) $$ is false in pseudo-Riemannian spaces of signature $(p,q)$, $p,q\ge 2$. The proof uses a certain convexity of the unit sphere (not its compactness), and this property holds only for the Lorentzian signature.