# Surjectivity of Pseudo-Riemannian exponential map on geodesically complete manifolds

Suppose one has a geodesically complete pseudo-Riemannian manifold $$M$$ i.e. the exponential map is defined for all tangent vectors on the manifold. Can one make a sensible statement about whether (or when) the exponential map restricted to some arbitrary tangent space

$$$$\exp : \text{T}_p M \to M$$$$

for some $$p \in M$$ will be surjective?

• Maybe you could be a bit more specific about what kind of 'sensible statement' you're looking for. It's known not to be surjective in every case. A well-known example in which surjectivity fails is the exponential map on $\mathrm{SL}(n,\mathbb{R})$ endowed with its bi-invariant pseudo-Riemannian metric. Of course, this is a problem of great interest in relativity, where there are many known cases of failure of surjectivity, each having some physics interpretation in terms of gravitational fields and signaling. – Robert Bryant Dec 31 '18 at 17:56
• Yes, you are right. This question is too broad. I will need to do some more research before I can get more specific about the problem in question. Thanks anyway. – iolo Jan 3 at 13:54