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

\begin{equation} \exp : \text{T}_p M \to M \end{equation}

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

  • 3
    $\begingroup$ 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. $\endgroup$ – Robert Bryant Dec 31 '18 at 17:56
  • $\begingroup$ 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. $\endgroup$ – iolo Jan 3 at 13:54

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.