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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?

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    $\begingroup$ 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. $\endgroup$ Dec 31, 2018 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, 2019 at 13:54
  • $\begingroup$ @RobertBryant Robert, in the case of Lie groups, is there a property that explains the lack of surjectivity of the exponential? $\endgroup$ Jun 17, 2019 at 23:40

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