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To state the context, I am familiar with the Hopf-Rinow theorem.

My request is three fold,

  • I would like to know of general classes of geodesically incomplete spaces. I basically want to see lots of examples for this.

  • I want to know of techniques of proving and testing for geodesic incompleteness or completeness.

  • I want to know of methods of proving and testing for existence of maximal extension of curves.

{Confusingly in quite a bit of literature I have seen the adjective "inextensible" being used when actually they seem to want to mean "maximally extended". Is there some subtle point here that I am missing?}

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    I have been told (but don't know how to prove) that every noncompact surface admits a metric of positive sectional curvature. However, the only noncompact surface admitting a complete metric of positive curvature is $\mathbb{R}^2$. Hence, most of the above metrics will be incomplete.2010-12-09
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    For a silly example, take a proper open subset of a complete connected Riemannian manifold. By Hopf-Rinow, this can't be geodesically complete.2010-12-09
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    I think that @Akhil's example is a very good one to think about.2010-12-09
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    @Akhil Thanks for this simple class of examples (should have thought of that!)2010-12-10
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    @Jason Apologies for the rudimentary state of my understanding. Can you kindly explain what is the connection between positive sectional curvature and metric incompleteness? (If you can give a reference etc)2010-12-10
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    @Anirbit - I'll try to find some references for the first thing I said - a professor just said something like that to me in passing conversation. The second comes from the Soul Theorem and Perelman's proof of the Soul Conjecture. I was trying to indicate a way to test for incompleteness of a metric on a noncompact surface. If your noncompact surface is not diffeomorphic to $\mathbb{R}^n$, and if for some metric every point on your surface has positive curvature, then the metric on it must be incomplete.2010-12-10

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