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Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. What about geodesics or the exponential map in points away from the boundary? I am actually intrested in the description of the injectivity radius if there is one. Do you know a good reference? Hope for answers.

Greetings, Phillip

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You might want to have a look at the following paper and its references:

MR1226829 (94c:53053) Alexander, Stephanie B.(1-IL); Berg, I. David(1-IL); Bishop, Richard L.(1-IL) Cut loci, minimizers, and wavefronts in Riemannian manifolds with boundary. Michigan Math. J. 40 (1993), no. 2, 229–237. 53C22 (53C20)

This has some basic information about geodesics on manifolds with boundary that you might find good for starting out in finding out what is known on these questions.

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