Let $(M,g)$ is a Riemannian manifold. (1)If $D_p$ is the largest domain on which $\exp_p$ can be a diffeomorphism, then is the set $$D=\bigsqcup_{p\in M} D_p$$ open in $TM$? (2)Likewise, if we denote $r_p$ the injectivity radius at $p\in M$, then is the set $$D_1=\bigsqcup_{p\in M} \{v\in T_pM: |v|_g< r_p\}$$ open in $TM$?
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Is the function $p \to r_p$ continuous? – Asaf Shachar Apr 24 '16 at 17:15
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Sorry, I don't know. But, I think this is also a good question. – Hang Apr 25 '16 at 12:55
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It's not obvious to me how $D_p$ is defined, because I'm not sure how to make precise the notion of "largest" among open subsets of $T_pM$ on which $\exp_p$ is a diffeomorphism. – mollyerin Apr 26 '16 at 19:52
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@mollyerin Thanks. $D_p$should be $seg^o(p)$ i.e. the segment domain at $p$ minus the cut locus(cf. Petersen, Riemmanian Geometry(GTM 171),pp.139-141). I think the Lemma 15 in this reference shows that it is the largest open set we desire. – Hang May 01 '16 at 15:43
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@Henry This is at least a precise definition of $D_p$, so I will think some about your question (1). (I am still not happy with the word "largest" here; I'll remark there can certainly be open domains of $T_pM$ on which $\exp_p$ is a diffeomorphism that are not contained within $seg^o(p)$. Picture the circle for easy examples.) – mollyerin May 03 '16 at 07:30
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If $M$ is complete, then $D_1$ is open, which follows essentially from the fact that $p \mapsto r_p$ is continuous. For instance, consider the map $f : TM \to TM$ given by $$ f : v \mapsto \frac{v}{r_p}, $$ where $p$ is the footpoint of $v$; $f$ is continuous, and your $D_1$ is the inverse image under $f$ of the open set $$ \lbrace v \in TM : ||v|| < 1 \rbrace. $$
