Im trying to get some intuition for convex neighbourhoods which is neighbourhoods ##U## such that for any two points ##p## and ##q## in U there exists a unique geodesic connecting ##p## and ##q## staying within ##U##.(adsbygoogle = window.adsbygoogle || []).push({});

QUESTION 1: These kind of neighbourhoods can be shown to always exist for Riemannian manifolds something i find a bit puzzling. Can someone offer some intuition on why such neighbourhoods must always exist?

QUESTION 2: Furhermore -- and more technical -- are their existence fundamentally related to the exponential map in the following way?

For any Riemannian manifold there always exists neighbourhoods ##U## of ##p## and ##V## of the origin of ##T_pM## such that ##\exp_p: V \to U## is a diffeomorphism (it is smoothly invertible). This means that for a point ##q## in ##U##, we are able to get to the unique tangent vector ##v \in T_pM## with the property that the unique geodesic whose tangent vector at ##p## is ##v## satisfies ##\gamma_v(1) =q##. In other words, there must exist a unique geodesic going through ##p## and ##q## -- for were it not unique, then we would not be able to pick out a unique vector ##v## with the above property and thus ##\exp_p## would not be a diffeomorphism within this region.

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# Intuition and existence for convex neighbourhoods

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