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A Note on Hyperconvexity in Riemannian Manifolds
Published online by Cambridge University Press: 20 November 2018
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Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.
By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) 
V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.
Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.
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- Research Article
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- Copyright © Canadian Mathematical Society 1959
References
1.
Whitehead, J. H. C., Convex regions in the geometry of paths, Quart. J. Math., Oxford, Ser. 8 (1932), 33–42.Google Scholar
2.
Michael, E. A., Convex structures and continuous selections.
Can. J. Math., 11 (1959), 556–575.Google Scholar
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