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The volume of geodesic disks in a Riemannian manifold. (English) Zbl 0586.53006
Let M denote an analytic Riemannian manifold of dimension \(>2\). For every unit vector \(x\in T_ mM\) and sufficiently small radius r the authors call \[ D^ x_ m(r)=\exp_ m\{y\in T_ mM| \| y\| \leq r\quad and\quad y\perp x\} \] a geodesic disc. It is the purpose of the paper to provide complete formulas for the (n-1)-dimensional volume \(V^ x_ m(r)\) of \(D^ x_ m(r)\) if M is a two-point homogeneous space, and to show that these spaces are characterized by the functions \((m,x,r)\mapsto V^ x_ m(r)\).
Reviewer: H.Reckziegel

MSC:
53B20 Local Riemannian geometry
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References:
[1] B. Y. Chen, L. Vanhecke: Differential geometry of geodesic spheres. J. Reine Angew. Math. 325 (1981), 28-67. · Zbl 0503.53013
[2] A. Gray, L. Vanhecke: Riemannian geometry as determined by the volumes of small geodesic balls. Acta Math. 142 (1979), 157-198. · Zbl 0428.53017
[3] A. Gray, L. Vanhecke: The volumes of tubes about curves in a Riemannian manifold. Proc. London Math. Soc. 44 (1982), 215-243. · Zbl 0491.53035
[4] O. Kowalski, L. Vanhecke: Ball-homogeneous and disk-homogeneous Riemannian manifolds. Math. Z. 180 (1982), 429-444. · Zbl 0476.53023
[5] H. S. Ruse A. G. Walker, T. J. Willmore: Harmonic Spaces. Cremonese, Roma, 1961. · Zbl 0134.39202
[6] L. Vanhecke, T. J. Willmore: Interaction of tubes and spheres. Math. Ann. 263 (1983), 31-42. · Zbl 0491.53034
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