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


53B20 Local Riemannian geometry
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