Hitting distributions of small geodesic spheres. (English) Zbl 0651.58037

The author generalizes to the n-dimensional case a result of M. A. Pinsky [Ann. Inst. Henri Poincaré, Probab. Stat. 21, 39-46 (1985; Zbl 0559.58026)] on the behaviour of the harmonic measure operator, to geodesic spheres with small radius, for a 2-dimensional Riemannian manifold. In the above cited paper Pinsky’s formula gives as corollary a characterization to surfaces with constant curvature. The analogous result of the author characterizes Einstein manifolds. Also a result is proved which gives us a sufficient condition for scalar curvature to be constant.


58J65 Diffusion processes and stochastic analysis on manifolds
53B20 Local Riemannian geometry
53C20 Global Riemannian geometry, including pinching


Zbl 0559.58026
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