Hsu, Pei Probabilistic methods in differential geometry. (English) Zbl 0698.58051 Stochastic processes, Semin., San Diego/CA (USA) 1989, Prog. Probab. 18, 123-134 (1990). [For the entire collection see Zbl 0686.00017.] Let M be a Riemannian manifold with metric g. The metric g determines a second order elliptic operator called the Laplace-Beltrami operator, which locally has the representation \[ \Delta =\frac{1}{\sqrt{\det g}}\frac{\partial}{\partial x^ i}(\sqrt{\det g}g^{ij}\frac{\partial}{\partial x^ j}). \] The process associated with half of the Laplace-Beltrami operator is called the Riemannian Brownian motion on M. The author studies the geometric properties of M by investigating the behavior of the Riemannian Brownian motion, because for some geometric problems the probabilistic approach is nicer and clearer than the analytic approach. Reviewer: S.Eloshvili MSC: 58J65 Diffusion processes and stochastic analysis on manifolds 53C20 Global Riemannian geometry, including pinching 60J65 Brownian motion 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:Riemannian manifold; Laplace-Beltrami operator; Riemannian Brownian motion Citations:Zbl 0686.00017 PDFBibTeX XML