Volume comparison and its applications in Riemann-Finsler geometry
In Riemannian geometry, every tangent space is endowed with an inner product, and the resulting inner product spaces are all isometric to the Euclidean model. In Finsler geometry, each tangent space gets a Minkowski norm. The resulting Minkowski spaces are typically non-isometric to each other. The geometry of a Finsler structure can be captured by two intrinsic curvatures. One is the flag curvature, which measures how the Minkowski spaces are fitted together. The other is a new quantity introduced by the author, called the mean-covariation. It measures how the non-isometric Minkowski models are changing. The mean-covariation turns out to be a key ingredient if one wants to generalize the Bishop-Gromov volume comparison theorem to Finsler geometry. In this paper, the author carries that generalization out, and gives some applications to topology.
scientific article; zbMATH DE number 1044182
scientific article