Geometric Structures as Determined by the Volume of Generalized Geodesic Balls

Several authors have studied the Taylor expansion for the volume of geodesic balls under the exponential mapping of an analytic Riemannian manifold $ (M, {\cal G}) $ . A more general structure $ (M, D{\cal G}) $ , where D is a torsion-free and Ricci-symmetric connection, appears in several geometric situations. We study the Taylor expansion in this case, where all metric notions are Riemannian, while now the exponential mapping is induced from the connection D. We give many applications, in particular in different hypersurface theories.