Kinematics of Hypersurfaces in Riemannian Manifolds

In this work we study the kinematics of an m-dimensional continuum moving in an (m+1)-dimensional ambient space, modelled by the motion of a hypersurface M in a Riemannian manifold N. Focusing on the codimension of the continuum relative to the ambient space rather than on its dimension, we provide a unified framework for either curves moving on surfaces, surfaces moving in a 3-dimensional space (Euclidean or Riemannian), or hypersurfaces moving in a Riemannian space of arbitrary dimension. Further, the use of general geometric structures and a coordinate-free language facilitate the description of more general continua. We present formulae for the variation of geometric quantities of the hypersurface, some of which generalize those given for surfaces in Euclidean space (Kadianakis, N. in J. Elasticity 16:1–17, 2010). The main point in the present work is the use of kinematical quantities which result from a generalized version of the polar decomposition theorem for surfaces introduced by Man, C.-S. and Cohen, H. (in J. Elast. 16:97–104, 1986) and adapted here for hypersurfaces. We apply our results to motion along the normal and to pure strain motion. Finally, we discuss the relation of our results to Differential Geometry and Physics. Specifically, we provide an application for the case of a curve moving on a 2-dimensional manifold, which models a 1-dimensional continuum moving on a surface. The later application is presented independently of the embedding of the surface in a larger space.