Kinematics of submanifolds and the mean curvature normal
This article provides a rigorous and interesting generalization of a transport theorem relative to the motion of a submanifold embedded in a semi-Riemannian manifold of finite dimension. This theorem, which is classical in fluid mechanics, then presents a more general and unified aspect, the submanifold being composed of discrete points, lines, or other surfaces. In that general case one has to consider the normal space at any point of the submanifold and a generalization of the mean curvature normal mapping of the case of two-dimensional surfaces embedded in the three-dimensional Euclidean space. The transport theorem is then written very simply by means of a generalization of the Riemannian divergence of a vector. Interesting complements and illustrations are given in the appendix. One may however express two criticisms concerning this work for which further applications are announced by the author. First it seems that the proof could be made more directly using a more intrinsic formalism of differential geometry. On the other hand, the proofs and results would profit from being worked out in the more general formalism of cohomology and homology, namely by using ''co-edges'' and ''edges'', in particular. Moreover, the application of de Rham's theorem for non-simply-connected domains (which are not considered in this paper) would give an interesting extension for some important applications in mathematical physics.
scientific article; zbMATH DE number 3996537
scientific article