Pseudo—Finsler Hypersurfaces

Abstract

In the first three sections of the present chapter we apply the general theory of pseudo—Finsler submanifolds developed in Chapters 2 and 3 to the particular case of pseudo—Finsler hypersurfaces. In this setting we present concepts and results owed to Berwald [3], Brown [1], [2], Matsumoto [4], Rund [1]–[5] and Varga [1], [2]. Also, we introduce the induced horizontal flag curvature of a pseudo—Finsler hypersurface ${\mathbb{F}^m}$ and compare it with both the intrinsic horizontal flag curvature of ${\mathbb{F}^m}$ and the horizontal flag curvature of the ambient manifold ${\widetilde F^{m + 1}}$. In particular, when ${\widetilde F^{m + 1}}$ is a pseudo—Minkowski space we stress the role of the structure equations induced by the Rund connection of ${\widetilde F^{m + 1}}$. In the last section of the chapter we present a Minkowskian approach to the theory of pseudo—Finsler hypersurfaces. Being based on a Minkowskian unit normal vector field depending on position only, the theory we develop here is completely different from what we presented till now throughout the book.