The flag curvature of a submanifold of a Randers–Minkowski space in terms of Zermelo data

Abstract

The main result of this paper is an expression of the flag curvature of a submanifold of a Randers–Minkowski space (V , F ) in terms of invariants related to its Zermelo data (h, W ). More precisely, these invariants are the sectional curvature and the second fundamental form of the positive definite scalar product h and some projections of the wind W. This expression allows for a promising characterization of submanifolds with scalar flag curvature in terms of Riemannian quantities, which, when a hypersurface is considered, seems quite approachable. As a consequence, we prove that any h-flat hypersurface S has scalar F-flag curvature and the metric of its Zermelo data is conformally flat. As a tool for making the computation, we previously reobtain the Gauss–Codazzi equations of a pseudo-Finsler submanifold using anisotropic calculus.