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 $(\mathcal{V},F)$ 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.

arXiv comment30 pages, v2: we added some references, a missing $\varphi (v)$ in (66) and (67) and a missing square in the last line of (68). As a consequence, the expression of Corollary 20 is now simpler