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

TL;DR

This paper derives an explicit formula for the flag curvature of submanifolds in Randers-Minkowski spaces using Zermelo data, linking Finsler and Riemannian invariants to characterize scalar flag curvature.

Contribution

It provides a new expression for flag curvature in terms of Riemannian invariants and proves properties of flat hypersurfaces in Randers-Minkowski spaces.

Findings

Flag curvature expressed via Zermelo data invariants.

Any flat hypersurface has scalar flag curvature.

The metric of Zermelo data is conformally flat.

Abstract

The main result of this paper is an expression of the flag curvature of a submanifold of a Randers-Minkowski space $({\mathscr V},F)$ 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.