Natural volume forms on pseudo-Finslerian manifolds with $m$-th root metrics

Abstract

We define natural volume forms on $n$-dimensional oriented pseudo-Finslerian manifolds with non-degenerate $m$-th root metrics. Our definitions of the natural volume forms depend on the parity of the positive integer $m>1$. The advantage of the stated definitions is their algebraic structure. The natural volume forms are expressed in terms of Cayley hyperdeterminants. In particular, the computation of the natural volume form does not require the difficult integration over the domain within the indicatrix in the tangent space $T_x M^n$ of the pseudo-Finslerian manifold at a point $x$.