Volume forms in Finsler spaces (Q2716459)

Let \((M^n,F)\) be a Finsler \(n\)-manifold. In the coordinate system \((x^i,X^j=\partial/ \partial x^j)\) for some neighborhood in \(M^n\), let \(I_x= \{X\in T_xM^n|F(x)\leq 1\}\) be the unit indicatrix in \(T_xM^n\). Define the symmetric, positive definite, twice contravariant tensor \(K^{ij}(x)= (n+1) \int_I X^iX^j dX/ \int_I dX\), \(dX=dX^1 dX^2\dots dX^n\), called the osculating Riemannian metric for \((M^n,F)\).NEWLINENEWLINENEWLINELet \(k(x)dx\) be the volume form arising from \(K^{ij}\) and \(\omega(x)dx\) the Busemann volume form: \(\omega(x)= \kappa_n/ \int_I dX\) where \(\kappa_n\) is the volume of the unit sphere in \(\mathbb{R}^n\) [\textit{H. Buseman}, Ann. Math. (2) 48, 234-267 (1947; Zbl 0029.35301)]. Then \(\omega(x)= k(x)\) if and only if \((M^n,F)\) actually is Riemannian with metric \(K^{ij}(x)\). In particular, let \((M^n,F)\) be a Berwald manifold with the volume invariant \(\nu(x)=k(x)/ \omega(x)\). Then \(\nu(x)\) is constant.