Finsler metrics and the degree of symmetry of a closed manifold (Q2898371)

Let \(M\) be a connected compact smooth manifold. The degree of symmetry of \(M\) is the maximum of the dimensions of the groups of isometries of all possible Riemannian metrics on \(M\). The degree of symmetry is an important topological invariant of closed manifolds. If \(\dim (M)=n\), then the degree of symmetry \(N(M)\;\mathrm{is} \leq \frac12n(n+1)\). Equality holds if and only if \(M\) admits Riemannian metrics of positive constant sectional curvature, that is, \(M\) is diffeomorphic to the sphere \(S^n\) or the real projective space \(\mathbb RP^n\).NEWLINENEWLINEIn this paper, the author studies the relationship between Finsler metrics and the degree of symmetry of a closed manifold. If \(F\) is a Finsler metric on \(M\), then the group of transformations \(I(M,F)\) of \(M\) is a Lie transformation group. The author proves that the degree of symmetry \(N(M)\) of a compact connected manifold \(M\) is the maximum of the dimensions of the group of isometries of possible Finsler metrics on \(M\). Also, if \(M\) is not diffeomorphic to a compact rank-one Riemannian symmetric space, then there exists a non-Riemannian Finsler metric \(F\) on \(M\) such that its group of isometries \(I(M,F)\) has dimension \(N(M)\).