LOCALLY SYMMETRICAL FINSLER-SPACES IN NEGATIVE CURVATURE

E. Cartan introduced in 1926 the Riemannian locally symmetric spaces, as the spaces whose curvature tensor is parallel. They also owe their name to the fact that, for each point, the geodesic reflexion is a loc al isometry. The aim of this Note is to announce a strong rigidity res ult for Finsler spaces. Namely, we show that a negatively curved local ly symmetric (in the first sense above) Finsler space is isometric to a Riemann locally symmetric space.