Geometry of generalized Einstein manifolds

A formula linking the horizontal Laplacian $\overline{\mathrm{\Delta }}\phi$ of a function φ on the fibre bundle W of unitary tangent vectors to a Finslerian compact manifold without boundary $(M\text{,}g)$, to the square of a symmetric 2-tensor and Finslerian curvature. From it an estimate, under a certain condition, is obtained for the function $\lambda \text{:}\overline{\mathrm{\Delta }}\phi =\lambda \phi$. If $\lambda =nk$ where k is a positive constant and M simply connected, then M is homeomorphic to an n-sphere. Let $F^{○}(g_t)$ be a deformation of $(M\text{,}g)$ preserving the volume of W. One proves that the critical points $g_0\in F^{○}(g_t)$ of the integral $I(g_t)$ of a certain Finslerian scalar curvature on W define a generalized Einstein manifold. One calculates the second variationals at the critical points first in the general case, then, for an infinitesimal conformal deformation and one shows that in certain cases one has $I^{″}(g_0)?0$. We also study the case when the scalar curvature is non-positive constant. To cite this article: H. Akbar-Zadeh, C. R. Acad. Sci. Paris, Ser. I 339 (2004).