GENERALIZED EINSTEIN MANIFOLDS

A Finslerian manifold is called a generalized Einstein manifold (GEM) if the Ricci directional curvature R(u,u) is independent of the direct ion. Let F-0(M,g(t)) be a deformation of a compact n-dimensional Finsl erian manifold preserving the volume of the unitary fibre bundle W(M). We prove that the critical points g(0) is an element of F-0(g(t)) of the integral I(g(t)) on W(M) of the Finslerian scalar curvature (and c ertain functions of the scalar curvature) define a GEM. We give an est imate of the eigenvalues of Laplacian Delta defined on W(M) operating on the functions coming from the base when (M, g) is of minima fibrati on with a constant scalar curvature (H) over tilde admitting a conform al infinitesimal deformation (CID). We obtain lambda greater than or e qual to (H) over tilde/(n - 1)(Delta f-=lambda f). If M is simply conn ected and lambda = (H) over tilde(n - 1), then (M, g) is Riemannian an d is isometric to an n-sphere. We first calculate, in the general case , the formula of the second variationals of the integral I(g(t)) for g = g(0), then for a CID we show that for certain Finslerian manifolds, I ''(g(0)) > 0. Applications to the gravitation and electromagnetism in general relativity are given. We prove that the spaces characterizi ng Einstein-Maxwell equations are GEMs.