Jacobi fields on manifold with random curvature

Contemporary geometry presents a lot of results concerning 2D manifolds and surfaces with positive and negative curvatures however it remains unclear how to find interesting general properties for manifolds and surfaces with sign-indefinite curvature. An instructive idea for that was suggested in 1960'th by Ya.B.Zeldovich in context of a cosmological problem. In terms of Riemannian geometry, he paid attention that if Gaussian curvature along a geodesic line can be considered as a random process with zero mean then Jacobi field along this geodesic line grow exponentially as if effective mean of curvature would be negative. This geometric interpretation was formulated and elaborated about 10 years ago by V.G.Lamburt, E.R.Rozendorn, D.D.Sokoloff and V.N.Tutubalin. The point however is that a Riemannian metric can be prolongated from such a geodesic line only to a very narrow strip integral curvature of which remains finite. We demonstrate how to improve the concept in order to consider Zeldovich's effect in a whole Riemannian manifold with random curvature.