The Integral Of Mixed Scalar Curvature Along A Leaf Of Foliation

. By using the method of matrix Riccati ODE, we study Riemannian manifolds with foliations (distributions), whose mixed Ricci or scalar curvature is nonnegative and the norms of integrability tensors are bounded above. We obtain the integral formula for mixed scalar curvature along the complete leaf, rigidity of totally geodesic distribution, splitting of foliation with "large" dimension. 1. Main results For two complementary orthogonal distributions T 1 and T 2 on TM we define the structural tensors B 1 : T 2 \Theta T 1 ! T 1 ; B 2 : T 1 \Theta T 2 ! T 2 by formulae, where P i : TM ! T i are orthoprojectors B 1 (y; x) := P 1 (r x y); B 2 (x; y) := P 2 (r y x): (1) Obviously, the equation B i = 0 means that T i is tangent to totally geodesic foliation. The equalities B 1 = B 2 = 0 mean by de Rham decomposition theorem that M is locally a product L 1 \Theta L 2 , where TL 1 = T 1 ; TL 2 = T 2 . Tensors (1) (as well as Gray's tensors O; T [Gra2] or O'Neill's tensors A; T [O'Ne]) cont..