Foliated manifolds and closed manifolds of non-positive sectional curvature

Let ( X , g T X ) (X,gTX) be a closed Riemannian manifold of dimension n n , and let F F be an integrable subbundle of T X TX . Let k F k^F be the leafwise scalar curvature associated to g F = g T X | F gF=gTX|F . Let Y Y be a closed Riemannian manifold of non-positive sectional curvature. We show that if either T X TX or F F is spin, and there is a smooth map g : X → Y g:X→ Y so that g ∗ [ X ] ≠ 0 g[X]≠ 0 in H n ( Y ; Q ) Hₙ(Y;Q) , then k F ≤ 0 kF≤ 0 . It extends the result of Gromov-Lawson [Inst. Hautes Études Sci. Publ. Math. 58 (1983), pp. 83–196] that a manifold which represents a non-trivial homology class in a compact K ( π , 1 ) K(π ,1) -manifold of non-positive sectional curvature, should not carry positive scalar curvature, to the case of foliations.