In this paper, we prove that if every totally real bisectional curvature of an n(≥ 3)-dimensional complete Kähler submanifold of a complex projective space of constant holomorphic sectional curvature c is greater than c 4(n2−1)n(2n− 1), then it is totally geodesic. Mathematics Subject Classifications: 53C50, 53C55, 53C56.

Abstract Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to leaves. Our proof uses hypoelliptic sub-Laplacian on forms recently introduced by two authors in Baudoin and Grong (Ann Glob Anal Geom 56(2):403–428, 2019).

For $$n \ge 2$$ , we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for manifolds. As in the case, primary result superrigidity theorem certain representations lattices. The proof requires developing new general tools not needed...