On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, let $f\colon(M,g)\to(N,g')$ be a projective submersion of an $m$-dimensional Riemannian manifold $(M,g)$ onto an $(m-1)$-dimensional Riemannian manifold $(N,g')$. Then $(M,g)$ is locally the Riemannian product of the sheets of two integrable distributions $\operatorname{Ker}f_*$ and $(\operatorname{Ker}f_*)^\bot$ whenever $(M,g)$ is one of the two following types: (a) a complete manifold with $\operatorname{Ric}\geqslant0$ (b) a compact oriented manifold with $\operatorname{Ric}\leqslant0$.