An Integral Formula for Singular Distributions

The work is devoted to the differential geometry of singular distributions (i.e., of varying dimension) on a Riemannian manifold. We define such distributions as images of the tangent bundle with smooth endomorphisms. We introduce an operator of the divergence type and prove a new divergence theorem. Then we derive a Codazzi type equation for a pair of transverse singular distributions, generalizing the known one for an almost product structure. Tracing our Codazzi type equation gives an expression of a modified divergence of the sum of mean curvature vectors in terms of invariants of distributions including the modified mixed scalar curvature, which allows us to obtain some splitting results. Applying our divergence type theorem to the above expression with a modified mixed scalar curvature, we obtain an integral formula for a pair of transversal singular distributions; the formula generalizes the well-known one with many applications.