Integral formulas for a Riemannian manifold with orthogonal distributions
In this article, the author studies distributions on Riemannian manifolds which are represented as the sums of multiple pairwise orthogonal distributions. By a theorem of Reeb, the integral of the mean curvature of a codimension-one foliation on a compact Riemannian manifold is zero. There are many works subsequent to Reeb's formula for foliations, distributions and more generally for multiple distributions [\textit{K. Andrzejewski} et al., in: Geometry and its applications. Selected papers based on the presentations at the 2nd international workshop on geometry and symbolic computation, Haifa, Israel, May 15--18, 2013. Cham: Springer. 73--82 (2014; Zbl 1344.53020)]. The author defines the mixed scalar curvature for multiple pairwise orthogonal distributions, which generalizes the one defined by himself in [\textit{V. Rovenski}, ``Integral formulas for a Riemannian manifold with several orthogonal complementary distributions'', Glob. J. Adv Res. Class. Mod. Geom. 10, No. 1, 32--42 (2021)] for the case where the sum is the whole tangent bundle of the ambient manifold. Based on this notion, the author obtained several integral formulas for curvatures of multiple pairwise orthogonal distributions, which are applied to various examples.
scientific article; zbMATH DE number 7478459
scientific article