An integral formula for a Riemannian manifold with \(k > 2\) singular distributions
The so-called Integral Formula for foliated manifolds was first studied by G. Reeb since 1950. For any codimension one foliation on a closed Riemannian manifold, the first known integral formula of G. Reeb shows that the integral mean curvature of the leaves vanishes [\textit{G. Reeb}, C. R. Acad. Sci., Paris 231, 101--102 (1950; Zbl 0040.24401)]. In 2014, \textit{K. Andrzejewski} et al. studied the second integral formula for a codimension-one foliation with a unit normal \(N\) to the leaves [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)]. In the paper [\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 et al. also showed that, there exists a series of integral formulas for codimension-one foliations and these integral formulas were obtained by applying the divergence theorem to suitable vector fields. \textit{P. Popescu} and the author studied an new integral formula for two singular orthogonal distributions which has published in [Result. Math. 75, No. 1, Paper No. 18, 19 p. (2020; Zbl 1442.53023)]. In early 2021, the author continued to research the integral formulas for a Riemannian manifold \((M, g)\) with \(k>2\) regular pairwise orthogonal complementary distributions which was published in [the author, ``Integral formulas for a Riemannian manifold with several orthogonal complementary distributions'', Global J. Adv. Res. 10, 32--42 (2021)]. The purpose of this paper is a continuation of the [Zbl 1442.53023; the author, ``Integral formulas for a Riemannian manifold with several orthogonal complementary distributions'', Global J. Adv. Res. 10, 32--42 (2021)] for the case of \(k > 2\) singular pairwise orthogonal distributions which are not necessarily generating the tangent bundle \(TM\) of the given Riemannian manifold \((M, g)\). Excluding Section 1 in which the history of the research problem is introduced, the rest of the paper consists of 2 sections. In Section 2, the author introduces some notions defined on a Riemannian manifold \((M, g)\) such as the allowed system of \(k \geq 2\) endomorphisms \((P_1, \ldots, P_k)\) of \(TM\) on \((M, g)\), the mixed scalar \(P\)-curvature as well as the \(P\)-divergence of a vector field on \((M, g)\). All these notions are used in the last part. Section 3 is devoted to stating and proving the main results of the paper. The first result for a closed Riemannian \((M, g)\) is Theorem 13, a new integral formula involving the mixed scalar \(P\)-curvature. In fact, this integral formula is generalized from Theorem 3.6 of the paper [Zbl 1442.53023]. Theorem 16 is the last result of the paper in which the author characterizes a sufficient condition about \(P\)-autoparallel distributions which implies splitting of manifolds in the regular case. For the entire collection see [Zbl 1468.53002].
scientific article; zbMATH DE number 7391065