Abstract
A Riemannian manifold endowed with k>2 orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for k=2) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections.
Highlights
Distributions on a manifold appear in various situations and are used to build up notions of integrability, and of a foliated manifold, e.g., [1,2]
The best known result in this direction is the Decomposition theorem of de Rham, which states that “if each distribution Di is parallel with respect to the Levi-Civita connection of M, any point p ∈ M has a neighborhood U, which is isometric to a product M1 × . . . × Mk of Riemannian manifolds such that the submanifolds, which are parallel to the factor Mi, correspond to integral manifolds of the distribution Di|U
We introduce the mixed scalar curvature of (M, g, D1, . . . , Dk) with respect to a non-Levi-Civita linear connection and represent this kind of curvature using fundamental tensors of the distributions and the divergence of a geometrically interesting vector field
Summary
Distributions on a manifold (that is subbundles of the tangent bundle) appear in various situations and are used to build up notions of integrability, and of a foliated manifold, e.g., [1,2]. Dk) with respect to a non-Levi-Civita linear connection and represent this kind of curvature using fundamental tensors of the distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems (sometimes called Liouville type theorems, e.g., [12,13]) and integral formulas (when M is compact or a certain vector field is compactly supported on M) for some classes of almost multi-product manifolds. The mixed scalar curvature for two orthogonal complementary distributions (D, D⊥) on a Riemannian manifold (Mm, g) with a linear connection ∇ ̄ is defined in [11] by Mathematics 2021, 9, 229.
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