On a Metric Affine Manifold with Several Orthogonal Complementary Distributions

Vladimir Rovenski and Sergey E. Stepanov
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Vladimir Rovenski: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel
Sergey E. Stepanov: Department of Mathematics, Finance University, 49-55, Leningradsky Prospect, 125468 Moscow, Russia

Mathematics, 2021, vol. 9, issue 3, 1-11

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.

Keywords: almost multi-product structure; mixed scalar curvature; integral formula; statistical connection; semi-symmetric connection; multiply twisted product; splitting (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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