On isometric immersions of almost $k$-product manifolds

TL;DR

This paper establishes an optimal inequality relating intrinsic and extrinsic invariants of Riemannian almost $k$-product manifolds immersed in another manifold, connecting mixed scalar curvature, mean curvature, and Chen's $delta$-invariants.

Contribution

It introduces the first relationship between intrinsic and extrinsic invariants for isometric immersions of Riemannian almost $k$-product manifolds, including a new optimal inequality involving mixed scalar curvature.

Findings

Derived an optimal inequality involving mixed scalar curvature and mean curvature.

Applied results to multiply twisted and warped products, improving existing inequalities.

Provided conditions for non-immersion and non-existence of compact leaves in foliated submanifolds.

Abstract

A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called the mixed curvature is also related to the extrinsic geometry of a Riemannian almost $k$-product manifold. Our inequality also contains mixed scalar curvature type invariants related to B.-Y Chen's $\delta$-invariants. Applications are given for isometric immersions of multiply twisted and warped products (we improve some known optimal inequalities by replacing the sectional curvature with our invariant) and to problems of non-immersion and non-existence of compact leaves of foliated submanifolds.