Abstract
In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D=TF⊕span(N), and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation.
Highlights
Foliations, which are defined as partitions of a manifold into collections of submanifolds of the same dimension, called leaves, appeared in the 1940s in the works of G
Integral formulas are the centerpiece of extrinsic geometry of foliations and are useful in several geometric situations: characterizing foliations, whose leaves have a given geometric property; prescribing the higher mean curvatures of the leaves of a foliation; minimizing functionals such as volume defined for tensor fields on a foliated manifold
We prove a series of integral formulas for a codimension-one foliated subRiemannian manifold ( M, D, F, g) with D = T F ⊕ span( N )
Summary
Foliations, which are defined as partitions of a manifold into collections of submanifolds of the same dimension, called leaves, appeared in the 1940s in the works of G. G. Reeb published a paper [10] on extrinsic geometry of foliations, in which he proved that the integral of the mean curvature of the leaves of any codimension-one foliation on any closed Riemannian manifold equals zero, Z. Integral formulas are the centerpiece of extrinsic geometry of foliations and are useful in several geometric situations: characterizing foliations, whose leaves have a given geometric property; prescribing the higher mean curvatures of the leaves of a foliation; minimizing functionals such as volume defined for tensor fields on a foliated manifold. In [4], the Newton transformations Tr ( A N ) of the shape operator A N of the leaves (with a unit normal vector field N) were applied to a codimension-one foliated (n + 1)-dimensional Riemannian manifold The results (in Sections 3 and 4) can be interpreted as integral formulas for codimension-one foliations of a sub-Riemannian manifold
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