Integral Formulas for Almost Product Manifolds and Foliations
Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-10-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/19/3645 |
| _version_ | 1797478151542013952 |
|---|---|
| author | Vladimir Rovenski |
| author_facet | Vladimir Rovenski |
| author_sort | Vladimir Rovenski |
| collection | DOAJ |
| description | Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula> distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds. |
| first_indexed | 2024-03-09T21:27:54Z |
| format | Article |
| id | doaj.art-c1cd2268de9c4f229f7d71b40609f838 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:27:54Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c1cd2268de9c4f229f7d71b40609f8382023-11-23T21:05:01ZengMDPI AGMathematics2227-73902022-10-011019364510.3390/math10193645Integral Formulas for Almost Product Manifolds and FoliationsVladimir Rovenski0Department of Mathematics, University of Haifa, Haifa 3498838, IsraelIntegral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula> distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds.https://www.mdpi.com/2227-7390/10/19/3645distributionfoliationmixed scalar curvaturealmost product structure |
| spellingShingle | Vladimir Rovenski Integral Formulas for Almost Product Manifolds and Foliations Mathematics distribution foliation mixed scalar curvature almost product structure |
| title | Integral Formulas for Almost Product Manifolds and Foliations |
| title_full | Integral Formulas for Almost Product Manifolds and Foliations |
| title_fullStr | Integral Formulas for Almost Product Manifolds and Foliations |
| title_full_unstemmed | Integral Formulas for Almost Product Manifolds and Foliations |
| title_short | Integral Formulas for Almost Product Manifolds and Foliations |
| title_sort | integral formulas for almost product manifolds and foliations |
| topic | distribution foliation mixed scalar curvature almost product structure |
| url | https://www.mdpi.com/2227-7390/10/19/3645 |
| work_keys_str_mv | AT vladimirrovenski integralformulasforalmostproductmanifoldsandfoliations |