Integral Formulae and Applications for Compact Riemannian Hypersurfaces in Riemannian and Lorentzian Manifolds Admitting Concircular Vector Fields

Mona Bin-Asfour, Kholoud Saad Albalawi () and Mohammed Guediri
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Mona Bin-Asfour: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia
Kholoud Saad Albalawi: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia
Mohammed Guediri: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Mathematics, 2025, vol. 13, issue 10, 1-15

Abstract: This paper investigates compact Riemannian hypersurfaces immersed in ( n + 1 ) -dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space R n + 1 , Euclidean sphere S n + 1 , and de Sitter space S 1 n + 1 .

Keywords: Euclidean space; sphere; de Sitter space; spacelike hypersurfaces; conformal vector fields; concircular vector fields; Ricci curvature; spaces of constant sectional curvature (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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