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Peer-reviewedCC BY 4.0Mathematics

Characterization of Compact Riemannian and Spacelike Hypersurfaces in Einstein Manifolds

  • Mohammed Guediri1
  1. 1Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
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Abstract

We explore rigidity results for compact Riemannian and spacelike hypersurfaces in Einstein manifolds. We show that for a compact Riemannian (or spacelike) hypersurface (M,g) in an Einstein Riemannian (or Lorentzian) manifold (M¯,g¯) which admits a closed conformal vector field ξ¯ (assumed to be timelike in the Lorentzian case), the hypersurface (M,g) is an extrinsic sphere in (M¯,g¯) if and only if it is not totally geodesic and the integral of the Ricci curvature of the hypersurface, taken in the direction of the tangential component ξ⊤ of the restriction ξ of ξ¯ to M, satisfies a certain lower bound. As an application of this result, we investigate compact spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes. Additionally, other related results are also provided.

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Cite

BibTeXRIS
BibTeX
@article{guediri2026characterization,
  title = {{Characterization of Compact Riemannian and Spacelike Hypersurfaces in Einstein Manifolds}},
  author = {Mohammed Guediri},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14213881},
  url = {https://www.mdpi.com/2227-7390/14/21/3881},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Characterization of Compact Riemannian and Spacelike Hypersurfaces in Einstein Manifolds
AU  - Mohammed Guediri
PY  - 2026
DA  - 2026/10/24
JO  - Mathematics
DO  - 10.3390/math14213881
UR  - https://www.mdpi.com/2227-7390/14/21/3881
AB  - We explore rigidity results for compact Riemannian and spacelike hypersurfaces in Einstein manifolds. We show that for a compact Riemannian (or spacelike) hypersurface (M,g) in an Einstein Riemannian (or Lorentzian) manifold (M¯,g¯) which admits a closed conformal vector field ξ¯ (assumed to be timelike in the Lorentzian case), the hypersurface (M,g) is an extrinsic sphere in (M¯,g¯) if and only if it is not totally geodesic and the integral of the Ricci curvature of the hypersurface, taken in the direction of the tangential component ξ⊤ of the restriction ξ of ξ¯ to M, satisfies a certain lower bound. As an application of this result, we investigate compact spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes. Additionally, other related results are also provided.
SN  - 2227-7390
ER  -