Version 1
: Received: 12 November 2023 / Approved: 13 November 2023 / Online: 13 November 2023 (10:21:04 CET)
How to cite:
Guediri, M.; Deshmukh, S. Hypersurfaces in a Euclidean Space with a Killing Vector Field. Preprints2023, 2023110799. https://doi.org/10.20944/preprints202311.0799.v1
Guediri, M.; Deshmukh, S. Hypersurfaces in a Euclidean Space with a Killing Vector Field. Preprints 2023, 2023110799. https://doi.org/10.20944/preprints202311.0799.v1
Guediri, M.; Deshmukh, S. Hypersurfaces in a Euclidean Space with a Killing Vector Field. Preprints2023, 2023110799. https://doi.org/10.20944/preprints202311.0799.v1
APA Style
Guediri, M., & Deshmukh, S. (2023). Hypersurfaces in a Euclidean Space with a Killing Vector Field. Preprints. https://doi.org/10.20944/preprints202311.0799.v1
Chicago/Turabian Style
Guediri, M. and Sharief Deshmukh. 2023 "Hypersurfaces in a Euclidean Space with a Killing Vector Field" Preprints. https://doi.org/10.20944/preprints202311.0799.v1
Abstract
An odd-dimensional sphere admits a killing vector field, induced by the transform of the unit normal by the complex structure of the ambiant Euclidean space. In this paper, we study orientable hypersurfaces in a Euclidean space those admit a unit Killing vector field and find two characterizations of odd-dimensional spheres. In first result, we show that a complete and simply connected hypersurface of Euclidean space Rn+1, n>1 admits a unit Killing vector field ξ that leaves the shape operator S invariant and has sectional curvatures of plane sections containing ξ positive satisfies S(ξ)=αξ, α mean curvature if and only if n=2m−1, α is constant and the hypersurfaces is isometric to the sphere S2m−1(α2). Similarly, we find another characterization of unit sphere S2(α2) using the smooth function σ=g(S(ξ),ξ) on the hypersurface.
Keywords
Euclidean space; Hypersurface; Killing vector field
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
