Hypersurfaces in a Euclidean space with a Killing vector field

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 studied orientable hypersurfaces in a Euclidean space that admits a unit Killing vector field and finds two characterizations of odd-dimensional spheres. In the first result, we showed that a complete and simply connected hypersurface of Euclidean space $ \mathbb{R}^{n+1} $, $ n > 1 $ admits a unit Killing vector field $ \xi $ that leaves the shape operator $ S $ invariant and has sectional curvatures of plane sections containing $ \xi $ positive which satisfies $ S(\xi) = \alpha \xi $, $ \alpha $ mean curvature if, and only if, $ n = 2m-1 $, $ \alpha $ is constant and the hypersurface is isometric to the sphere $ S^{2m-1}(\alpha^2) $. Similarly, we found another characterization of the unit sphere $ S^2(\alpha^2) $ using the smooth function $ \sigma = g(S(\xi), \xi) $ on the hypersurface.

Keywords

• euclidean space
• hypersurface
• killing vector field