Constant mean curvature hypersurfaces with constant angle in semi-Riemannian space forms

详细信息    查看全文

• 作者：Matias Navarro^a ; ^{matias.navarro@correo.uady.mx" class="auth_mail" title="E-mail the corresponding author} ; Gabriel Ruiz-Herná ; ndez^b ; ^{gruiz@matem.unam.mx" class="auth_mail" title="E-mail the corresponding author} ; Didier A. Solis^a ; ^{didier.solis@correo.uady.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53B25 ; 53B30 ; 53C42 ; 53C50
• 刊名：Differential Geometry and its Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：49
• 期：Complete
• 页码：473-495
• 全文大小：467 K

文摘

We study constant angle semi-Riemannian hypersurfaces M immersed in semi-Riemannian space forms, where the constant angle is defined in terms of a closed and conformal vector field Z in the ambient space form. We show that such hypersurfaces belong to the class of hypersurfaces with a canonical principal direction. This property is a type of rigidity. We further specialize to the case of constant mean curvature (CMC) hypersurfaces and characterize them in two relevant cases: when the hypersurface is orthogonal to Z then it is totally umbilical, whereas if Z   is tangent to the hypersurface then it has zero Gauss–Kronecker curvature and either its mean curvature vanishes or the ambient is a semi-Euclidean space. We also treat in detail the surface case, giving a full characterization of the constant angle CMC surfaces immersed in all three dimensional space forms. They are isoparametric surfaces with constant principal curvatures when the ambient is flat. If the mean curvature of the surface is not 203074ab6703ebf3c">$\pm 2/\sqrt{3}$ they are either totally umbilic or totally geodesic. In particular when the surface has zero mean curvature it is totally geodesic.