Circles and spheres in pseudo-Riemannian geometry
In the present paper pseudo-Riemannian submanifolds \(M_{\alpha}\) of a pseudo-Riemannian manifold \(\bar M_{\beta}\) are studied. \(\alpha\) (resp. \(\beta\)) is the index of the metric tensor of \(M_{\alpha}\) (resp. \(\tilde M_{\beta})\). When \(M_{\alpha}\) is totally umbilical and has parallel mean curvature vector field it is called an extrinsic sphere. A theorem of \textit{K. Nomizu} and \textit{K. Yano} in cases of submanifolds in a Riemannian manifold [Math. Ann. 210, 163-170 (1974; Zbl 0273.53039)] is generalized by the present authors to the following result. Let \(M_{\alpha}\) be a pseudo-Riemannian submanifold in a pseudo-Riemannian manifold \(M_{\beta}\) and \(\epsilon_ 0=+1\quad or\quad -1,\epsilon_ 1=+1,-1\quad or\quad 0\) \((2-2\alpha \leq \epsilon_ 0+\epsilon_ 1\leq 2n-2\alpha -2).\) For any positive constant k the following conditions (a) and (b) are equivalent, (a) every circle in \(M_{\alpha}\) with \(g(X,X)=\epsilon_ 0\) and \(g(\nabla_ XX,\nabla_ XX)=\epsilon_ 1k^ 2\) is a circle in \(\tilde M_{\beta}\), (b) \(M_{\alpha}\) is an extrinsic sphere. Here X is the unit tangent vector field of the circle and \(\nabla\) is the covariant differentiation of \(M_{\alpha}\). A theorem of \textit{K. Sakamoto} [Tohoku Math. J., II. Ser. 29, 25-56 (1977; Zbl 0357.53035)] is also generalized to pseudo-Riemannian cases.
scientific article; zbMATH DE number 4157422