Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature
The maximal spacelike submanifolds of a pseudo-Riemannian manifold of constant curvature are investigated. Let \(N_ p^{n+p}(c)\) denote an \((n+p)\)-dimensional pseudo-Riemannian manifold of constant curvature \(c\) and index \(p\). Let \(M\) be an \(n\)-dimensional complete Riemannian manifold which is isometrically immersed in \(N_ p^{n+p}(c)\) and maximal, i.e. has vanishing mean curvature. Let \(S\) be the square of the length of the second fundamental form. Three main results are proved. \par (i) If \(c\ge 0\) then the immersion is totally geodesic \((S=0)\) and \(M\) has constant curvature \(c\). \par (ii) If \(c<0\) then \(0\le S\le -npc.\) In the case \(c=-1/r^ 2,\) the ambient space \(N_p^{n+p}(c)\) is a pseudo-hyperbolic space \(H_p^{n+p}(r).\) \par (iii) The only complete connected maximal spacelike submanifolds of dimension \(n\) in \(H_p^{n+p}(1)\) satisfying \(S=np\) are the submanifolds \(H^{n_1}(\sqrt{n_1/n})\times\ldots\times H^{n_{p+1}}(\sqrt{n_{p+1}/n}),\) which can be described as follows. Let \(H^ m(r)\) denote the component of \(H^m_0(r)\) through \((0,0,\ldots,0,r)\). Let \(n_1,\ldots,n_{p+1}\) be positive integers with \(n=n_1+\ldots+n_{p+1}.\) For \(x_ i\in H^{n/i}(\sqrt{n_i/n}),\) \(x=(x_1,\ldots,x_{p+1})\) is a vector in \({\mathbb{R}}_{p+1}^{n+p+1}\) with \(<x,x>=-1.\) This defines an isometric immersion of \par \[ H^{n_1}(\sqrt{n_ 1/n})\times\ldots\times H^{n_{p+1}}(\sqrt{n_{p+1}/n})\) \text{ into } \(H_p^{n+p}(1). \] These results can be compared with the work of \textit{S. S. Chern}, \textit{M. P. do Carmo}, and \textit{S. Kobayashi} [Functional analysis and related fields, Conf. Chicago 1968, 59--75 (1970; Zbl 0216.44001)] on minimal submanifolds of Riemannian spheres. Result (i) above generalizes the theorem of \textit{S.-Y. Cheng} and \textit{S.-T. Yau} [Ann. Math. (2) 104, 407--419 (1976; Zbl 0352.53021)] that complete maximal spacelike hypersurfaces in Minkowski space are totally geodesic.
scientific article; zbMATH DE number 4117401