Classification and construction of complete hypersurfaces satisfying R(X,Y) R\(=0\). (English) Zbl 0578.53036

Let (M,g) be a differentiable Riemannian manifold of dimension n. We assume that this manifold has the property \(R(X,Y)\cdot R=0\) for tangent vectors X,Y, where the curvature endomorphism R(X,Y) operates on R as a derivation of the tensor algebra at each point of the manifold. A Riemannian manifold with this property is called semisymmetric. Let us assume that the semisymmetric manifold (M,g) is an immersed hypersurface in \({\mathbb{R}}^{n+1}.\)
The main results of this paper can be stated as follows: Theorem I. Let \(M^ n\) be a complete semisymmetric immersed hypersurface in \({\mathbb{R}}^{n+1}\). Then \(M^ n\) is one of the following types. 1. \(M^ n\) is of zero curvature, and it is of the form \(M^ n=c\times {\mathbb{R}}^{n-1}\), where c is a curve in a hyperplane \({\mathbb{R}}^ 2\) and \({\mathbb{R}}^{n-1}\) is orthogonal to \({\mathbb{R}}^ 2\). 2. \(M^ n\) is a straight cylinder of the form \(M^ n=S^ k\times {\mathbb{R}}^{n-k}\) described in Nomizu’s theorem. 3. \(M^ n\) is purely trivial of the form \(M^ n=M^ 2\times {\mathbb{R}}^{n-2}\), where \(M^ 2\) is a hypersurface in a 3-dimensional Euclidean subspace \({\mathbb{R}}^ 3\) and \({\mathbb{R}}^{n- 2}\) is orthogonal to \({\mathbb{R}}^ 3\). 4. \(M^ n\) is purely parabolic of the form \(M^ n=M^ k\times {\mathbb{R}}^{n-k}\), where \(M^ k\) is an irreducible pure parabolic hypersurface in a Euclidean subspace \({\mathbb{R}}^{k+1}\) and \({\mathbb{R}}^{n-k}\) is orthogonal to \({\mathbb{R}}^{k+1}\). 5. \(M^ n\) is purely hyperbolic of the form \(M^ n=M^ 3\times {\mathbb{R}}^{n-3}\), where \(M^ 3\) is a purely hyperbolic irreducible hypersurface in a 4-dimensional Euclidean subspace \({\mathbb{R}}^ 4\) and \({\mathbb{R}}^{n-3}\) is orthogonal to \({\mathbb{R}}^ 4\). 6. \(M^ n\) satisfies the relation k(p)\(\leq 2\) (k denoting the rank of the Weingarten map) and it is mixed with \({\mathcal V}_ 0,{\mathcal V}_ t,{\mathcal V}_ p,{\mathcal V}_ h\) parts. [These notations are specified in the paper.]
Theorem II. A complete semisymmetric immersed hypersurface with \(K>0\) is one of the following types. 1. \(M^ n\) is a cylinder \(M^ n=S^{k- 1}\times {\mathbb{R}}^{n-k}\) described in Nomizu’s theorem. 2. \(M^ n\) is purely trivial of the form \(M^ n=M^ 2\times {\mathbb{R}}^{n-2}\) described above in point 3.
Theorem III. Let \(M^ n\) be a complete immersed semisymmetric hypersurface with \(| K| \geq \epsilon >0\) for a constant \(\epsilon\). Then \(M^ n\) is also one of the types described in the above theorem.
Reviewer: G.Tsagas


53C40 Global submanifolds
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces