## 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

### MSC:

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