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On three-dimensional hypersurfaces with type number two in $$\mathbb{H}^4$$ and $$\mathbb{S}^4$$ treated in intrinsic way. (English) Zbl 1057.53044
Slovák, Jan (ed.) et al., The proceedings of the 23th winter school “Geometry and physics”, Srní, Czech Republic, January 18–25, 2003. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 72, 107-126 (2004).
A Riemannian manifold $$(M,g)$$ is said to be of constant conullity two if the tangent space $$T_xM$$ at any point $$x\in M$$ admits an orthogonal decomposition: $$T_xM=T_x^1M\oplus T_x^0M$$, where $$\dim T^1_xM=2$$ and $$T^0_xM$$ is the nullity space of the curvature tensor $$R_x$$.
In this paper the authors study 3-dimensional Riemannian manifold of constant conullity two with their natural classification according to the number of asymptotic foliations. They limit to the case $$c<0$$ and study the isometric immersions of the corresponding classes of manifolds in hyperbolic 4-space. They show how the maximal number of nontrivial isometric deformations of the immersed manifold corresponds to its number of asymptotic foliations. In this article, they also investigate a similar problem in $$S^4$$.
For the entire collection see [Zbl 1034.53002].

##### MSC:
 53C40 Global submanifolds 53B25 Local submanifolds
##### Keywords:
conullity two; rigidity