Ruy Tojeiro

Commuting Codazzi tensors and the Ribaucour transformation for submanifolds

In this paper we develop a general theory for the Ribaucour transformation of sub-manifolds. In the process, a beautiful correspondence between Ribaucour transforms of a submanifold and Codazzi tensors that commute with its second fundamental form is revealed.

On deformable hypersurfaces in space forms

We first extend the classical Sbrana-Cartan theory of isometrically deformable euclidean hypersurfaces to the sphere and hyperbolic space. Then we construct and characterize a large family of hypersurfaces which admit a unique deformation. This is used to show, by means of explicit examples, that different types of hypersurfaces in the Sbrana-Cartan classification can be smoothly attached. Finally, among other applications, we discuss the existence of complete deformable hypersurfaces in hyperbolic space.

Submanifolds of codimension two attaining equality in an extrinsic inequality

We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector.

Copolarity of isometric actions

We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity k representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Sainelson.

ON A CLASS OF SUBMANIFOLDS CARRYING AN EXTRINSIC TOTALLY UMBILICAL FOLIATION

In this paper we give a conformal classification of all euclidean submanifolds carrying a parallel principal curvature normal under the intrinsic additional assumption that the associated conformal conullity is involutive and the leaves are extrinsic spheres in the submanifold in the sense of Nomizu. We also provide several applications of this result.

The Ribaucour transformation for flat Lagrangian submanifolds

We study the Ribaucour transformation for flat Lagrangian submanifolds in complex flat space and complex projective space. As a consequence, we obtain a process to generate a new family of such submanifolds from a given one. Analytically, this provides a method to construct new solutions of the corresponding systems of PDEs from a given one. We also show that such transformation always comes with a permutability formula.

Hypersurfaces of two space forms and conformally flat hypersurfaces

We address the problem of determining the hypersurfaces

PSEUDO-PARALLEL SUBMANIFOLDS WITH FLAT NORMAL BUNDLE OF SPACE FORMS

f:M^{n} \rightarrow \mathbb {Q}_s^{n+1}(c)