On a generic symmetry defect hypersurface

We show that symmetry defect hypersurfaces for two generic members of the irreducible algebraic family of n-dimensional smooth irreducible subvarieties in general position in C²ⁿ are homeomorphic and they have homeomorphic sets of singular points. In particular symmetry defect curves for two generic curves in C² of the same degree have the same numer of singular points

On a singular variety associated to a polynomial mapping from \C^n to \C^{n-1}

We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping G : \C^{n} \to \C^{n - 1} where $n \geq 2$. We prove that in the case G : \C^{3} \to \C^{2}, if $G$ is a local submersion but is not a fibration, then the homology and the intersection homology with total perversity (with compact supports or closed supports) in dimension two of the variety ${\mathcal{V}}_G$ is not trivial. In the case of a local submersion G : \C^{n} \to \C^{n - 1} where $n \geq 4$, the result is still true with an additional condition

On the simplicity of multigerms

We prove several results regarding the simplicity of germs and multigerms obtained via the operations of augmentation, simultaneous augmentation and concatenation and generalised concatenation. We also give some results in the case where one of the branches is a non stable primitive germ. Using our results we obtain a list which includes all simple multigerms from $\mathbb C^3$ to $\mathbb C^3$.Comment: 26 pages, to appear in Mathematica Scandinavica. Second version adds two families that were missing in Table

Horo-tight spheres in Hyperbolic space

We study horo-tight immersions of manifolds into hyperbolic spaces. The main result gives several characterizations of horo-tightness of spheres,\ud answering a question proposed by T. Cecil and P. Ryan.\ud For instance, we prove that a sphere is horo-tight if and only if it is tight in the hyperbolic sense. For codimension bigger than one, it follows that horo-tight spheres in hyperbolic space are metric spheres. We also prove that horo-tight hyperspheres are characterized by the property that\ud both of its total absolute horospherical curvatures attend their minimum value. We also introduce the notion of weak horo-tightness: an immersion is weak horo-tight\ud if only one of its total absolute curvature attends its minimum.\ud We prove a characterization theorem for weak horo-tight hyperspheres

Total absolute horospherical curvature of submanifolds in hyperbolic space

We study the horospherical geometry of submanifolds in hyperbolic space. The main result is a formula for the total absolute horospherical curvature of $M,$ which implies, for the horospherical geometry, the analogues of classical inequalities of the Euclidean Geometry. We prove the horospherical Chern-Lashof inequality for surfaces in $3$-space and the horospherical Fenchel and Fary-Milnor's theorems