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W. Domitrz

Publications and source records attributed to W. Domitrz.

8 recordsLinked to original sources
arXiv2023

On singularities of the Gauss map components of surfaces in ${\mathbb R}^4$

The Gauss map of a generic immersion of a smooth, oriented surface into $\mathbb R^4$ is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in $\mathbb R^4$. Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and $\mathcal J$-holomorphic curves with respect to an orthogonal complex structure $\mathcal J$ on $\mathbb R^4$. Finally, we get some formulas of Gauss-Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface.

math.DG↗
arXiv2021

The Geometry of the Wigner Caustic and a Decomposition of a Curve Into Parallel Arcs

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls - the important object to study the dynamics of a quantum particle in the optical lattice potential.

math.DG↗
arXiv2021

The Geometry of the Secant Caustic of a Planar Curve

The secant caustic of a planar curve $M$ is the image of the singular set of the secant map of $M$. We analyse the geometrical properties of the secant caustic of a planar curve, i.e. the number of branches of the secant caustic, the parity of the number of cusps and the number of inflexion points in each branch of this set. In particular, we investigate in detail some of the geometrical properties of the secant caustic of a rosette, i.e. a smooth regular oriented closed curve with non-vanishing curvature.

math.DG↗
arXiv2015

Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space

For a generic embedding of a smooth closed surface $M$ into $\mathbb R^4$, the subset of $\mathbb R^4$ which is the affine $λ$-equidistant of $M$ appears as the discriminant set of a stable mapping $M \times M \to \mathbb R^4$, hence their stable singularities are $A_k, \, k=2, 3, 4,$ and $C_{2,2}^{\pm}$. In this paper, we characterize these stable singularities of $λ$-equidistants in terms of the bi-local extrinsic geometry of the surface, leading to a geometrical study of the set of weakly parallel points on $M$.

math.DG↗
arXiv2014

Singularities of affine equidistants: projections and contacts

Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine $λ$-equidistants of $n$-dimensional closed submanifolds of $\mathbb R^q$, for $q\leq 2n$, whenever $(2n,q)$ is a pair of nice dimensions.

math.DG↗
arXiv2009

Volume preserving subgroups of A and K and singularities in unimodular geometry

For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for A-equivalence with volume-preserving target diffeomorphisms and A-stable maps f and for K-equivalence with volume-preserving source diffeomorphisms and K-simple maps f. On the other hand, there are A-stable maps f with infinite-dimensional moduli space for A-equivalence with volume-preserving source diffeomorphisms.

math.DG↗