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Wojciech Domitrz

Publications and source records attributed to Wojciech Domitrz.

16 recordsLinked to original sources
arXiv2024

Bi-Lagrangian structures and the space of rays

This paper focuses on local curvature invariants associated with bi-Lagrangian structures. We establish several geometric conditions that determine when the canonical connection is flat, building on our previous findings regarding divergence-free webs. Addressing questions raised by Tabachnikov, we provide complete solutions to two problems: the existence of flat bi-Lagrangian structures within the space of rays induced by a pair of hypersurfaces, and the existence of flat bi-Lagrangian structures induced by tangents to Lagrangian curves in the symplectic plane.

math.DG↗
arXiv2022

Local invariants of divergence-free webs

The objects of our study are webs in the geometry of volume-preserving diffeomorphisms. We introduce two local invariants of divergence-free webs: a differential one, directly related to the curvature of the natural connection of a divergence-free 2-web introduced by S. Tabachnikov (1992), and a geometric one, inspired by the classical notion of planar 3-web holonomy defined by W. Blaschke and G. Thomsen (1928). We show that triviality of either of these invariants characterizes trivial divergence-free web-germs up to equivalence. We also establish some preliminary results regarding the full classification problem, which jointly generalize the theorem of S. Tabachnikov on normal forms of divergence-free 2-webs. They are used to provide a canonical form and a complete set of invariants of a generic divergence-free web in the planar case. Lastly, the relevance of local triviality conditions and their potential applications in numerical relativity are discussed.

math.DG↗
arXiv2020

Unimodular families of symmetric matrices

We introduce the volume-preserving equivalence among symmetric matrix-valued map-germs which is the unimodular version of Bruce's $\mathcal{G}$-equivalence. The key concept to deduce unimodular classification out of classification relative to $\mathcal{G}$-equivalence is symmetrical quasi-homogeneity, which is a generalization of the condition for a $2 \times 2$ symmetric matrix-valued map-germ in Corollary~2.1 (ii) by Bruce, Goryunov and Zakalyukin. If a $\mathcal{G}$-equivalence class contains a symmetrically quasi-homogeneous representative, the class coincides with that relative to the volume-preserving equivalence (up to orientation reversing diffeomorphism in case if the ground field is real). By using that we show that all the simple classes relative to $\mathcal{G}$-equivalence in Bruce's list coincides with those relative to the volume preserving equivalence. Then, we classify map-germs from the plane to the set of $2 \times 2$ and $3 \times 3$ real symmetric matrices of corank at most $1$ and of $\mathcal{G}_e$-codimension less than $9$ and we show some of the normal forms split into two different unimodular singularities. We provide several examples to illustrate that non simplicity does not imply non symmetrical quasi-homogeneity and the condition that a map-germ is symmetrically quasi-homogeneous is stronger than one that each component of the map-germ is quasi-homogeneous. We also present an example of non symmetrically quasi-homogeneous normal form relative to $\mathcal{G}$ and its corresponding formal unimodular normal form.

math.DG↗
arXiv2020

Singular Improper Affine Spheres from a given Lagrangian Submanifold

Given a Lagrangian submanifold $L$ of the affine symplectic $2n$-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension $2n$, both of whose sets of singularities contain $L$. Although these improper affine spheres (IAS) always present other singularities away from $L$ (the off-shell singularities studied in our previous paper), they may also present singularities other than $L$ which are arbitrarily close to $L$, the so called singularities "on shell". These on-shell singularities possess a hidden $\mathbb Z_2$ symmetry that is absent from the off-shell singularities. In this paper, we study these canonical IAS obtained from $L$ and their on-shell singularities, in arbitrary even dimensions, and classify all stable Lagrangian/Legendrian singularities on shell that may occur for these IAS when $L$ is a curve or a Lagrangian surface.

math.DG↗
arXiv2019

Order of tangency between manifolds

We study the order of tangency between two manifolds of same dimension and give that notion three quite different geometric interpretations. Related aspects of the order of tangency, e.g., regular separation exponents, are also discussed.

math.AG↗
arXiv2018

The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications

In [31,32,33] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surfaces with boundary. As a corollary of our results we obtain Fukuda-Ishikawa's theorem. We also study geometry of the affine extended wave fronts for planar closed non singular hedgehogs (rosettes). In particular, we find a link between the total geodesic curvature on the boundary and the total singular curvature of the affine extended wave front, which leads to a relation of integrals of functions of the width of a resette.

math.DG↗
arXiv2016

On local invariants of singular symplectic forms

We find a complete set of local invariants of singular symplectic forms with the structurally stable Martinet hypersurface on a $2n$-dimensional manifold. In the $\mathbb C$-analytic category this set consists of the Martinet hypersurface $Σ_2$, the restriction of the singular symplectic form $ω$ to $TΣ_2$ and the kernel of $ω^{n-1}$ at the point $p\in Σ_2$. In the $\mathbb R$-analytic and smooth categories this set contains one more invariant: the canonical orientation of $Σ_2$. We find the conditions to determine the kernel of $ω^{n-1}$ at $p$ by the other invariants. In dimension $4$ we find sufficient conditions to determine the equivalence class of a singular symplectic form-germ with the structurally smooth Martinet hypersurface by the Martinet hypersurface and the restriction of the singular symplectic form to it. We also study the singular symplectic forms with singular Martinet hypersurfaces. We prove that the equivalence class of such singular symplectic form-germ is determined by the Martinet hypersurface, the canonical orientation of its regular part and the restriction of the singular symplectic form to its regular part if the Martinet hypersurface is a quasi-homogeneous hypersurface with an isolated singularity.

math.DG↗
arXiv2016

Symplectic singularity of curves with semigroups (4,5,6,7), (4,5,6) and (4,5,7)

We study the local symplectic algebra of curves with semigroups $(4,5,6,7)$, $(4,5,6)$ and $(4,5,7)$. We use the method of algebraic restrictions to parameterized curves as in \cite{D1}. A new discrete invariant for algebraic restrictions to parameterized quasi-homogeneous curves is introduced. This invariant together with the method of algebraic restriction can distinguish different symplectic orbits of quasi-homogeneous curves.

math.AG↗
arXiv2014

Even Dimensional Improper Affine Spheres

There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dimensions: the former class corresponds to the center-chord transform of a pair of Lagrangian submanifolds while the latter is related to special Kähler manifolds. Furthermore, we show that the improper affine spheres obtained in this way are solutions of certain exterior differential systems. Finally, we also discuss the problem of realization of simple stable Legendrian singularities as singularities of these improper affine spheres.

math.DG↗
arXiv2012

The Wigner caustic on shell and singularities of odd functions

We study the Wigner caustic on shell of a Lagrangian submanifold L of affine symplectic space. We present the physical motivation for studying singularities of the Wigner caustic on shell and present its mathematical definition in terms of a generating family. Because such a generating family is an odd deformation of an odd function, we study simple singularities in the category of odd functions and their odd versal deformations, applying these results to classify the singularities of the Wigner caustic on shell, interpreting these singularities in terms of the local geometry of L.

math-ph↗
arXiv2012

Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

We define the Global Centre Symmetry set (GCS) of a smooth closed m-dimensional submanifold M of R^n, $n \leq 2m$, which is an affinely invariant generalization of the centre of a k-sphere in R^{k+1}. The GCS includes both the centre symmetry set defined by Janeczko and the Wigner caustic defined by Berry. We develop a new method for studying generic singularities of the GCS which is suited to the case when M is lagrangian in R^{2m} with canonical symplectic form. The definition of the GCS, which slightly generalizes one by Giblin and Zakalyukin, is based on the notion of affine equidistants, so, we first study singularities of affine equidistants of Lagrangian submanifolds, classifying all the stable ones. Then, we classify the affine-Lagrangian stable singularities of the GCS of Lagrangian submanifolds and show that, already for smooth closed convex curves in R^2, many singularities of the GCS which are affine stable are not affine-Lagrangian stable.

math.SG↗
arXiv2011

Symplectic $T_7$, $T_8$ singularities and Lagrangian tangency orders

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $T_7$ singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical $A-D-E$ singularities of planar curves, $S_5$ singularity and $T_7$ singularity. We also give the geometric description of these symplectic singularities.

math.SG↗
arXiv2011

Symplectic Sμ singularities

We study the local symplectic algebra of the 1-dimensional isolated complete intersection singularity of type Sμ. We use the method of algebraic restrictions to classify symplectic Sμ singularities. We distinguish these symplectic singularities by discrete symplectic invariants. We also give the geometric description of them.

math.SG↗
arXiv2009

Local symplectic algebra of quasi-homogeneous curves

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of an analytic curve is a finite dimensional vector space. We also show that the action of local diffeomorphisms preserving the curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain the complete symplectic classification of curves with the semigroups (3,4,5), (3,5,7), (3,7,8).

math.SG↗
arXiv2008

Reductions of locally conformal symplectic structures and de Rham cohomology tangent to a foliation

We propose a produre of reduction a locally conformal symplectic structure. This procedure of reduction can be applied to wide class of submanifolds. There are no local obstructions for this procedure. But there are global obstructions. We find a necessary and sufficient condition when this reduction holds in terms of the special kind of de Rham cohomology class (tangent to the characteristic foliation) of the Lee form.

math.SG↗