arXiv subjects

Katarzyna Grabowska

Publications and source records attributed to Katarzyna Grabowska.

At least 19 recordsLinked to original sources

arXiv2026

Sasaki structures on general contact manifolds

We extend the notion of a Sasakian structure from the classical setting of a cooriented contact manifold, where it is given by a compatibility between a contact form $η$ and a Riemannian metric $g_M$ on $M$, to the case of an arbitrary contact structure understood as a contact distribution. In the cooriented case, this compatibility can be equivalently expressed by the fact that the symplectic form $ω=\mathrm{d}(s^2η)$ and the cone metric $g(x,s)=\mathrm{d} s\otimes\mathrm{d} s+s^2g_M(x)$ define a Kähler structure on the cone $\mathcal{M}=M\times\mathbb{R}_+$. Since general contact structures admit canonical realizations as homogeneous symplectic structures $ω$ on principal $\mathbb{R}^\times$-bundles $P\to M$, it is natural to interpret Sasakian geometry in full generality in terms of suitable homogeneous Kähler structures on $P$. We characterize homogeneous Kähler structures on symplectizations $(P,ω)$ associated with arbitrary contact structures on $M$, and show that they canonically determine a two-sheeted covering $\tilde M$ of $M$ equipped with a contact form. This reduces the problem to the cooriented case and leads to a notion of a generalized Sasakian structure on $M$ associated with a homogeneous Kähler structure on $(P,ω)$. Moreover, since products of Kähler manifolds are again Kähler, our framework naturally yields a concept of a product of Sasakian manifolds. The whole constructions are intrinsic and conceptual, avoiding any ad hoc choices.

math.DG↗

arXiv2026

Equivalence functors in graded supergeometry

It has recently been proved that the category of N-manifolds of degree $n$, that is, $\mathbb N$-graded supermanifolds of degree $n$ for which the parity agrees with the gradation, is equivalent to the category of purely even $n$-tuple vector superbundles equipped with a suitable action of the symmetric group $S_n$ permuting the vector bundle structures. This equivalence may be interpreted as a `desuperization' of N-manifolds. In the present paper, we place this result within a broader framework of graded structures on supermanifolds and explicitly describe several canonical equivalences between the corresponding categories in a purely geometric, constructive manner. The desuperization equivalence functor appears as a composition of some of these canonical equivalences. Our constructions are entirely canonical and rely on standard tools of supergeometry, including iterated tangent functors, parity reversion in vector superbundles, and the interpretation of $n$-tuple vector bundles in terms of commuting Euler vector fields associated with the underlying vector bundle structures.

math.DG↗

arXiv2025

Graded supermanifolds and homogeneity

We introduce the concept of a homogeneity supermanifold, which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates carry prescribed (real) homogeneity degrees. This structure defines a sheaf of graded algebras on the supermanifold, regarded as an additional geometric structure. The guiding principle of this approach is that grading is ultimately related to homogeneity. Assigning homogeneity degrees to coordinates in a consistent way is equivalent to fixing a global vector field, the weight vector field. This approach is simple and substantially more general than most existing approaches to graded manifolds. In particular, the homogeneity degrees may be arbitrary real numbers, and the resulting category includes compact supermanifolds. We systematically study homogeneity submanifolds, homogeneity Lie supergroups, tangent and cotangent lifts of homogeneity structures, homogeneous distributions and codistributions, as well as related notions such as double homogeneity. The main achievements of this framework include proofs of the homogeneous Poincaré Lemma, the homogeneous Frobenius Theorem, and the homogeneous symplectic Darboux Theorem, results that are of independent interest even in the purely even case.

math.DG↗

arXiv2025

Nijenhuis operators on Banach fibration

In the infinite-dimensional Banach setting, we consider general smooth Banach fibrations $τ:M\to M_0$ and `$(1,1)$-tensors' $N:TM\to TM$ that are projectable (in the obvious sense) onto Nijenhuis operators $N_0:TM_0\to TM_0$ on $M_0$. We prove that the vanishing of the Nijenhuis torsion of $N_0$ is equivalent to the fact that the Nijenhuis torsion of $N$ takes only vertical values, i.e., values in $ker(Tτ)$. Consequences for almost complex structures on (real) Banach manifolds are also derived. As canonical examples, we define tangent lifts $d_T(N_0):TT M_0\to TT M_0$ of Nijenhuis operators $N_0$ in the Banach category, and prove that they are automatically projectable for the canonical fibrations $τ_{M_0}:TM_0\to M_0$. Finally, we comment on the projectability in the case of Banach homogeneous manifolds $τ:G\to G/K$, studied recently by some authors.

math.DG↗

arXiv2025

Reductions: precontact versus presymplectic

We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden-Weinstein-Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous) symplectic structures. A group action by contactomorphisms is lifted to a Hamiltonian action on the corresponding symplectic manifold, called the symplectic cover of the contact manifold. In contrast to the majority of the literature in the subject, our approach includes general contact structures (not only co-oriented) and changes the traditional view point: contact Hamiltonians and contact moment maps for contactomorphism groups are no longer defined on the contact manifold itself, but on its symplectic cover. Actually, the developed framework for reductions is slightly more general than purely contact, and includes a precontact and presymplectic setting which is based on the observation that there is a one-to-one correspondence between isomorphism classes of precontact manifolds and certain homogeneous presymplectic manifolds.

math.SG↗

arXiv2025

Dirac structures in nonholonomic mechanics

The concept of a Dirac algebroid, which is a linear almost Dirac structure on a vector bundle, was designed to generate phase equations for mechanical systems with linear nonholonomic constraints. We apply it to systems with magnetic-like or gyroscopic potentials, that were previously described by means of almost Poisson structures. The almost Poisson structures present in the literature in this context were constructed using constraints, metrics and information about magnetic or gyroscopic potential present in the Hamiltonian function of the system. The Dirac algebroid we use is constructed out of constraints and canonical geometric structures of the underlying bundles and is universal in the sense that it is independent on the particular Hamiltonian or Lagrangian. We provide examples showing that using the same Dirac structure we can describe systems with different potentials, magnetic or mechanical, added freely to a function generating the dynamics.

math-ph↗

arXiv2024

The regularity and products in contact geometry

We study regular contact manifolds $(M,η)$ whose Reeb vector field is complete and prove that they are canonically principal bundles with the structure group $S^1$ or $\mathbb{R}$. For compact $M$, our proof is very short and elementary and covers the celebrated Boothby-Wang theorem, but we do not assume compactness from the very beginning. However, to prove our result in full generality we use some topological tools adapted to smooth fibrations. In the second part of the paper, we describe a natural concept of contact products of general contact manifolds as well as a product of principal contact manifolds, which exists if the periods of the Reeb vector fields are commensurate, and corresponds to the construction of products of prequantization bundles of symplectic manifolds.

math.SG↗

arXiv2024

Jacobi algebroids and Jacobi sigma models

The definition of an action functional for the Jacobi sigma models, known for Jacobi brackets of functions, is generalized to \emph{Jacobi bundles}, i.e., Lie brackets on sections of (possibly nontrivial) line bundles, with the particular case of contact manifolds. Different approaches are proposed, but all of them share a common feature: the presence of a \emph{homogeneity structure} appearing as a principal action of the Lie group $\mathbb{R}^{\times}=\mathrm{GL}(1;\mathbb{R})$. Consequently, solutions of the equations of motions are morphisms of certain \emph{Jacobi algebroids}, i.e., principal $\mathbb{R}^{\times}$-bundles equipped additionally with a compatible Lie algebroid structure. Despite the different approaches we propose, there is a one-to-one correspondence between the space of solutions of the different models. The definition can be immediately extended to \emph{almost Poisson} and \emph{almost Jacobi brackets}, i.e., to brackets that do not satisfy the Jacobi identity. Our sigma models are geometric and fully covariant.

math-ph↗

arXiv2024

Contact geometric mechanics: the Tulczyjew triples

We propose a generalization of the classical Tulczyjew triple as a geometric tool in Hamiltonian and Lagrangian formalisms which serves for contact manifolds. The rôle of the canonical symplectic structures on cotangent bundles in Tulczyjew's case is played by the canonical contact structures on the bundles $J^1L$ of first jets of sections of line bundles $L\to M$. Contact Hamiltonians and contact Lagrangians are understood as sections of certain line bundles, and they determine (generally implicit) dynamics on the contact phase space $J^1L$. We also study a contact analog of the Legendre map and the Legendre transformation of generating objects in both contact formalisms. Several explicit examples are offered.

math.SG↗

arXiv2024

Contactifications: a Lagrangian description of compact Hamiltonian systems

If $η$ is a contact form on a manifold $M$ such that the orbits of the Reeb vector field form a simple foliation $\mathcal{F}$ on $M$, then the presymplectic 2-form $dη$ on $M$ induces a symplectic structure $ω$ on the quotient manifold $N=M/\mathcal{F}$. We call $(M,η)$ a $\textit contactification$ of the symplectic manifold $(N,ω)$. First, we present an explicit geometric construction of contactifications of some coadjoint orbits of connected Lie groups. Our construction is a far going generalization of the well-known contactification of the complex projective space $\mathbb{C}P^{n-1}$, being the unit sphere $S^{2n-1}$ in $\mathbb{C}^{n}$, and equipped with the restriction of the Liouville 1-form on $\mathbb{C}^n$. Second, we describe a constructive procedure for obtaining contactification in the process of the Marsden-Weinstein-Meyer symplectic reduction and indicate geometric obstructions for the existence of compact contactifications. Third, we show that contactifications provide a nice geometrical tool for a Lagrangian description of Hamiltonian systems on compact symplectic manifolds $(N,ω)$, on which symplectic forms never admit a `vector potential'.

math.SG↗

arXiv2024

Tulczyjew triple on the Atiyah algebroid with connection

The Tulczyjew triple on a principal bundle with connection is constructed in a convenient trivialisation. A reduction by the structure group is performed leading to the triple on the trivialised Atiyah algebroid and a presentation of this algebroid via a double vector bundle morphism. The dynamics of physical systems with configuration manifolds having the structure of a principal bundle with connection or the Atiyah algebroid is discussed and applied to the example of an axially symmetric body confined to a sphere.

math-ph↗

arXiv2023

Regular contact manifolds: a generalization of the Boothby-Wang theorem

A regular contact manifold is a manifold $M$ equipped with a globally defined contact form $η$ such that the topological space $M/\mathcal{R}$ of orbits (trajectories) of the Reeb vector field $\mathcal{R}$ of $η$ carries a smooth manifold structure, so the canonical projection $p:M\to M/\mathcal{R}$ is a smooth fibration. We show that, under the additional assumption that $\mathcal{R}$ is a complete vector field, this fibration is actually either an $S^1$- or an $\mathbb{R}$-principal bundle. Moreover, there exists a unique symplectic form $ω$ on $M/\mathcal{R}$ such that $p^*(ω)=\mathrm{d}η$ which is $ρ$-integral in the $S^1$-bundle case, where $ρ$ is the minimal period of the $S^1$-action, so the symplectic manifold $(M/\mathcal{R},ω)$ admits a prequantization. We do not assume that $M$ is compact.

math.SG↗

arXiv2022

Lifting statistical structures

We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a statistical manifold structure, this time on the higher tangent bundles, with the only difference that the metric tensor is pseudo-Riemannian. What is more, natural lifts of potentials (called also divergence or contrast functions) turn out to be again potentials, this time for the lifted statistical structures. We propose an analogous procedure for lifting statistical structures on Lie algebroids and lifting contrast functions which are defined on Lie groupoids. In particular, we study in detail Lie groupoid structures of higher tangent bundles of Lie groupoids. Our geometric constructions of lifts are illustrated by explicit examples, including some important statistical models and potential functions on Lie groupoids.

math.DG↗

arXiv2022

A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory

We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain GL(1,R)-principal bundle P\to M which is equipped with a homogeneous symplectic form ω. In other words, our understanding of contact geometry is that it is not an `odd-dimensional cousin' of symplectic geometry but rather a part of the latter, namely `homogeneous symplectic geometry'. This understanding of contact structures is much simpler than the traditional one and very effective in applications, reducing the contact Hamiltonian formalism to the standard symplectic picture. We develop in this language contact Hamiltonian mechanics in autonomous, as well as time-dependent case, and the corresponding Hamilton-Jacobi theory. Fundamental examples are based on canonical contact structures on the first jet bundles J^1(L) of sections of line bundles L, which play in contact geometry a fundamental role similar to that played by cotangent bundles in symplectic geometry.

math.SG↗

arXiv2022

VB-structures and generalizations

Motivated by properties of higher tangent lifts of geometric structures, we introduce concepts of weighted structures for various geometric objects on a manifold F equipped with a homogeneity structure. The latter is a smooth action on F of the monoid of multiplicative reals. Vector bundles are particular cases of homogeneity structures with the action being the multiplication of vectors by reals, and weighted structures on them we call VB-structures. In the case of Lie algebroids and Lie groupoids, the weighted structures include the concepts of VB-algebroids and VB-groupoids, intensively studied recently in the literature. Investigating various weighted structures, we prove some interesting results about their properties.

math.DG↗

arXiv2020

Solvable Lie Algebras of Vector Fields and a Lie's Conjecture

We present a local and constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such Lie algebras. Also infinite-dimensional analytical solvable and transitive Lie algebras of vector fields whose derivative ideal is nilpotent can be adapted to this scheme.

math.DG↗

arXiv2020

Information geometry on groupoids: the case of singular metrics

We use the general setting for contrast (potential) functions in statistical and information geometry provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. We study the case when the two-form is degenerate and show how in sufficiently regular cases one reduces it to a pseudometric structures. Transversal Levi-Civita connections for Riemannian foliations are generalized to the Lie groupoid/Lie algebroid case.

math.DG↗

arXiv2020

Transitive nilpotent Lie algebras of vector fields and their Tanaka prolongations

Transitive local Lie algebras of vector fields can be easily constructed from dilations of $\mathbb{R}^n$ associating with coordinates positive weights (give me a sequence of $n$ positive integers and I will give you a transitive nilpotent Lie algebra of vector fields on $\mathbb{R}^n$). It is interesting that all transitive nilpotent local Lie algebra of vector fields can be obtained as subalgebras of nilpotent algebras of this kind. Starting with a graded nilpotent Lie algebra one constructs graded parts of its Tanaka prolongations inductively as `derivations of degree 0, 1, etc. Of course, vector fields of weight $k$ with respect to the dilation define automatically derivations of weight $k$, so the Tanaka prolongation is in this case never finite. Are they all such derivations given by vector fields or there are additional `strange'ones? We answer this question. Except for special cases, derivations of degree 0 are given by vector fields of degree 0 and the Tanaka prolongation recovers the whole algebra of polynomial vectors defined by the dilation. However, in some particular cases of dilations we can find `strange' derivations which we describe in detail

math.DG↗