We study affine Jacobi structures on an affine bundle $π:A\to M$, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on $A$ and Lie algebroid structures on the vector bundle $A^+=\bigcup_{p\in M}Aff(A_p,\R)$ of affine functionals. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited (strongly-affine or affine-homogeneous Jacobi structures) over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finite-dimensional Lie algebra.
We study Jacobi structures on the dual bundle $A^\ast$ to a vector bundle $A$ such that the Jacobi bracket of linear functions is again linear and the Jacobi bracket of a linear function and the constant function 1 is a basic function. We prove that a Lie algebroid structure on $A$ and a 1-cocycle $ϕ\in Γ(A^\ast)$ induce a Jacobi structure on $A^\ast$ satisfying the above conditions. Moreover, we show that this correspondence is a bijection. Finally, we discuss some examples and applications.
The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alternative proof of the splitting theorem of homogeneous Poisson structures.
A Jacobi structure $J$ on a line bundle $L\to M$ is weakly regular if the sharp map $J^\sharp : J^1 L \to DL$ has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5-dimensional nilmanifold i…
We show that a suitable notion of Dirac-Jacobi structure on a generic line bundle $L$, is provided by Dirac structures in the omni-Lie algebroid of $L$. Dirac-Jacobi structures on line bundles generalize Wade's $\mathcal E^1 (M)$-Dirac structures and unify generic (i.e.~non-necessarily coorientable) precontact distributions, Dirac structures and local Lie algebras with one dimensional fibers in the sense of Kirillov (in particular, Jacobi structures in the sense of Lichnerowicz). We study the main properties of Dirac-Jacobi structures and prove that integrable Dirac-Jacobi structures on line-b…
We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an alternate proof of the main result of [13], showing that the characterization of vector bundles by means of their Lie algebras of homogeneous differential operators also holds for vector bundles of rank 1 and over any base manifolds.
Let (M,g) be a Riemannian manifold and G a nondegenerate g-natural metric on its tangent bundle T M . In this paper we establish a relation between the Jacobi operators of (M,g) and that of (T M,G). In the case of a Riemannian surface (M,g), we compute explicitly the spectrum of some Jacobi operators of (TM,G) and give necessary and sufficient conditions for (T M,G) to be an Osserman manifold.
In this paper, we use the affine Hermitian-Yang-Mills flow to prove a generalized Donaldson-Uhlenbeck-Yau theorem on flat Higgs bundles over a class of non-compact affine Gauduchon manifolds.
The connection between Jacobi fields and odular structures of affine manifold is established. It is shown that the Jacobi fields generate the natural geoodular structure of affinely connected manifolds.
We show that the total space of any affine $\mathbb{C}$-bundle over $\mathbb{CP}^1$ with negative degree admits an ALE scalar-flat Kähler metric. Here the degree of an affine bundle means the negative of the self-intersection number of the section at infinity in a natural compactification of the bundle, and so for line bundles it agrees with the usual notion of the degree.
Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure affine quantum group. A particular attention is given to the study of the specific properties of quantum affine connections and several purely quantum phenomena appearing in the context of quantum affine bundles. Various interesting constructions are presented. In particular, the…
We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold $S$ controls the coisotropic deformation problem of $S$ under both Hamiltonian and Jacobi equivalence.
Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle is passed down to Jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure between the groups of automorphisms of related Weil algebras
We define certain extensions of Jacobi groups of $A_n$, prove an analogue of Chevalley Theorem for their invariants, and construct a Dubrovin Frobenius structure on it orbit space.
We introduce a class of Higgs bundles called cyclic which lie in the Hitchin component of representations of a compact Riemann surface into the split real form of a simple Lie group. We then prove that such a Higgs bundle is equivalent to a certain class of solutions to the affine Toda equations. We further explain this relationships in terms of lifts of harmonic maps.
In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geometry of higher-order total differentials, to appear in Cahiers de Topologie et Géométrie Différentielle Catégoriques] we have introduced another synthetic approach to jet bundles in terms of infinitesimal spaces D_{n}'s, and have compared it with the former approach both in the …
A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles have been recognized as a proper geometrical tool in a frame-independent description of many important systems. Different approaches to the (special) double affine bundles are compared and carefully studied together with the problems of double vector bundle models and hulls, duality…
We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontriv…
We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalised connection.
In this paper, for a variety of nonholonomic (reducible) Hamiltonian systems, we first give to various distributional Hamiltonian systems, by analyzing carefully the dynamics and structures of the nonholonomic Hamiltonian systems. Secondly, we derive precisely the geometric constraint conditions of the induced distributional two-form for the nonholonomic dynamical vector field, which are called the Type I and Type II of Hamilton-Jacobi equations. Thirdly, we generalize the above results for the nonholonomic reducible Hamiltonian systems with symmetries, as well as with momentum maps, and prove…
As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.
The goal of this paper is to introduce the lifting theory that has an important role in geometry. Therefore, using the lifts of differential geometric structures we show that tangent bundle TM of paracomplex manifold M admits para-complex torsion-free affine connection.
A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mechanics are provided.
Natural analogs of Lie brackets on affine bundles are studied, based on natural examples from differential geometry and analytical mechanics. In particular, a close relation to Lie algebroids and, by a sort of duality, to affine analogs of Poisson structures is established as well as affine versions of the complete lift of a section and the Cartan exterior calculus.
Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed. The developed concepts are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics.
We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce the notion of quasi-Jacobi bialgebroid and we prove that each twisted Jacobi manifold has a quasi-Jacobi bialgebroid canonically associated. Moreover, the double of a quasi-Jacobi bialgebroid is a Courant-Jacobi algebroid. Several examples of twisted Jacobi manifolds and twisted D…
We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible, brings …
Based on ideas of W. M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which…
In this paper, we present a relation between Jacobi-Reeb dynamics and the dynamics associated with a mechanical Hamiltonian system with respect to a linear Poisson structure on a vector bundle. For this purpose, we will use the so-called Jacobi bundle metrics induced by the mechanical Hamiltonian system. These constructions extend classical results on the relation between standard mechanical Hamiltonian systems on cotangent bundles and Reeb dynamics.
Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal $G$-bundle $E_G$ over $M$ admits a Hermitian-Einstein structure if and only if $E_G$ is polystable. A polystable flat principal $G$--bundle over $M$ admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for …
We prove a nonholonomic version of the classical Mauper\-tuis-Jacobi principle which transforms an autonomous mechanical nonholonomic problem, determined by a kinetic minus potential energy and a distribution, in a kinetic nonholonomic problem over a fixed level set of the Lagrangian energy. To prove this result we introduce an appropriate contact bundle structure clarifying the geometric equivalence between both problems. By using the nonholonomic Maupertuis-Jacobi principle, we prove that the regular solutions of a mechanical nonholonomic problem starting from a fixed point and in the same l…
Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as odd Jacobi brackets on the supermanifolds associated with the vector bundles. Jacobi bialgebroids are defined in the same manner. A lifting procedure of elements of this Grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding bracket…
Starting from a bundle E over R, the dual of the first jet bundle, which is a co-dimension 1 sub-bundle of the cotangent bundle of E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type (1,1) tensor R on E to both of these manifolds. We discuss how an interplay between these lifted tensors leads to the identification of related distributions on both manifolds. The integrability of these distributions, a coordinate free condition, is shown to produce exactly Forbat's conditi…
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 `…
We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalised connections on such bundles.
We develop a general framework for the quantization of bosonic and fermionic field theories on affine bundles over arbitrary globally hyperbolic spacetimes. All concepts and results are formulated using the language of category theory, which allows us to prove that these models satisfy the principle of general local covariance. Our analysis is a preparatory step towards a full-fledged quantization scheme for the Maxwell field, which emphasises the affine bundle structure of the bundle of principal U(1)-connections. As a by-product, our construction provides a new class of exactly tractable loc…
The standard formulation of Jacobi manifolds in terms of differential operators on line bundles, although effective at capturing most of the relevant geometric features, lacks a clear algebraic interpretation similar to how Poisson algebras are understood to be the algebraic counterpart of Poisson manifolds. We propose a formalism, based on the dimensioned algebra technology recently developed by the author, to capture the algebraic counterparts of Jacobi manifolds as dimensioned Poisson algebras. Particularly, we give a generalisation of the functor $\text{C}^\infty:\textsf{Smooth}\to \textsf…
In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi-Nijenhuis structures.
For the Jacobian resulting from the previously considered problem of the path integral reduction in Wiener path integrals for a mechanical system with symmetry describing the motion of two interacting scalar particles on a manifold that is the product of a smooth compact finite-dimensional Riemannian manifold and a finite-dimensional vector space, a geometric representation is obtained. This representation follows from the formula for the scalar curvature of the original manifold endowed by definition with a free isometric smooth action of a compact semisimple Lie group. The derivation of this…
The dually flat structure of statistical manifolds can be derived in a non-parametric way from a particular case of affine space defined on a qualified set of probability measures. The statistically natural displacement mapping of the affine space depends on the notion of Fisher's score. The model space must be carefully defined if the state space is not finite. Among various options, we discuss how to use Orlicz-Sobolev spaces with Gaussian weight. Such a fully non-parametric set-up provides tools to discuss intrinsically infinite-dimensional evolution problems.
By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the modified Riemannian extension metric on the cotangent bundle is both spacelike and timelike projective Osserman. Since any rank 1 symmetric space is affine projective Osserman, this provides additional information concerning the cotangent bundle of a rank 1 Riemannian symmetric space with the modified Riemannian extension metric. We construct other examples of…
Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Elie Cartan extended these concepts for other types of connections on a manifold: Euclidean, Galilean and Minkows…
We give characterizations of affine transformations and affine vector fields in terms of the spray. By utilizing the Jacobi type equation that characterizes affine vector fields, we prove some rigidity theorems of affine vector fields on compact or forward complete non-compact Finsler manifolds with non-positive total Ricci curvature.
We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds. As an application we have that a natural circle bundle over the Kuranishi moduli space of a Calabi-Yau threefold is a Lorentzian proper affine hypersphere.
The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray $S:TM\to TTM$. The flow lines of the vector field $\ka_{TM}øTS:TTM\to TTTM$ project to the Jacobi fields on $TM$. This could be called the Jacobi flow.
A frame independent formulation of analytical mechanics in the Newtonian space-time is presented. The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds, is used. Lagrangian and hamiltonian generating objects, together with the Legendre transformation independent on inertial frame are constructed.
Given a complex manifold $X$, any Kähler class defines an affine bundle over $X$, and any Kähler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact Kähler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Szőke and to the existence question for good complex…
We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence of Hermitian-Einstein metrics on Kähler manifolds, and the extension of this theorem by Li-Yau to the non-Kähler complex case of Gauduchon metrics. Our definition of stability involves only flat vector subbundles (and not singular subsheaves), and so is simpler than the complex ca…
The Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron-monopole system.
We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.
An affine Cartan calculus is developed. The concepts of special affine bundles and special affine duality are introduced. The canonical isomorphisms, fundamental for Lagrangian and Hamiltonian formulations of the dynamics in the affine setting are proved.
