Tangent Lifts Research Articles

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 \((\mathbb {R},\cdot )\) of multiplicative reals. Vector bundles are particular cases of homogeneity structures and weighted structures on them we call \(\mathrm{V\!B}\)-structures. In the case of Lie algebroids and Lie groupoids, the weighted structures include the concepts of \(\mathrm{V\!B}\)-algebroids and \(\mathrm{V\!B}\)-groupoids, intensively studied recently in the literature. Investigating various weighted structures, we prove some interesting results about their properties.

Characterization of symplectic forms induced by some tangent G-structures of higher order

Let (M, ω) be a symplectic manifold induced by an integrable G-structure P on M . In this paper, we characterize the symplectic manifolds induced by the tangent lifts of higher order r ≥ 1 of G-structure P, from M to TrM .

Local properties of exponential contractive systems

In this technical note, we prove that exponential contraction can be deduced from the behaviour of the tangent lift dynamics on a neighbourhood of the zero section, instead of on the whole tangent bundle. The result holds for general nonlinear systems evolving on Riemannian manifolds.

Generalized variational calculus for continuous and discrete mechanical systems

In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory... This generalized variational calculus is based on two main notions: the tangent lift of curves and the notion of complete lift of a vector field. Both concepts are also adapted for the case of skew-symmetric algebroids, therefore, our formalism easily extends to the case of Lie algebroids and nonholonomic systems (see also [20]). Hence, this framework automatically includes reduced mechanical systems subjected or not to constraints. Finally, we show that our formalism can be used to tackle the case of discrete mechanics, including reduced systems, systems subjected to constraints and discrete optimal control theory.

Tangent lifts of bi-Hamiltonian structures

We construct two Poisson structures πTM and π̃TM on the tangent bundle TM to a Poisson manifold (M,π) using Lie algebroid (T*M, qM, M). Next, we construct the Poisson manifold (TM,πTM,λ) from a bi-Hamiltonian manifold (M,π1,π2). This structure can be considered as a deformation of the Poisson structure related to an algebroid structure. We show that the bi-Hamiltonian structure from M can be transferred to the manifold TM. Moreover we present how to construct the Casimir functions for structures πTM, πTM,λ, π̃Tg*, and π̃Tg*,λ from the Casimir functions for π1 and π2 and discuss some particular examples.

Tangent lifts of higher order of multiplicative Dirac structures

The tangent lifts of higher order of Dirac structures and some properties have been defined in [9] and studied in [11]. By the same way, the tangent lifts of higher order of Poisson structures have been studied in [10] and some applications are given. In particular, the authors have studied the nature of the Lie algebroids and singular foliations induced by these lifting. In this paper, we study the tangent lifts of higher order of multiplicative Poisson structures, multiplicative Dirac structures and we describe the Lie bialgebroid structures and the algebroid-Dirac structures induced by these prolongations.

Some properties of tangent Dirac structures of higher order

Let $M$ be a smooth manifold. The tangent lift of Dirac structure on $M$ was originally studied by T. Courant in [3]. The tangent lift of higher order of Dirac structure $L$ on $M$ has been studied in [10], where tangent Dirac structure of higher order are described locally. In this paper we give an intrinsic construction of tangent Dirac structure of higher order denoted by $L^{r}$ and we study some properties of this Dirac structure. In particular, we study the Lie algebroid and the presymplectic foliation induced by $L^{r}$.

Geodesic Reduction via Frame Bundle Geometry

A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation.

Generalized submersiveness of second-order ordinary differential equations

We generalize the notion of submersive second-order differential equations by relaxing the condition that the decoupling stems from the tangent lift of a basic distribution. It is shown that this leads to adapted coordinates in which a number of first-order equations decouple from the remaining second-order ones.

Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids

Generalized Schouten, Frolicher–Nijenhuis and Frolicher–Richardson brackets are defined for an arbitrary Lie algebroid. Tangent and cotangent lifts of Lie algebroids are introduced and discussed and the behaviour of the related graded Lie brackets under these lifts is studied. In the case of the canonical Lie algebroid on the tangent bundle, a new common generalization of the Frolicher–Nijenhuis and the symmetric Schouten brackets, as well as embeddings of the Nijenhuis–Richardson and the Frolicher–Nijenhuis bracket into the Schouten bracket, are obtained.

Tangent lifts of Poisson and related structures

The derivation $d_T$ on the exterior algebra of forms on a manifold $M$ with values in the exterior algebra of forms on the tangent bundle $TM$ is extended to multivector fields. These tangent lifts are studied with applications to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

Tangent and frame bundle harmonic lifts

Tangent and frame bundle harmonic lifts

Tangent Dirac structures

The lift of a closed 2-form Omega on a manifold Q to a closed 2-form on TQ may be achieved by pulling back the canonical symplectic structure on T*Q by the bundle map Omega : TQ to T*Q. It is also known how to lift a Poisson structure on Q to a Poisson structure on TQ. The author calls these lifted structures 'tangent' structures. The notion of a Dirac structure is reviewed. This is a hybrid of Poisson and pre-symplectic structures, which may be thought of as a (singular) foliation of Q by pre-symplectic leaves. The main result of this paper is a single method which achieves the lift to TQ of either a Poisson or a pre-symplectic structure on Q. Natural involution on TTQ is shown to preserve the lift to TTQ of the lifted structure on TQ. The method is then applied to Dirac structures, with the result that a Dirac structure on Q has a tangent lift to TQ generalizing the lifts of Poisson and pre-symplectic structures.