VB-structures and generalizations

Abstract

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.

arXiv comment46 pages

References in corpus (24)

• Higher vector bundles and multi-graded symplectic manifolds
• Lie algebroids and Poisson-Nijenhuis structures
• Tangent Lifts of Poisson and Related Structures
• Lie algebroid structures on double vector bundles and representation theory of Lie algebroids
• Dirac Algebroids in Lagrangian and Hamiltonian Mechanics
• Algebroids - general differential calculi on vector bundles
• Vector bundles over Lie groupoids and algebroids
• Jacobi structures revisited
• Graded contact manifolds and contact Courant algebroids
• Graded bundles and homogeneity structures
• The graded Jacobi algebras and (co)homology
• VB-groupoids and representation theory of Lie groupoids
• Variational calculus with constraints on general algebroids
• Nonholonomic Constraints: a New Viewpoint
• Linear duals of graded bundles and higher analogues of (Lie) algebroids
• Brackets
• Higher order mechanics on graded bundles
• VB-algebroid morphisms and representations up to homotopy
• Differential graded contact geometry and Jacobi structures
• Duality functors for triple vector bundles
• Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
• Duality functors for $n$-fold vector bundles
• Lifting statistical structures
• New developments in geometric mechanics

Cited by in corpus (6)

• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems
• Various instances of Harish-Chandra pairs
• Lifting statistical structures
• Jacobi algebroids and Jacobi sigma models
• The geometric constraints on Filippov algebroids