Mathematics – Differential Geometry
Scientific paper
2011-12-04
Mathematics
Differential Geometry
40 pages, minor corrections
Scientific paper
It is developed a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal GL(1,R)-bundles. Gradings compatible with the GL(1,R)-action lead to the concept of a graded contact manifold, in particular a linear contact structure. Linear contact structures are proven to be exactly the canonical contact structures on first jets of line bundles. They give rise to linear Kirillov (or Jacobi) brackets and the concept of a principal Lie algebroid, a contact analog of a Lie algebroid. The corresponding cohomology operator is represented not by a vector field (a de Rham derivative) but a first-order differential operator. It is shown that one can view Kirillov or Jacobi brackets as homological Hamiltonians on linear contact manifolds. Contact manifolds of degree 2 are also studied as well as contact analogs of Courant algebroids. Lifting procedures to tangent and cotangent bundles are described and provide constructions of canonical examples of the structures in question.
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