Geodesics on a supermanifold and projective equivalence of super connections
Autor: | Gijs M. Tuynman, Thomas Leuther, Fabian Radoux |
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Jazyk: | angličtina |
Rok vydání: | 2012 |
Předmět: |
Tangent bundle
Mathematics - Differential Geometry Pure mathematics Geodesic Connection (vector bundle) Geodesic map Mathematical analysis General Physics and Astronomy Differential Geometry (math.DG) Tensor (intrinsic definition) Supermanifold FOS: Mathematics Vector field Geometry and Topology Mathematics::Differential Geometry Equivalence (measure theory) 58A50 53B10 53C22 Mathematical Physics Mathematics |
Popis: | We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field associated with the connection. Our (super) geodesics possess the same properties as the in the classical case: there exists a unique (super) geodesic satisfying a given initial condition and when the connection is metric, our supergeodesics coincide with the trajectories of a free particle with unit mass. Moreover, using our definition, we are able to establish Weyl's characterization of projective equivalence in the super context: two torsion-free (super) connections define the same geodesics (up to reparametrizations) if and only if their difference tensor can be expressed by means of a (smooth, even, super) 1-form. 20 pages |
Databáze: | OpenAIRE |
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