On curves and jets of curves and on supermanifolds

I placed a preprint on the arXiv entitled “On curves and jets of curves and on supermanifolds” [1]. The abstract can be found below

In this note we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature.

Quick overview
The k-th order tangent bundle on a classical manifold can be defined as equivalence classes of curves at the points of the manifold that agree up to velocity, acceleration, rate of change of acceleration on so on up to k-th order. The tangent bundle which is much more widely known, is the “1st order tangent bundle”. These higher order tangent bundles have found applications in higher derivative Lagrangian mechanics for example and appear in the work of Grabowski & Rotkiewicz [2].

The main question posed and answered in this preprint is “can we define the k-th order tangent bundles of a supermanifold is such a kinematic way?”

The problems with simply generalising the classical notions directly are two fold:

  1. Supermanifolds do not consist solely of topological point, thus point-wise constructions need some careful handling.
  2. The naive definition of a curve as a morphism of supermanifolds \(\mathbb{R} \rightarrow M \) totally misses the odd dimensions of the supermanifold \(M\).

To resolve these issues we look at a “superised” version of curves defined via the internal Homs, a notion from category theory. In sort we consider curves that are paramaterised by all supermanifolds, however all the constructions should be functorial in this parametrisation. This allows us to define what I call S-curves that can “feel” both the even and odd dimensions of a supermanifold. Moreover, this allows us to think in terms of S-points and parallel the classical constructions of jets of curves rather closely.

This allow us to define the k-th order tangent bundle of a supermanifold at first as a generalised supermanifold (a functor from the opposite category of supermanifolds to sets ) and then we show that this is representable, that is a genuine supermanifold. Moreover, locally everything looks the same as the classical higher order tangent bundles.

[1] Andrew James Bruce, On curves and jets of curves and on supermanifolds, arXiv:1401.5267 [math-ph].
[2] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. geom. Phys. 62 (2012), 21-36.

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