In a short preprint The super-Sasaki metric on the antitangent bundle, I explicitly show how to lift a Riemannian metric and an almost symplectic two-form on a manifold \(M\) to a Riemannian metric on the antitangent bundle \(\Pi T M\), which is, of course, a supermanifold.

This example was first given in
Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields, but in The super-Sasaki metric on the antitangent bundle I give more details and deduce some direct results.

In particular, I compare the construction with that of the Sasaki metric [1] on the tangent bundle of a Riemannian manifold. Indeed the construction that I give is really the natural analog of Sasaki’s construction to the setting of antitangent (aka shifted tangent or odd) bundles. Due to the anticommuting nature of the fibre coordinates on \(\Pi T M\), it is clear that directly lifting the metric will not work. One requires an antisymmetric component to the construction and this is provided for by an almost symplectic structure, i.e., a non-degenerate two form that is not necessarily closed.

It is well-known that differential forms on a manifold \(M\) are functions on the antitangent bundle \(\Pi T M\). Furthermore, the de Rham differential, the interior product and the Lie derivative can all be realised as vector fields on the antitangent bundle. In the short preprint, I examine the super-Sasaki metric on these vector fields. We get some interesting formula in this way that related the ‘super-picture’ with the more classical framework of the underlying Riemannian metric and differential forms.

It is worth noting that the classical Sasaki metric plays a role in geometric mechanics. One can equip the configuration space of a Lagrangian system with the Jacobi metric and then, in turn, the tangent bundle of the configuration space naturally comes equipped with the associated Sasaki metric. Trajectories can then be understood as geodesics on the configuration space itself or as geodesic on the tangent bundle of the configuration space. This makes me wonder if the construction of the super-Sasaki metric can play some role in supermechanics.

References
[1] Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds Tohoku Math. J. (2) 10 (1958),338-354.