{"id":5546,"date":"2020-01-28T09:40:19","date_gmt":"2020-01-28T09:40:19","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=5546"},"modified":"2020-01-28T09:40:19","modified_gmt":"2020-01-28T09:40:19","slug":"construction-of-a-metric-on-the-antitangent-bundle","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2020\/01\/28\/construction-of-a-metric-on-the-antitangent-bundle\/","title":{"rendered":"Construction of a metric on the antitangent bundle"},"content":{"rendered":"\n\n\n
\"board\"<\/td>\nIn a short preprint
\nThe super-Sasaki metric on the antitangent bundle<\/a>, 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. <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

In particular, I compare the construction with that of the Sasaki metric<\/a> [1] on the tangent bundle<\/a> of a Riemannian manifold<\/a>. Indeed the construction that I give is really the natural analog of Sasaki’s construction to the setting of antitangent<\/a> (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<\/a>, i.e., a non-degenerate two form that is not necessarily closed<\/a>. <\/p>\n

It is well-known that differential forms<\/a> on a manifold \\(M\\) are functions on the antitangent bundle \\(\\Pi T M\\). Furthermore, the de Rham differential<\/a>, the interior product<\/a> and the Lie derivative<\/a> can all be realised as vector fields<\/a> 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. <\/p>\n

It is worth noting that the classical Sasaki metric plays a role in geometric mechanics<\/a>. One can equip the configuration space<\/a> of a Lagrangian system<\/a> 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<\/a> 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. <\/p>\n

References<\/b>
\n[1] Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds Tohoku Math. J. (2)<\/em> 10 (1958),338-354.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … Continue reading Construction of a metric on the antitangent bundle<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20,13],"tags":[],"class_list":["post-5546","post","type-post","status-publish","format-standard","hentry","category-post-doc-luxembourg","category-research-work"],"_links":{"self":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5546","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/comments?post=5546"}],"version-history":[{"count":20,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5546\/revisions"}],"predecessor-version":[{"id":5566,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5546\/revisions\/5566"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=5546"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=5546"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=5546"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}