A regular Lie group action yield smooth sections of the tangent bundle and relatedness of vector fields, diffeomorphisms

Abstract

In this paper, we have concentrated on a group action on the tangent bundle of some smooth/differentiable manifolds which has been built from a regular Lie group action on such smooth/differentiable manifolds. Interestingly, elements of orbit space yield smooth sections of the tangent bundle having beautiful algebraic properties. Moreover, each of those smooth sections behaves nicely as a left-invariant vector field with respect to Lie group action by $G$. We have explained here a simple isomorphism between the set of such smooth sections and each tangent space of that smooth/differentiable manifold. Also we have discussed more about $F$-relatedness and have introduced vector field relatedness by notations $rel_{\mathfrak{X}(M)}(F), rel_{Diff(M)}(X)$, etc. which are sets based on both vector field related diffeomorphisms and diffeomorphism related vector fields. We have presented consequences based on the algebraic structure on $rel_{\mathfrak{X}(M)}(F), rel_{Diff(M)}(X)$, etc. sets and built some related group actions. We have placed some interrelationship between the both kinds of rel operations.