Open AccessA regular Lie group action yield smooth sections of the tangent bundle and relatedness of vector fields, diffeomorphisms
Author(s) -
Chidanand Badiger,
AUTHOR_ID,
T. G. Venkatesh,
AUTHOR_ID
Publication year - 2021
Publication title -
bulletin of the "transilvania" university of braşov. series iii, mathematics and computer science
Language(s) - English
Resource type - Journals
eISSN - 2810-2037
pISSN - 2810-2029
DOI - 10.31926/but.mif.2021.1.63.1.4
Subject(s) - tangent bundle , mathematics , lie group , vector field , diffeomorphism , vector bundle , pure mathematics , fundamental vector field , group (periodic table) , lie algebra , differentiable function , tangent , tangent space , mathematical analysis , geometry , physics , adjoint representation of a lie algebra , lie conformal algebra , quantum mechanics
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.