American Journal of Computational Mathematics, 2020, 10, 460-472 https://www.scirp.org/journal/ajcm ISSN Online: 2161-1211 ISSN Print: 2161-1203 DOI: 10.4236/ajcm.2020.103026 Sep. 27, 2020 460 American Journal of Computational Mathematics Operations and Actions of Lie Groups on Manifolds Sharmin Akter 1 , Mir Md. Moheuddin 1* , Saddam Hossain 2 , Asia Khatun 1 1 Department of CSE (Mathematics), Atish Dipankar University of Science and Technology (ADUST), Dhaka, Bangladesh 2 Department of Basic Science (Mathematics), World University of Bangladesh (WUB), Dhaka, Bangladesh Abstract As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this pa- per, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on n  . A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of conti- nuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds. Keywords Group (G), Abelian Group ( 1 2 2 1 gg gg = ), Subgroup (H Is a Subgroup of G), Co-Sets (gH), Lie Groups ( ( ) , G G Gxy xy × → ⋅  and G G → , 1 g g −  ), Smooth Mapping ( : G G G σ × → ) 1. Introduction In the present era, the study of the group related to the Lie group is essential for the sake of its comprehensive applications in several fields. Through the mathe- matical analysis, representations of groups on the manifold are vital due to it al- How to cite this paper: Akter, S., Mo- heuddin, M.Md., Hossain, S. and Khatun, A. (2020) Operations and Actions of Lie Groups on Manifolds. American Journal of Computational Mathematics, 10, 460-472. https://doi.org/10.4236/ajcm.2020.103026 Received: July 2, 2020 Accepted: September 24, 2020 Published: September 27, 2020 Copyright © 2020 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access