A Brief Review of Differentiable Manifolds and Their Applications
Creators: name: Gyanvendra Pratap Singh, Pragya Mishra
Publisher: name: Zenodo
Year: 2024

description: The main idea of this review paper is to make understandable the idea of differentiable manifolds using Lie groups . 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 paper, 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 eigen functions of differential operators on n . A Lie group is a group as well as differ-entiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct.This paper presents an overview of differentiable manifolds, focusing on their definition, prop-erties, and significance. Beginning with the basic definitions of smoothness and charts, we explore the construction of differentiable structures on topological spaces. We delve into tangent spaces, differential forms, and vector fields, elucidating their role in defining smooth functions and mappings between manifolds. Additionally, we discuss important theorems such as the inverse function theorem and Stokes’ theorem, showcasing their relevance in manifold theory. Finally, we touch upon applications in geometric model-ing, general relativity, and gauge theory, underscoring the manifold’s importance across disciplines. 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 continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical phys-ics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.Differentiable manifolds, a fundamental concept in differential geometry, find widespread applications across various fields of mathematics, physics, and engineering. This abstract provides a concise overview of some notable applications of differentiable manifolds.

As in theoretical physics, differentiable manifolds serve as the mathematical frame-work for formulating theories such as general relativity. The spacetime continuum is modeled as a four-dimensional differentiable manifold, where curvature encodes gravita-tional effects. This application has profound implications in understanding the dynamics of celestial bodies, black holes, and the structure of the universe. In robotics and control theory ifferentiable manifolds play a crucial role in modeling the configuration spaces of robotic systems. By representing the possible configurations of robot joints as points on a manifold, researchers can develop efficient algorithms for motion planning, trajectory optimization, and control synthesis. These applications find practical use in robotic ma-nipulation, autonomous vehicles, and industrial automation. In machine learning and data analysis manifold learning techniques aim to uncover the underlying structure of high-dimensional data by approximating it as a lower-dimensional manifold embedded in the ambient space. Algorithms such as t-SNE (t-distributed Stochastic Neighbor Em-bedding) and Isomap leverage differential geometry principles to perform dimensionality reduction and visualize complex datasets. These methods facilitate exploratory data ana-lysis, pattern recognition, and feature extraction tasks in machine learning.