Introduction to Lie Groups and Differential Geometry

Lie groups sit at the elegant intersection of algebra and geometry, serving as a vital framework for modern physics, robotics, and advanced mathematics. For many learners, however, the transition from standard calculus to the abstract language of manifolds and continuous symmetry groups can feel overwhelming. This course demystifies these advanced concepts, building your understanding from the ground up without requiring years of prior graduate study. You will transition from basic algebraic structures to a confident grasp of how continuous groups act on geometric spaces. By reading through detailed proofs, concrete matrix examples, and structured mathematical expositions, you will develop a deep intuition for both the algebraic and geometric sides of the theory. What you'll learn: - Understand the foundational definitions of topological groups, smooth manifolds, and Lie groups - Calculate Lie algebras from Lie groups using matrix exponentials and vector fields - Explore the relationship between Lie group homomorphisms and Lie algebra representations - Analyze the geometric properties of invariant metrics and connection forms on groups - Apply the concept of one-parameter subgroups to solve geometric flow problems - Examine how symmetric spaces and Riemannian geometry interface with group actions This course begins with a thorough introduction to essential terminology, including topological groups, smooth manifolds, and basic tangent spaces, before progressing to Lie algebras, exponential maps, and the geometry of homogeneous spaces. You will progress naturally from abstract definitions to concrete matrix examples and geometric applications. This course is designed for advanced undergraduate students, beginning graduate students, and self-directed learners in mathematics, physics, or engineering who want a solid, accessible introduction to Lie groups. No prior knowledge of differential geometry is required, though familiarity with linear algebra and multivariable calculus is highly recommended. Begin your journey into the mathematics of symmetry and geometry today.