Differential Geometry and Group Theory

This book offers a comprehensive and self-contained introduction to the geometric and algebraic foundations essential for modern theoretical physics, engineering, and materials science.It bridges two powerful mathematical frameworks-differential geometry and group theory-through a unified perspective centered on spatial transformations and coordinate invariance. The first part develops the tools of differential geometry from the ground up, starting with tensors and their transformation properties in Cartesian and Minkowski spaces, progressing to differential forms, affine connections, and covariant derivatives on curved manifolds.These concepts culminate in the derivation of the geodesic equation and Einstein’s field equations, providing a rigorous pathway to general relativity via the Einstein-Hilbert action. The second part introduces the structure and representation of groups, beginning with the axioms of group theory and advancing through discrete and continuous groups, Lie algebra