Differential Geometry and Group Theory
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Beschreibung
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 algebras, and representation theory. Special emphasis is placed on physical applications, including quantum mechanics, condensed matter systems, and symmetry principles underlying fundamental laws. Worked examples and character tables are presented in detail, alongside discussions of rotation groups (SO(2), SO(3)), SU(2), and the Lorentz and Poincaré groups.
Designed for graduate students, researchers, and final-year undergraduates, this textbook combines step-by-step derivations, intuitive explanations, and real-world applications, making it an indispensable resource for mastering the mathematical language of modern physics.
An encyclopaedic work that is self-contained and concise—compact, even. A useful collection of all these topics in one place, for novice and professional alike.
— Randall Kamien, University of Pennsylvania
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 algebras, and representation theory. Special emphasis is placed on physical applications, including quantum mechanics, condensed matter systems, and symmetry principles underlying fundamental laws. Worked examples and character tables are presented in detail, alongside discussions of rotation groups (SO(2), SO(3)), SU(2), and the Lorentz and Poincaré groups.
Designed for graduate students, researchers, and final-year undergraduates, this textbook combines step-by-step derivations, intuitive explanations, and real-world applications, making it an indispensable resource for mastering the mathematical language of modern physics.
Provides a unified approach linking differential geometry and group theory for advanced physical sciences Integrates real-world applications in quantum mechanics, condensed matter, and relativity throughout the text Offers a self-contained resource requiring only undergraduate-level calculus and physics as prerequisites
Autor*in
Alessio Zaccone
Themen in »Differential Geometry and Group Theory«
Tensor Analysis in Cartesian Spaces Tensors in Minkowski Space-Time Covariant Electrodynamics Applications Curvilinear Coordinates and Moving Frames Affine Connections on Curved Manifolds Covariant Derivatives in Differential Geometry Geodesic Equation in General Relativity Differential Manifolds and Topology Discrete Rotation Groups in Physics Abelian and Non-Abelian Groups Lie Algebra of Continuous Groups Irreducible Representations in Group Theory Character Tables for Discrete Groups Poincaré Group Representation Theory Lorentz Group and Lie Algebra