The Geometry of Physics

Preface Part I. Manifolds, Tensors and Exterior Forms: 1. Manifolds and vector fields 2. Tensors and exterior forms 3. Integration of differential forms 4. The Lie derivative 5. The Poincare Lemma and potentials 6. Holonomic and non-holonomic constraints Part II. Geometry and Topology: 7. R3 and Minkowski space 8. The geometry of surfaces in R3 9. Covariant differentiation and curvature 10. Geodesics 11. Relativity, tensors, and curvature 12. Curvature and topology: Synge's theorem 13. Betti numbers and De Rham's theorem 14. Harmonic forms Part III. Lie Groups, Bundles and Chern Forms: 15. Lie groups 16. Vector bundles in geometry and physics 17. Fiber bundles, Gauss-Bonnet, and topological quantization 18. Connections and associated bundles 19. The Dirac equation 20. Yang-Mills fields 21. Betti numbers and covering spaces 22. Chern forms and homotopy groups Appendix: forms in continuum mechanics.

[1]  H. Weyl.  The Theory Of Groups And Quantum Mechanics , 1931 .

[2]  B. Eckmann.  Harmonische Funktionen und Randwertaufgaben in einem Komplex , 1944 .

[3]  H. Samelson,et al.  Topology of Lie groups , 1952 .

[4]  G. Duff,et al.  HARMONIC TENSORS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY , 1952 .

[5]  Edmund Taylor Whittaker.  Reviews of Books and Recordings: A History of the Theories of Aether and Electricity , 1954 .

[6]  I. N. Sneddon,et al.  Finite Deformation of an Elastic Solid , 1954 .

[7]  K. O. Friedrichs,et al.  Differential forms on riemannian manifolds , 1955 .

[8]  K. Nomizu.  Lie groups and differential geometry , 1956 .

[9]  L. Ryder,et al.  Quantum Field Theory , 2001, Foundations of Modern Physics.

[10]  Shôshichi Kobayashi.  Fixed Points of Isometries , 1958, Nagoya Mathematical Journal.

[11]  C. Truesdell,et al.  The Classical Field Theories , 1960 .

[12]  Richard Phillips Feynman,et al.  The Theory Of Fundamental Processes , 1961 .

[13]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[14]  J. Milnor.  Topology from the differentiable viewpoint , 1965 .

[15]  Theodore Frankel,et al.  Critical Submanifolds of the Classical Groups and Stiefel Manifolds , 1965 .

[16]  L. Brillouin,et al.  Tensors in Mechanics and Elasticity , 1965 .

[17]  R. Courant,et al.  Introduction to Calculus and Analysis , 1991 .

[18]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[19]  L. Susskind,et al.  OBSERVABILITY OF THE SIGN CHANGE OF SPINORS UNDER 2$pi$ ROTATIONS. , 1967 .

[20]  H. K. Moffatt,et al.  The degree of knottedness of tangled vortex lines , 1969, Journal of Fluid Mechanics.

[21]  R. Hermann.  Differential geometry and the calculus of variations , 1970 .

[22]  F. W. Warner.  Foundations of Differentiable Manifolds and Lie Groups , 1971 .

[23]  J. Boyling.  An axiomatic approach to classical thermodynamics , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[24]  T. Lasinski,et al.  Strongly interacting particles , 1973 .

[25]  V. Arnold,et al.  Ordinary Differential Equations , 1973 .

[26]  N. Grossman.  Holonomic measurables in geodesy , 1974 .

[27]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[28]  J. Sniatycki.  Prequantization of charge , 1974 .

[29]  H. Lawson.  Minimal varieties in real and complex geometry , 1974 .

[30]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[31]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[32]  G. Galloway.  A generalization of Myers' theorem and an application to relativistic cosmology , 1979 .

[33]  D. Raine.  General relativity , 1980, Nature.

[34]  Louis Michel,et al.  SYMMETRY DEFECTS AND BROKEN SYMMETRY. CONFIGURATIONS - HIDDEN SYMMETRY , 1980 .

[35]  D. Bleecker,et al.  Gauge theory and variational principles , 1981 .

[36]  Geometry, particles and fields , 1983 .

[37]  Loring W. Tu,et al.  Differential forms in algebraic topology , 1982, Graduate texts in mathematics.

[38]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[39]  R. Bott.  Lectures on Morse theory, old and new , 1982 .

[40]  C. Truesdell.  The influence of elasticity on analysis: The classic heritage , 1983 .

[41]  Barry Simon,et al.  Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase , 1983 .

[42]  R. Abraham,et al.  Manifolds, Tensor Analysis, and Applications , 1983 .

[43]  Siddhartha Sen,et al.  Topology and geometry for physicists , 1983 .

[44]  M. Berry.  Quantal phase factors accompanying adiabatic changes , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[45]  S. Coleman,et al.  Aspects of Symmetry , 1985 .

[46]  Richard Phillips Feynman,et al.  Elementary Particles and the Laws of Physics , 1987 .

[47]  J. Roe.  Elliptic Operators, Topology and Asymptotic Methods , 1988 .

[48]  On Induced Representations , 1988 .

[49]  P. Bamberg,et al.  A course in mathematics for students of physics , 1990 .

[50]  C. N. Yang and contemporary mathematics , 1993 .

[51]  On the growth of waves on manifolds , 1993 .

[52]  Introduction to quantum field theory for mathematicians , 1995 .

[53]  R. Osserman.  Poetry of the Universe , 1995 .

[54]  S. Weinberg.  The Quantum Theory of Fields: THE CLUSTER DECOMPOSITION PRINCIPLE , 1995 .

[55]  Gerard 't Hooft.  In Search of the Ultimate Building Blocks , 1996 .

[56]  W. Hsiang,et al.  Lectures on Lie Groups , 1998 .

[57]  R. Ho.  Algebraic Topology , 2022 .