bob jantzen
differential geometry, cosmology and relativity notes

if you find these notes useful please contact me!

• Geometry of metric connections (Princeton 1973-1974)
• Bundles and gauge fields (UC Berkeley mid 1970s, Chapel Hill 1979)
   
• Spatially Homogeneous Cosmology:
  Background and Dynamics (UC Berkeley 1976)
   
• Rome theoretical physics lecture notes (1980s-1990s): GR and Cosmology
   
• Villanova University Differential Geometry Course/Independent Study Notes
  including 450 page book on Differential Geometry for Undergraduates
   
• Gravitoelectromagnetism: the User Manual [typeset]
   
• Lichnerowicz -de Rham Laplacian of a 3-form calculation

Over the years I wrote up (printing neatly by hand) a lot of notes of my self-study or (post)-lecture notes on differential geometry, relativity and gravitation, cosmology, gauge fields, etc. After 3 decades it occurred to me that maybe someone else might find something useful in them, which would not occur while they were sitting in a file cabinet. So in May-June 2006 I scanned many of them and put them on-line.

Geometry of metric connections and GR, differential forms,
variational approach to Einstein's equations

• Princeton Undergraduate Self Taught notes 1973-1974

Bundles, Kaluza-Klein and gauge fields (UC Berkeley mid 1970s, Chapel Hill 1979)

• See index.htm

Rome theoretical physics lecture notes: GR and Cosmology

These notes were written for the students in the course of theoretical physics of Remo Ruffini at the Department of Physics of the University of Rome "la Sapienza" during my many visits (lectures given in January, a week in March and May-June, irregularly) in the 1980s and 1990s, including a series of notes for the PhD course (in which Juergen Renn was a student) that were incorporated into ICM below. Most of the notes discuss cosmological models from a mathematical point of view, and some various aspects of perturbations and related topics, including some basic tools of general relativity, starting out with a little special relativity in the beginning.

• Special relativity notes (January, 1983?)
  (11 pages)
• Cosmological models: four lectures (January 1984)
  (35 pages)
• Cosmological models:  a short course (too short) (Spring 1985)
  (31 pages: to do)
• Lecture notes on harmonics (May-June 1986)
  (14 pages) Lifshitz 1
• Notes on the Lifshitz perturbation analysis (May 1984)
  (24 pages) Lifshitz 2
• Symmetry breaking in cosmology (May 1987)
  (SB:  40 pages) Lifshitz 3
•  
• Appreciating the article "General Relativity and Kinetic Theory" by Juergen Ehlers (May 1989)
  (42 pages)
• Introduction to Cosmological Models (March 1987)
  (38 pages) incorporated into ICM1 the next year
•  
• Some notes on Groups, Symmetries, Differential Geometry, and Splitting Techniques in the Context of General Relativity
  July 1995 [bundling of previous years notes]
  
  • Rough notes written on a moving train (March 1990):
    A Second Look at Tensors, Covariant Differentiation and Curvature
    (SL: 12 pages)
  • Curvilinear Coordinates and Curvature (January 1989)
    (CCC: 4 pages)
  • Introduction to Cosmological Models (ICM)
    
    • Part I: orthogonal coordinates on flat or constant curvature manifolds with metric, FRW geometry, simplest spacetime splittings, Gaussian normal coords, intrinsic and extrinsic curvature (March 1987, January 1988)
      (ICM1: 39 pages)
    • Part II: Symmetries and Lie groups (March 1988)
      (ICM2: 27 pages)
    • Part III: Differential geometry, classical mechanics, matrix groups, rigid body dynamics
      (ICM3: 19 pages)
    • Part IV: Fiber bundles, gauge groups (May 1988)
      (ICM4: 20 pages)
    • Part V: Anisotropic cosmological models, gravitational dynamics
      (ICM5: 6 pages)
  • Lie derivatives yet again (March 1991)
    (LD: 4 pages)
  • Spacetime splitting: Some notes on splitting techniques in general relativity (June 1994)
    (SS: 14 pages) see below
  • Symmetry breaking in cosmology (May 1987)
    (SB:  40 pages) see above
• Some notes on splitting techniques in general relativity (June 1994)
  (18 pages, plus some background notes)
  Continued (June 1996)
  (SS: 9 pages)
• Spacetime splitting '96: tensor harmonics and gravitoelectromagnetism
  (SS96: 14 pages)
  These were bundled with ICM1,2, SS1994, SB1987 for June 1996.

Villanova University Differential Geometry Course/Independent Study Notes

• Modern Mathematics and Physics, Spring 1984 [150 pages]
• MAT5600 Differential Geometry Rough Notes, Spring 1989 [?? pages]
• Differential Geometry Notes based on Undergraduate Linear Algebra and Multivariable Calculus, Spring 1991 [296 hand written pages, expanded and typeset here to nearly 500 pages PDF 2007-2009, after someone found them useful and roughly typeset them in LaTeX, giving me the start I never would have had otherwise to create such a document]
• Action of a Group on a Space, Fall 2001 [17 pages]
• Geodesics on the Torus 2009-2010, [50 some pages]; expanded to geodesics on the tubular helical surface of the cavatappi pasta shape, with screw symmetry rather than rotational symmetry.

Villanova University Differential Equations with Linear Algebra Notes

These don't fit into the topics of the rest of this page, but lie in the foundation so...

• dewla