dr bob's elementary differential geometry

by bob jantzen, Department of Mathematics and Statistics, Villanova University
with initial typesetting help from Hans Kuo, Taiwan [2007]

short hype and long title

Original handwritten notes for undergraduate Differential Geometry class Spring 1991.
Serendipitously typeset and expanded by bob for Spring 2008 class.
bob taught this course in Spring 2013, with special relativity incorporated into the mathematical examples and exercises and symmetry groups into the text and exercises.


News.
Working on the problems and problem solutions. Stalled again.

History.
LaTeX conversion done 2007-2008 with the tremendous help from Hans Kuo, without which this book would never exist. Raw latex and first pass polishing edit done while adding more concrete examples used in actually teaching the course Spring 2008. Rereading from start to finish needed to round out what has been done so far.
Stalled by research and teaching and life since June 2008
.
Reader correction November 25, 2008  [after memory stick crashed with a few latest changes made in Rome Summer 2008, then had to reconstruct them!]
Minor correction February 2009 [typo in problem 1.2.2 on p16, thanks, Chris! anybody else out there who can help proof this stuff?]

February 2010. Reorganization of all the section textfiles into chapter and appendix files. A student is doing an independent study so I am rereading the book slowly.

Kepler problem updated October 2011.Fall 2012. Getting ready for teaching this in Spring 2013, adding applications to relativity. Extensive editing in progress Fall 2012. The missing thread on symmetry groups is being woven into the text in various chapters.

Spring semester 2013 (thanks to Cole Johnston VU '14 for pushing me to offer this course), constant editing while teaching from the book. big push spring break the first week of March. Added: Appendix A.3 on curves in space and surface (and spacetime!) to review multivariable calculus. Appendix A.4 on surfaces in space (and spacetime) to review multivariable calculus. New special relativity Appendix A.2 after hyperbolic geometry appendix A.1, with a problem on the Frenet-Serret geometry of a timelike helix in 3-d spacetime in appendix A.3, to introduce immediately students to actual special relativity mathematics based only on their multivariable calculus background.
Black hole embedding diagram and parallel transport along symmetry circles leading to the GP-B geodetic gyro precession  developed to motivate the course. Geodesic deviation discussion illustrated by nearby meridians on surfaces of revolution, and geodesics nearby extremal parallels.

April 2013. Eduard Bachmakov (VU '14)  helps fix hyperref package to make PDF active hyperlinks, etc. Thanks, Eduard!

Fall 2013 And finally bob solves the equation cross-referencing to worked problems, giving each exercise a brief name.

Winter 2014 bob adds some problems dealing with the curvature and geodesic deviation on an ellipsoid of revolution, thanks to Charles Karney, an expert on geodesics on spheroidal models of the Earth.

March 2014 bob adds a section on the geodesics of the Schwarzschild black hole equatorial plane 3d Minkowski spacetime, and circular orbits (helices in the spacetime, comparable to the Lorentz helix in appendix C.


Figure scans of hand drawn figures (temporary?) These need to be replaced either by better hand drawn figures for now, or maple output when possible.

LaTeX files

Figure drawing files (no longer used):

Maple 11-17 .mw files [still to clean up]:


Geodesics on the Torus and other Surfaces of Revolution
Clarified Using Undergraduate Physics Tricks with Bonus:
Nonrelativistic and Relativistic Kepler Problems
(in progress Fall 2009)


Why not extend this to a screw-rotation symmetric helical tube, the shape of cavatappi pasta?

General Relativity, Cosmology and Pasta? a life of USA-Italy academic commuting (Talk April 2012)

"Geodesics on Surfaces with Helical Symmetry: Cavatappi Geometry," (2012).

"Cavatappi 2.0: More of the Same but Better" (2012).