dr bob's elementary differential geometry

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

short hype and long title

current book PDF:  dgn2025-11-15.pdf [revisiting this daily, thanks Dan!]
[last secure copy dgn2021-05-25.pdf]  [dgn = Differential Geometry Notes]

• table of contents and preface [200K]
• chapter 0 on index notation in matrix form for up to 2 index objects

History [reverse chronological order]

• Fall 2025, after a long hiatus that morphed into retirement, bob starts rereading his own book to see what might become of it. Finally forced to move the project to Overleaf since even though several laptops ago TexLive was somehow successfully installed, now bob is just too old to hassle with it. Overleaf is efficient and painless and an amazing collaborator! PCTeX still works for less sophisticated documents. While reading, finding some little editing glitches, needing slight rewording. Overleaf reporting multiple definitions of some cross-referencing. Big improvements being made inspired by the interest of a graduate student in ME.
  
  
• Spring 2021, finally TexLive comes to the rescue (thanks texhax!) to solve the PDF navigation issue.
  Maybe bob can play more with the exercise solutions now. 836 pages.
  
• Fall 2019, my version of TeXworks or TexnicCenter with MikTex no longer produces the hyperlinked table of contents in the exported PDF nor the links back to the original exercises in the main body from their solutions in an appendix. 
  
• Fall 2017, an independent study not using this book awakened interest in it again, so some editing changes are in the works.
  (version: Feb 1, 2015, it has grown to about 750 pages plus 50 some pages in an incomplete solution manual [now I am working on that], about 10 MB altogether)
  
• 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.
• 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.
• Fall 2013 And finally bob solves the equation cross-referencing to worked problems, giving each exercise a brief name.
• Spring 2013: MAT5600 is over, editing stopped for summer travel.
• April 2013. Eduard Bachmakov (VU '14)  helps fix hyperref package to make PDF active hyperlinks, etc. Thanks, Eduard!
• 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.
• bob taught this course MAT5600 in Spring 2013, with special relativity incorporated into the mathematical examples and exercises and symmetry groups into the text and exercises.
• 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.
• October 2011. Kepler problem updated.
• 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.
• February 2009. Minor correction  [typo in problem 1.2.2 on p16, thanks, Chris! anybody else out there who can help proof this stuff?]
• 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!]
• Spring 2008. First class taught with PDF text.
• LaTeX conversion done 2007-2008 with the tremendous help from Hans Kuo who found the handwritten notes online and converted them to a LaTeX document, 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.
• Original handwritten notes for undergraduate Differential Geometry class Spring 1991.
  Serendipitously typeset and expanded by bob for Spring 2008 class.

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.

• dgnfigps.zip
  .ps figure scan files: scan0001-scan0032 (part 1), scan0040-scan0115 (part 2)
• dgnfigjpg.zip
  .jpg figure scan files: scan0001-scan0032 (part 1), scan0040-scan0115 (part 2) [for windows previewing thumbnails]

LaTeX files

• dgn.tex LaTeX main file [dgntexfiles]

Figure drawing files (no longer used):

• dgnfigdrp.zip LaTeX picture environment coding  (no longer necessary, incorporated into latex chapter files)
  [limited, circles not of arbitrary radius, so PicTeX preferrable in some cases]
• dgnfigmp.zip metapost coding to produce eps output (no longer necessary)
• Maple plot exports [this seems to be the easiest solution for me, but the filesize is way to big in the Maple EPS output]:
  [dgnepsfiles.zip, dgnepsfiles2013.zip]

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

• maple/index.htm

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)

• see its own web page

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).