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While you could, of course, go get a Mathematics textbook on the subject (the famous one by Spivak is 5 thick volumes!), even this might not help because what you really need to know is how the continuous concepts can be applied to discrete things you can compute on (like meshes) - which is a field known as "Discrete Differential Geometry" (DDG). But, its hard to leard DDG until you know normal Differential Geometry.

Fortunately, there are a few short primers written by graphics people for graphics. No you don't have to read the whole list. I recommend you start with a look at the slides from #3 (Ch1 Ch2->Ch2->reader:DDG/06-Slides-2.pdf]]), and reading #1 (here)

1. Garland's notes http://mgarland.org/class/geometry/topics/diffgeom.pdf. These are quite terse and concise, but complete in they start out with the continuous stuff.
2. Garland refers to Meyer et al http://multires.caltech.edu/pubs/diffGeoOps.pdf, a technical paper introducing some of the first rogorous attempts to do discrete differential geometry. This effort then evolved into the SIGGRAPH course.
3. The notes from the discrete differential geom course http://ddg.cs.columbia.edu/
4. The section of the surface parameterization course, gets at the surface ideas from a different direction, his slides are quite nice reader:DDG/DifferentialGeometryPrimer.pdf, although they lose some without the notes/annotations/animations. The actual notes themselves are a bit math heavy, but have nice pictures reader:DDG/ParamCourseNotesCh2.pdf.
5. chapter 5 of the 2007 mesh class (a bit intense, but gets at the differential stuff)
6. Simon R's notes on Differential geometry from the line drawings class (at the beginning) http://www.cs.princeton.edu/gfx/proj/sg08lines/lines-2-diffgeom.pdf
7. Simon R's paper on how he computes curvature is nice since it explains things well http://www.cs.princeton.edu/gfx/pubs/_2004_ECA/index.php
8. Hilbert's famous "Geometry and the Imagination" is a suprisingly accessible text. The chapter on Differential Geometry is reader:DDG/Hilbert_GeomAndImagination_Ch4.pdf

We'll do the readings for Differential Geometry in a "read-behind" fashion. After Monday's lecture, you should try to read something about differential geometry in order to help it all sink in. In order to make sure it sinks in, there will be a "homework assignment" that everyone has to do.

Notes on Differential Geometry

My hand-written notes on Differential Geometry (used to give the lectures in class) are: