University of California at San Diego

Differential Geometry - by J. Fillmore

Math150A - Winter 2001 - is divided into fourteen "lectures" with exercises.

     Course information, references, outline, historical
  0  Introduction - The theorems of Gauss-Bonnet and Poincaré-Hopf
  1  Curves in space - Local theory
  2  Curves in space - Some global theorems
     Ref: R.A. Horn, On Fenchel's theorem, Amer. Math. Monthly 78 1971 380-381, MR0284918
  3  Surfaces in space - First fundamental form
  4a Surfaces in space - Second fundamental form
  4b Example - Surfaces of revolution
  5  Surfaces in space - The Gauss map
  6a Surfaces in space - Geodesics
  6b Surfaces in space - Intrinsic derivatives
  6c Example - Surfaces of revolution
  7  Abstract surfaces - Local notions
  8  Abstract surfaces - Global notions
  9  Abstract surfaces - Parallel transport
 10  Abstract surfaces - Theorem of Gauss and Bonnet
     Additional problems
     Additional problems from Spring 1971 course
     Problem set for "final exam"

Additions to Lectures as numbered
  6  Surfaces in space - Compatibility conditions: Weingarten and Codazzi-Mainardi
  6  Surfaces in space - Special parameter curves
  6  Surfaces in space - Liebmann's Theorem
  7  Abstract surfaces - Shortest arcs, Whitehead's Theorem
  7  Abstract surfaces - Geodesic polar coördinates, the exponential map
  9  Abstract surfaces - details of calculation of Lecture 9, p.6 ff.

Two additional lectures from an earlier course
 13a Abstract surfaces - Vector fields on surfaces
 13b Abstract surfaces - Theorem of Poincaré and Hopf
 14a Non-euclidean - Non-euclidean geometries
 14b Non-euclidean - Models of the Lobachevsky plane
 14c Non-euclidean - The Poincaré model
 14d Non-euclidean - Trigonometry

Additional references

     Calculating curvature and torsion with arbitrary parameterizations
     Weingarten's equations
     Principal directions and Rodrigues's formula
     Normal curvature and Meusner's theorem
     Gauss curvature in orthogonal and geodesic polar coördinates
     Jacobi's equation
     Geodesics on a sphere from the differential equations - fragments
          - see Lecture 6c Problem 6.12
     Exercise on conformal transformations and stereographic projection