Differential Geometry and Topology of Curves and Surfaces (Math 497C)
Spring 2004, Penn State
- Lecture Notes 0
Basics of Euclidean Geometry, Cauchy-Schwarz inequality.
- Lecture Notes 1
Definition of curves, examples, reparametrizations, length, Cauchy's integral formula, curves of constant width.
- Lecture Notes 2
Isometries of Euclidean space, formulas for curvature of smooth regular curves.
- Lecture Notes 3
General definition of curvature using polygonal approximations (Fox-Milnor's theorem).
- Lecture Notes 4
Curves of constant curvature, the principal normal, signed curvature, turning angle,
Hopf's theorem on winding number,
fundamental theorem for planar curves.
- Lecture Notes 5
Osculating circle, total curvature, convex curves.
- Lecture Notes 6
The four vertex theorem, isoperimetric inequality.
- Lecture Notes 7
Torsion, Frenet-Seret frame, helices, spherical curves.
- Lecture Notes 8
Definition of surface, differential map.
- Lecture Notes 9
Gaussian curvature, Gauss map, shape operator, coefficients of the first
and second fundamental forms, curvature of graphs.
- Lecture Notes 10
Interpretations of Gaussian curvature as a measure of local convexity,
ratio of areas, and products of principal curvatures.
- Lecture Notes 11
Intrinsic metric and isometries of surfaces, Gauss's Theorema Egregium,
Brioschi's formula for Gaussian curvature.
- Lecture Notes 12
Gauss's formulas, Christoffel symbols, Gauss and Codazzi-Mainardi equations,
Riemann curvature tensor, and a second proof of Gauss's Theorema Egregium.
- Lecture Notes 13
The covariant derivative and Lie bracket; Riemann curvature tensor and Gauss's formulas
revisited in index free notation.
- Lecture Notes 14
The induced Lie bracket on surfaces. Self adjointness of the shape operator, Riemann curvature tensor of surfaces, Gauss and Codazzi
Mainardi equations, and Theorema Egregium revisited.
- Lecture Notes 15
The definition of geodesic curvature, and the proof that it is intrinsic.
- Lecture Notes 16
Applications of the Gauss-Bonnet theorem.