# LECTURE NOTES

## 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.