# 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.