Petrunin A., Barrera S.Z. Differential geometry of curves and surfaces: a working approach

Pennsylvania State University, 2018. — 178 p.These notes are based on lectures at MASS program (Mathematics Advanced Study Semesters at Pennsylvania State University) Fall semester 2018. The course is designed for those who plan to do differential geometry in the future, or at least who want to have a solid ground to decide not to do it. The differential geometry of curves and surfaces is a classical subject that is introductory to differential geometry. This subject provides a collection of examples critical for further study, so it does not make sense to do differential geometry until one is a master in curves and surfaces. Differential geometry does geometry on top of several branches of mathematics including real analysis, differential equations, topology and few other branches of geometry, including elementary and convex geometry. The subject of differential geometry is huge, it is easy to imagine two professional differential geometers who can not find a single subject in the field which they are both slightly interested in. These are two reasons why it is hard to study and hard to teach.Curves
Definitions
Length.
Curvature.
Torsion.
Signed curvature.
Supporting curves.
Surfaces
Definitions.
Curvatures.
Curvature estimates.
Saddle surfaces.
Positive Gauss curvature.
Geodesics.
Spherical map.
Parallel transport
Local comparison.
Global comparison.
Review (App. A)
Metric spaces.
Continuity.
Regular values.
Multiple integral.
Initial value problem.
Lipschitz condition.
Uniform continuity.
Jordan’s theorem.
Connectedness.
Convexity.
Elementary geometry.
Triangle inequality for angles.
Semisolutions (App. B)
Bibliography