Differential Geometry

Revision Guide

Author
Affiliation

Dr. Silvio Fanzon

University of Hull

Published

27 Nov 2024

Revision Guide

Revision Guide document for the module Differential Geometry 661955 2024/25 at the University of Hull. If you have any question or find any typo, please email me at

S.Fanzon@hull.ac.uk

Full lenght Lecture Notes of the module available at

silviofanzon.com/2024-Differential-Geometry-Notes

Checklist

You should be comfortable with the following topics/taks:

You should be comfortable with the following topics/tasks:

Curves

  • Regularity of curves

  • Length, arc-length, and arc-length reparametrization

  • Calculating the curvature and torsion of unit speed curves from the definitions

  • Calculating the curvature and torsion of (possibly not unit speed) curves from the formulae

  • Calculating the Frenet frame of a unit-speed curve

  • Applying the Fundamental Theorem of Space Curves to determine if two curves coincide, up to a ridig motion

  • Proving that a curve is contained in a plane, and computing the equation of such plane

  • Proving that a curve is part of a circle

Topology: To be completed

Surfaces:

  • Regularity of surface charts

  • Computing reparametrizations

  • Computing a basis and the equation of the tangent plane

  • Calculating the standard unit normal of a surface chart

  • Calculating the differential of a smooth function between surfaces

  • Proving that a given level surface is regular, and computing its tangent plane

  • Proving that a given surface is ruled

  • Calculating the first fundamental form of a surface chart

  • Proving that a given map is a local isometry / conformal

  • Prove that a given parametrization is conformal

  • Calculating length and angles of curves on surfaces

  • Calculating the second fundamental form of a surface chart

  • Calculating the matrix of the Weingarten map, the principal curvatures, and principal directions of a surface chart

  • Calculating Gaussian and mean curvature of a surface chart

  • Calculating normal and geodesic curvature of a curve on a surface

  • Classifying points of a surface as elliptic, parabolic, hyperbolic, planar