Differential Geometry
Revision Guide
Revision Guide
Revision Guide for the Exam of 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
Full lenght Lecture Notes of the module available at
Recommended revision strategy
Make sure you are very comfortable with:
- The Definitions, Theorems, Proofs, and Examples contained in this Revision Guide
- The Homework questions
- The 2022/23 and 2023/24 Exam Papers questions.
- The Checklist below
Checklist
You should be comfortable with the following topics/taks:
Curves
- Regularity of curves
- Computing the length of a curve
- Computing arc-length function 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 from the definitions
- Calculating the Frenet frame of a (possibly not unit-speed) unit-speed curve from the formulas
- 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:
- Proving that a given collection of sets is a topology
- Proving that a given set is open / closed
- Proving that a given topology is discrete
- Comparing two topologies, and determining which one is finer
- Studying convergent sequences in topological space
- Proving that a given set with a distance function is a metric space
- Studying the topology induced by the metric
- Studying convergent sequences in metric space
- Proving that a topological space is Hausdorff
- Proving that a given function between topological spaces is continuous
- Studying the subspace topology of a given subset of a topological space
- Showing that a given topological space is connected / path-connected
- Proving that two given topological spaces are not homeomorphic, by making use of connectedness arguments
Surfaces:
- Regularity of surface charts
- Computing reparametrizations of surface charts
- Calculating the standard unit normal of a surface chart
- Given a surface chart, compute a basis and the equation of the tangent plane
- 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 vectors of a surface chart
- Calculating Gaussian and mean curvature of a surface chart
- Calculating normal and geodesic curvature of a unit-speed curve on a surface
- Calculating the normal and geodesic curvature of a (possibly not unit-speed) curve on a surface from the formulae
- Classifying surface points as elliptic, parabolic, hyperbolic, planar, umbilical