Edinburgh Mathematics Programme
Level 3 Module
311 DGe Differential Geometry (S2)
Introduction
This course develops the traditional mathematics of curves and
surfaces in a contemporary way using the calculus of differential
forms. We also study integration of differential forms and Stokes'
Theorem.
Aims
- To gain familiarity with the calculus of differential forms.
- To understand the classical quantities describing the shape of
curves and surfaces.
- To understand the Gauss-Bonnet theorem.
Prerequisites
- MS0225 LAG
- MS0097 SVC
Syllabus summary
Plane and space curves, the Frenet-Serret equations.
Differential forms on R^n. Maps to Euclidean space and the
structure equations. Application to surfaces.
First and second fundamental forms, mean and Gauss curvature.
The Theorem Egregium. Geodesics. Integration of forms and Stokes'
theorem. Euler characteristic and the Gauss-Bonnet Theorem.
Texts
There are comprehensive course notes covering the theory.
Syllabus
- Curves in R^n, tangent vector, regularity. Reparametrisation.
Arc-length and arc-length parametrisation.
(1.5)
- Analysis of curves in E^3. Curvature.
The Frenet frame of a biregular curve
and the Frenet-Serret equations.
Formulae for curvature and torsion.
The equivalence problem and its solution.
The case of E^2 discussed briefly. (3)
- Vectors as derivative operators.
Differential forms in R^n. Wedge product. Exterior
derivative, $d^2=0$.
Forms as alternating multilinear functions of vectors.
The point that differential forms are a
"coordinate independent calculus" to be emphasised, but without
labouring proofs. (Pull-back of forms by a map treated very lightly.)
Connections with Vector Calculus (not examinable).
(4.5)
- Euclidean space E^n as R^n with the standard inner product.
Analysis of a map R^m -> E^n in terms of a "moving frame"
e_1,...,e_n. The associated coframe field. Connection forms and the
first and second structure equations. (1.5)
- Surfaces in R^3, regularity. Parametrisation and
reparametrisation. Normals and orientation. (1.5)
- Interior multiplication of 1-forms to obtain symmetric bilinear
forms. The first and second fundamental forms of a surface.
Principal curvatures (as eigenvalues of II relative to I),
Gauss and mean curvature. Computational formulae for a parametrised
surface. (2)
- Adapted frames for a surface. The structure equations.
Freedom in adapted frame. Gauss and Codazzi equations.
Fundamental forms in these terms. (1)
- Results on surfaces: Taylor expansion as z=f(x,y) in suitable
coordinates, Euler's Theorem, SFF=0 implies plane,
everywhere umbilic implies sphere. (2)
- Computation of Gauss curvature by solving structure equations.
The idea of isometric surfaces. Proof of Theorem Egregium.
Hyperbolic space and its curvature ("abstract
Riemannian geometry" treated very lightly).
(1)
- Geodesics on a surface. The geodesic equations. Analysis for a
surface of revolution and hyperbolic space (2)
- Integration of differential forms over regions in R^3, curves and
surfaces (details not laboured). Statement of general Stokes' theorem
and
its relation to vector calculus. Integration of a function on a
surface via the area form. (2)
- Stationary area if and only if H=0. For a geodesic triangle
A+B+C-pi = integral of K. Euler characteristic, and statement and
sketched proof of the Gauss-Bonnet Theorem. (2)
Notes and Links
- Differential forms are primarily introduced as a calculational
tool, with the idea that they constitute a
"coordinate independent calculus" discussed but not laboured.
- A link should be made between geodesics and the general
calculus of variations.
- The syllabus takes a very particular route through this material
so as to arrive at the Gauss-Bonnet theorem, which is usually not
covered at this level. This is achieved by treating some
theoretical aspects lightly and taking the view always that a local
surface is an R^3-valued function on a domain in R^2.
Outcomes
- An ability to perform simple manipulations with forms; being able
to relate these to the standard differential formulae of 3 dimensions (grad,
div, curl) if these have been covered in other courses.
- An ability to manipulate the Frenet-Serret equations and calculate
curvature and torsion.
- An understanding of the fundamental forms of a surface (I, II) and
its principal curvatures. An ability to compute simple examples.
- An understanding of geodesics, and ability to calculate the geodesic
equations for a surface of revolution.
- Ability to translate the "general Stokes' theorem" into the examples
of vector calculus.
TNB 20 Jan 2002. (Minor amendments AC 1 Feb 2002.)