This module is not currently running in 2024 to 2025.
The main aim is to give an introduction to the basics of differential geometry, keeping in mind the recent application in mathematical physics and the analysis of pattern recognition.
The synopsis may include:
(a) Theory of curves: Regular plane and space curves. Tangent vectors. Arclength parameterisation. Curvature and Euclidean invariants. The Frenet formula.
(b) Geometry of surfaces: Regular parameterised surface. The tangent plane. Curvature of a curve on a surface. First and second fundament forms. Shape operator. Gaussian curvature and mean curvature.
(c) Geodesics and Minimal surfaces: The Christoffel symbols. Geodesics. The Euler-Lagrange equations. The Gauss-Bonnet Theorem. Minimal surfaces.
Possible other topics may include: Evolution of curves and surfaces as integrable systems: Invariant curve evolution. The mean curvature flow. Riemannian metrics, connections, curvatures and geodesics.
In addition, for M-level students, the connection with integrable systems; curves evolution in Riemannian manifolds with constant curvature and Moving frames.
42-48 lectures and example classes
80% Examination, 20% Coursework
V A Toponogov, "Differential Geometry of Curves and Surfaces A Concise Guide'’, Birkha¨user Boston, 2006 (E-book available in the Templeman Library).
M P Carmo, “Differential Geometry of Curves and Surfaces A Concise Guide’’, Prentice-Hall, 1976.
V Y Rovenskii, “Geometry of Curves and Surfaces with MAPLE’’, Birkha¨user Boston, 2000.
See the library reading list for this module (Canterbury)
On successful completion of this module, students will:
a) understand basic geometric objects such as curves and surfaces;
b) be able to construct and manipulate the Frenet frames for plane and space curves;
c) be able to analyse surfaces in three-dimensional space by calculating various quantities, e.g., the first and second fundamental forms, Gauss curvatures and mean curvatures;
d) gain an understanding of basic geometric concepts such as geodesics and minimal surfaces.
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