This module is not currently running in 2024 to 2025.
Differential geometry studies geometrical objects using analytical methods. It originates in classical mechanics. Modern differential geometry has made a huge impact in the development of nonlinear mathematical physics including integrable systems and string theory. Nowadays differential geometry is at the centre of the analysis of pattern recognition, image processing and computer graphics.
Indicative specific subtopics are:
• Theory of curves. Plane and space curves. Euclidean invariants of curves. Frenet frame.
• Theory of surfaces. Metrics on regular surface. Curvature of a curve on a surface. Gaussian curvature and mean curvature. Covariant derivative and geodesics. The Euler-Lagrange equations. Minimal surfaces.
• Evolution of curves and surfaces as integrable systems: Invariant curve evolution. The mean curvature flows. The connection with integrable systems. The modified Korteweg de-Vries equation.
• Curves in Riemannian manifolds: Riemannian metrics, connections, curvatures and geodesics. Curves evolution in Riemannian manifold with constant curvature.
• Modern applications.
i. 2D and 3D projective geometry and application to multiple view geometry in computer vision;
ii. Moving frames, invariant signatures in pattern recognition;
iii. Poisson manifold and Hamiltonian systems.
42-48 hours.
80% Examination, 20% Coursework
R Hartley & A Zisserman, Multiple View geometry in computer vision. (Cambridge university press, 2nd ed, 2003) (B)
R Kimmel, Numerical geometry of images, theory, algorithms and applications. (Springer Verlag, 2003) (B)
PJ Olver, Lectures on moving frames. (preprint, University of Minnesota, 2008) (B)
C Rogers & WK Schief, Bäcklund and Darboux transformations: Geometry and modern
applications in soliton theory. (Cambridge University Press, 2002) (B)
IA Taimanov, Lecture on differential geometry. (EMS series of Lectures in Mathematics, 2008) (R)
See the library reading list for this module (Canterbury)
The Intended Subject Specific Learning Outcomes. On successful completion of this module students will:
(i) understand basic geometric objects such as curves and surfaces and be able to determine their intrinsic properties
(ii) be able to derive the geometric evolution equations for curves and surfaces and understand the connection with nonlinear integrable systems
(iii) have broadened their experience with the basic concepts in Riemannian geometry such as metrics, connections and curvatures
(iv) have developed awareness of modern applications to mathematical physics, computer vision and image processing
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