 Student Records
Programme & Unit Catalogues

## MA50039: Differential geometry of curves & surfaces

6
Masters UG & PG (FHEQ level 7)
Semester 1
CW25EX75 Supplementary Assessment: Like-for-like reassessment (where allowed by programme regulations) Description: Aims:
This course will use methods from multi-dimensional analysis to develop the local differential geometry of curves and surfaces in Euclidean space. In this way, the course provides an introduction to an area of active research in mathematics.

Learning Outcomes:
At the end of the course, the students will be able to apply the methods of calculus with confidence to geometrical problems. They will be able to compute the curvatures of curves and surfaces and understand the geometric significance of these quantities. Through their coursework, the students will be able to demonstrate a deep understanding of an area of differential geometry.

Skills:
Analytic skills T/F A; Problem solving T/F A; Written communication F A.

Content:
Topics will be chosen from the following:
* Parametrization, tangent spaces, tangent maps.
* Euclidean motions.
* Curves: length of curves; arc-length; normal fields; curvatures and torsion; normal connection; parallel transport; Frenet curves, Frenet formulae; fundamental theorem; isoperimetric inequality; four-vertex theorem.
* Surfaces: induced metric; conformal parametrization; Gauss map;shape operator; mean, Gauss and principal curvatures; curvature line parametrization; covariant derivative/Levi-Civita connection; Koszul's formulae; curvature tensor; Gauss-Weingarten equations; Gauss-Codazzi equations; Bonnet's theorem.
* Curves on surfaces: geodesics; geodesic curvature; geodesic polar coordinates; geodesics as local length minimizers; Minding's theorem; Clairaut's theorem; normal curvature; Euler's theorem; Meusnier's theorem; asymptotic lines; curvature lines; Rodrigues' equation; Joachimsthal's theorem; integration on srufaces; Gauss-Bonnet theorem.
* Special surfaces: minimal surfaces; surfaces of constant mean or Gauss curvature; ruled surfaces; developable surfaces. Programme availability:

#### MA50039 is Optional on the following programmes:

Department of Mathematical Sciences

NB. Programmes and units are subject to change at any time, in accordance with normal University procedures.