F10PG - Geometry

Course leader(s):

• Anatoly Konechny (Edinburgh, January)

Syllabus

1.1 Definition and symmetry of 3-dimensional Euclidean , 1.2 pace, parametrised curves in Euclidean space, arc length, curvature and torsion.

2. Differential forms (2.1 Vector fields as derivative operations, differential1-forms, line integrals, forms of higher degree, exterior derivative, 2.2 , 2.3 Moving frames and structure equations:Definition of a moving frame in Euclidean space, connection forms, first and second structure equations.)

3. Surfaces (3.1 Surfaces described by maps into Euclidean space, normal vectors and tangent vectors, adapted frames, first and second fundamental forms, Gaussand mean curvature, Gauss and Codazzi equation., 3.2 , 3.3 Curvature and geodesics:The meaning of curvature, Theorema Egregium, definition of geodesic, introduction to Riemannian geometry.)

Learning outcomes

By the end of the course, students should be able to do the following:

• Compute speed, arc-length, curvature, torsion, and Frenet frame of a given curve, and interpret the result of the computations geometrically in simple examples.
• Perform computations involving vector fields and forms.
• Compute normal and tangent vector fields, first and second fundamental forms, and Gauss and mean curvature for simple parametrised surfaces.
• Determine whether a given curve on a parametrised surface is geodesic or not.
• Calculate connection forms, Gauss curvature, and geodesic equations for surfaces with given first fundamental form.
• Demonstrate their geometric intuition of the aforementioned notions, in particular by being able to provide examples satisfying required properties.
• Use the theory behind the aforementioned notions to prove simple new results on curves and surfaces.

Further details

Curriculum explorer: Click here

SCQF Level: 10

Credits: 15