Differential Geometry

Aim

The aim of the course is to provide knowledge of the geometry of curves and surfaces. The course integrates concepts from different parts of mathematics, such as linear algebra, calculus and differential equations. It also provides intuitive examples for many concepts in linear algebra, calculus and differential equations. These examples are fundamental to physics and mechanics: they play a role in our understanding of the movements of particles and the theory of relativity. After completing this course, students should be able to:

• Determine and calculate curvature of curves in different coordinate systems.
• Parametrise surfaces and use the metric tensor. Calculate isometries.
• Treat geodesic curves and parallel translation
• Calculate and analyse curvature of surfaces in different settings.
• Know the concept of tensor and recognise tensors that are used in mechanics, image processing and theory of relativity.
• Apply geometry of curves and surfaces to computer aided graphics

Content

Curves: tangents, curvature and torsion. Contact. Different types of curves. Regular surfaces: tangent plane. The first fundamental form: normal and geodesic curvature. Geodesics and parallel transport. Gauss' formulae. The second fundamental form: Weingarten's equation, principal, Gauss and mean curvature. Minimal and developable surfaces. Riemann's and Ricci's tensors, Codazzi-Mainardi's equations. Gauss' ''Theorema Egregium''. Isometrical and conformal mappings. Gauss-Bonnet theorem. Applications to CAD.