Differential Geometry — DIG

Instructor:  Dr. Szilárd SZABÓ

Text:  B. Csikós, Differential Geometry (BSM Lecture Notes)

Prerequisite: Linear algebra (linear spaces, linear maps, basis, matrix operations, eigenvalues, eigenvectors); multivariable calculus (differentiation and integration of functions in several variables, inverse function theorem); analytical geometry (inner product and cross product, equations and parameterizations of straight lines and planes), point set topology (definition of a topological space, continuous maps).

Course description:
Differential geometry is one of the most powerful branches of contemporary mathematics, indispensable not only for today's geometry, but also for modern analysis, calculus of variations, theoretical physics, etc. This course gives an introduction to the subject. We start with the study of curves in a Euclidean space. We define the curvatures of a curve and clarify their geometrical meaning. The second part of the course is devoted to the investigation of hypersurfaces in a Euclidean space. We prove the most important formulae and theorems of surface theory and then apply the general theory to the study of special surfaces. At the end of the course we get acquainted with differentiable manifolds and some ideas of Riemannian geometry.

Topics:
Curves in  Rⁿ. Length of a curve, natural parameterization. Osculating k-planes, Frenet frame, curvatures. Osculating circle of a planar curve, evolutes, involutes and parallel curves. The “Umlaufsatz” and its applications.

 Hypersurfaces in  Rⁿ. Normal curvature, Meusnier's theorem. Fundamental forms, Weingarten map, principal curvatures and principal directions, Euler's formula. Special surfaces in   R³ -  surfaces consisting of umbilical points, surfaces of revolution, ruled surfaces, developable surfaces, Dupin's theorem. Gauss and Codazzi-Mainardi equations, “Theorema Egregium”.

Riemannian manifolds. Differentiable manifolds, Lie groups. Tangent vectors, the Lie algebra of vector fields. Affine connections. Riemannian manifolds and the Levi-Civita connection. The torsion and the curvature tensor, Bianchi identities and other symmetries of the Riemann curvature tensor, sectional curvature, space forms. Ricci tensor, scalar curvature. Geodesic curves.