*Instructor*: Dr. Balázs CSIKÓS

*Text*: Balázs 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 ** R**^{n}. 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 **R**^{n}.
Normal curvature, Meusnier's theorem. Fundamental forms, Weingarten map,
principal curvatures and principal directions, Euler's formula. Special
surfaces in ** R**^{3} -
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.