Differential Geometry

Entry requirements

60 credits in mathematics (or 40 credits in mathematics and 40 credits in physics) including Several Variable Calculus M and Linear Algebra II. Several Variable Calculus M may be replaced by Geometry and Analysis III or Several Variable Calculus.

Learning outcomes

In order to pass the course (grade 3) the student should be able to

• give an account of important differential geometric concepts and definitions,
• formulate and explain the meaning of important results and theorem,
• describe the main features of the proofs of central theorem and perform proof of simpler differential geometry,
• use the theory, methods and techniques of the course to solve mathematical problems.

Content

Tangent vector. Tangent bundle. Curves. Curvature and torsion. Frenet’s equations. Surfaces. The fundamental forms. Curvature. Theorema Egregium. Vector fields and covariant derivative. Geodetic curves. Two-dimensional Riemannian geometry. Briefly about the global theory of surfaces, n-dimensional Riemannian theory, space-time and Einstein’s equations.

Instruction

Lectures and problem solving sessions

Assessment

Assignments combined with oral examination.