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;
• exemplify and interpret important concepts in specific cases;
• formulate important results and theorems covered by the course;
• describe the main features of the proofs of important theorems;
• express problems from relevant areas of applications in a mathematical form suitable for further analysis;
• use the theory, methods and techniques of the course to solve mathematical problems;
• present mathematical arguments to others.

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.