Riemannian Geometry

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/03 Geometry.

Examination method

Oral exam.

Learning
objectives

The course aims to present an introduction to Riemannian geometry, to provide the basic notions and techniques (mainly of an analytical nature) and to discuss some of the fundamental theorems, with particular attention to the relations between curvature and topology.

Syllabus

Review of tensor calculus. Riemannian metrics. The Levi-Civita connection and the covariant derivative. The Riemann tensor. Theory of geodesics, the exponential map and applications. Outline of the theory of submanifolds. Distance functions. Comparison theorems. Relations between curvature and topology. Splitting/soul theorems. Manifolds of positive curvature.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

know and understand the basic concepts and tools of the discipline;
are able to apply the knowledge acquired to tackle problems on manifolds;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the subject matter, clarity of exposition, rigour in the use of language, confidence in using the notions acquired.

Riemannian Geometry