# Manifolds

## Content

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## Description

**Lectures [Thursdays, 13:00-15:00]**

- Lecture 1: Submanifolds of Euclidean space, the Implicit Function Theorem, abstract manifolds.
- Lecture 2: Tangent vectors and the tangent bundle, vector bundles.
- Lecture 3: Vector fields and flows, Lie derivative, the Frobenius Theorem.
- Lecture 4: Differential forms, integration on manifolds.
- Lecture 5: de Rham cohomology.
- Lecture 6: Riemannian metrics, connections, the Levi-Civita connection.
- Lecture 7: Geodesics, the exponential map.
- Lecture 8: Curvature and integrability, Riemannian curvature, spaces with constant curvature.
- Lecture 9: Geometry of hypersurfaces of Euclidean space, the Gauss-Bonnet Theorem.
- Lecture 10: Curvature and topology: Chern-Weil theory, second variation arguments, the Bochner technique.

**Assessment**

This module is assessed in two assignments.

#### Prerequisites

Working knowledge of metric spaces, linear algebra, group theory, vector calculus, topological spaces, the fundamental group, homology.

#### The Team

- Lecturer: Alexandre Minets