Manifolds

Content

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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.

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