Differential Calculus on Manifolds

Course guarantor:

Matěj Tušek

Lecturer:

Matěj Tušek

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Smooth manifold, tangent space differential forms, tensors, Riemannian metrics and manifold, covariant derivative, parallel transport, orientation of manifold, itegration on manifold and Stokes theorem.

Requirements:

A good knowledge of linear algebra and multivariable differential and integral calculus, a basic knowledge of topological notions (e.g., in the extent of the courses 01MAN1-2, 01ANA/B3-4, 01LAL1-2, and 01TOP held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. Smooth manifolds 2. Tangent and cotangent space 3. Tensors, differential forms 4. Orientation of manifold, integration on manifold 5. Stokes theorem 6. Riemannian manifold.

Syllabus of tutorials:

Study Objective:

Knowledge: To get acquainted with basic notions of differential geometry with emphasis on mathematical details.

Abilities: Consequently, to be able to self-study advanced physical (not only) literature.

Study materials:

key references:

[1] J.M. Lee: Introduction to Smooth Manifolds, Springer, 2003.

recommended references:

[2] J. M Lee: Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.

[3] M. Spivak: Calculus on Manifolds, Addison-Wesley Publishing Company, 1965.

[4] F. Morgan: Riemannian Geometry: A Begginer's Guide, Jones and Bartlett Publishers, 1993.

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans: