Differential Geometry

Annotation

This course introduces students to the main concepts of modern differential geometry and topology, including smooth manifolds, Lie groups, fiber bundles, tensor fields and differential forms, connections, de Rham cohomology, K-theory, etc. The main theorems are proved and key examples are considered. Emphasis is put on the concepts and statements that are used in modern theoretical and mathematical physics. 

Fundamental structures and theorems of this course will be used in the quantum field theory and gravity courses,  basic mathematical courses such as "Lie groups and Lie algebras", "Conformal geometry and Riemann surfaces", "Homological algebra", and various special courses.

Course plan

1. Separability, compactness, Urysohn lemma, Tietze theorem,  partition of unity, maps of Hausdorff compacts.
2. Smooth manifolds, smooth maps, diffeomorphisms, existence of a smooth partition of unity; three approaches to defining a tangent vector and a differential of a smooth map.
3. Submanifolds and their constructions (the preimage of a regular value and the image of an embedding). Weak Whitney theorem. Manifolds with boundaries. Orientation of manifolds,  induced orientation. 
4. Geometry of Lie groups. Classical subgroups of a matrix group as submanifolds and Lie subgroups.
5. Locally trivial bundles, structure group, basic examples: linear bundles, principal bundles, covers.
6. Tensor fields and tensor operations, symmetric and skew-symmetric tensors,  coordinate representation of tensors.
7. Affine connection and covariant differentiation. Riemannian metric and Levi-Civita connection. Geometric interpretation of Levi-Civita connection, derivational formulas. Cartan method of moving frames.
8. Parallel transport and  geodesics: existence, exponential map, local existence and uniqueness theorem for geodesics.  Riemann curvature.
9. Differential forms and their coordinate representation. Differentiation and integration, De Rham cohomology, Stokes theorem.
10. Lie bracket  of vector fields. Lie algebra of a Lie group, exponential map, Maurer-Cartan forms.
11. Connection in general bundles. Covariant derivatives.  Horizontal distributions and parallel transport.
12.  Isomorphism and stable isomorphism of vector bundles, direct sum, basic concepts of K-theory.

Literature

1. Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. Modern geometry—methods and applications. Graduate Texts in Mathematics, 93. Springer-Verlag, New York, Part I – 1992, Part II – 1984.
2. Helgason, S. Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, 2001.
3. Lee, Jeffrey M. Manifolds and Differential Geometry. Graduate Studies in Mathematics, 107. American Mathematical Society, Providence, RI, 2009.
4.  Novikov, S. P.; Taimanov, I. A. Modern geometric structures and fields. Graduate Studies in Mathematics, 71. American Mathematical Society, Providence, RI, 2006.