Differential geometry

computer science
mathematics
physics

elective courses

Linear Algebra and Multivariable Calculus, Basic notions of topology. Differential geometry of curves and surfaces in R^3.

Classroom

Abstract smooth manifolds, smooth maps. Tangent vectors and derivative of a smooth map. Vector fields as differentatials and flows. Lie bracket. Tangent bundle. Ventor bundles and operations on them. Tensor fields. Foliations and Frobenius theorem. Differential forms, exterior derivative and the Stokes theorem. Covariant derivative and affine connection, parallel transport and geodesics. Curvature tensor. Levi-Civita connection on Riemannian manifold. Ricci tensor.Geodesically complete manifolds. Manifolds of constantt curvature (space form problem). Lie groups and algebras.

1. Smooth atlas and coordinate systems. Abstract smooth manifolds (also with boundary), submanifolds, smoth maps and difeomorphisms.

2. Algebra (sheaf) of smooth functions on a manifold. Smooth partition of unity.

3. Tangent vectors as equivalence classes of curves, differentals on the algebra of smooth functions. Derivative of a smooth map. Submersions, immersions and embeddings.

4. Tangent bundle. Vector fields as sections, differentia operators and flows. Lie algebra of vector fields.

5. Vector bundles and their morhisms. Extension of constructions from linear algebra to vector bundles. Stuctures defined on vector bundles: (orintation, comples, metric snd symplectic forms).

6. Tensor fields. Differential forms, exteror derivative. Integration of forms and the Stokes theorem.

7. Distibutions, foliations and contact structures. The Frobenius theorem.

8. Differentiation of vector fields. Covariant derivative and affine connection. Parallel transport and geodesics. Curvature tensor.

9. Riemannian manifolds and Riemannian connection. Sectional and scalar curvature. Ricci tensor.

10. Geodesically complete manifolds. Hopf – Rinow theorem.

11. Riemannian manifolds of constatnt curvature. (Space Form Problem).

12. Lie groups. One-parameter subgroups and the exp map. Algebra of left-invariant vector firlds. Correspondence between groups and algebras (info).

Aubin, T. "A Course in Differential Geometry". AMS, Graduate Studies in Mathematics, vol. 27, 2001.

Baer, Ch. "Differential Geometry" https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Lehre/Lehrmaterialien/skript-DiffGeo-engl.pdf

Lee, J.M. "Manifolds and Differential Geometry." AMS Graduate Studies in Mathematics Volume: 107; 2009

Spivak,M. "A Comprehensive Introduction to Differential Geometry. Volumes I-V", Publish or Perish, 1999.

Sternberg, S.. Lectures on Differential Geometry. Prentice–Hall, Englewood Cliffs, N.J., 1964.

A student understands: :

- Notions of an abstract smooth manifold, tangent vectors and their various interpretations. A differential of a smooth map. The role played by the algebra of smooth functions.

- Variuos interpretations of vector fileds and their Lie bracket.

- Constructions on vector bundles as generalization of linear algebra. Structures on vector bunldes (orientation, metric, symplectic etc.)

- Why one integrates differential forms on manifolds. Geometric sense of the Stokes theorem.

- Notion of covariant derivative and parallel transport as additional structures on manifolds.

- How Riemannian metric defines a compatible connection.

A students knows examples of:

- abstract manifolds: projective spaces, abstract surfaces, construction of manifolds as orbit spaces of group actions. Lie groups.

- manifolds of constant curvature; in particular hyperbolic geometry.

- parallel transport and geodesics on some manifolds.

A student is able to demonstrate geometric arguments on drawings and expose mathematical reasoning in written form.

Final grade based on an essay and written exam consisting of quiz and problems.