Geometria różniczkowa  / Differential Geometry (2020L)

Wykład fakultatywny na Wydziale MIM UW

Prowadzący, miejsce i czas Program Pomocnik studenta - notatki i zadania  Literatura i dowiązania

Prowadzący, miejsce i czas

Prowadzący: Stefan Jackowski

Poniedziałki, 8:30-12:00

Zoom, Moodle

Tematy wykładów

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

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

3. Tangent vectors as equivalence classes of curves and differentials on the algebra of smooth functions. Derivative of a smooth map. Submersion, immersion and embedding.

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

5. Vector bundles and their morphisms. Extension of constructions from linear algebra to vector bundles. Structures defined on vector bundles: (orientation, complex, metric and symplectic forms).

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

7. Distributions, 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. HopfRinow theorem.

11. Riemannian manifolds of constant 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).

Pomocnik studenta - notatki i zadania

Literatura i dowiązania

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

Baer, Ch. "Differential Geometry"

Białynicki-Birula, A. "Geometria różniczkowa II. " Skrypt wykładów.

Grabowska, K. "Notatki do wykładu Geometria Różniczkowa" Wydział Fizyki UW, 2019.

Kijowski, J. "Geometria różniczkowa jako narzędzie nauk przyrodniczych" CSZ Politechnika Warszawska.

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.

Stefan Jackowski
Aktualizacja: 2021-01-26

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