Geometry of Lie groups and algebras (geometric structures on Lie groups, Lie algebra, the action of Lie group on a manifold, vector representations). Hodge theory (Hodge decomposition, de Rham-Laplace operator, harmonics). Topological methods (Cohomology a homology groups, homotopy, fundamental group, homotopy equivalence, Poincare lemma). Fibre bundles (vector bundles, covariant derivative). Geometry of gauge fields (inner degrees of freedom, gauge symmetry, the action and field equations). Characteristic classes (invariant symmetric polynomials, Chern-Weil theorem, characteristic classes, Euler form). Curvature splitting on submanifolds (first and second fundamental form, orthogonal projection of the curvature, Gauss, Weingarten, and Codazzi–Mainardi equations, extrinsic curvature for hypersurfaces, Gauss's Theorema Egregium for 2-surfaces).

Knowledge of the differential geometry at the level of the course NTMF059 is assumed.

The course is scheduled on **Wednesdays at 14:00–16:20 in lecture room T1**.

The lectures are given mostly in Czech. However, all topics given in Czech are available in English as recordings from previous years. New topics, not recorded before, will be given in English.

Lectures are given in person. New topics will be recorded. Together with recordings from previous years, the enrolled students will have access to recordings of all topics.

- Geometry of Lie groups and algebras
- Lie groups, construction of Lie algebra, exponential mapping, Killing metric, structure constants. Bi-invariant metric, measure, covariant derivative. Adjoint representations. The action of Lie group on a manifold, flows and their generators. Representations on vector spaces.
- Hodge theory
- Scalar product on forms, Hodge dual, coderivative, de Rham-Laplace and Beltrami-Laplace operators. Hodge decomposition, potential and copotential, harmonics, cohomology.
- Topological methods
- Cohomology a homology groups, homotopy, fundamental group, homotopy equivalence, homotopy operator, contraction, Poincare lemma.
- Fibre bundles
- Abstract fibre bundles. Vector bundles and their geometry, covariant derivative, vector potential and curvature. Objects on the gauge-algebra bundle.
- Geometry of gauge fields
- Inner degrees of freedom and their description in terms of vector bundles. Gauge symmetry. Gauge group and gauge algebra bundles. Gauge and Yang-Mills fields. The action and field equations. Electromagnetic and charged fields.
- Characteristic classes
- Invariant symmetric polynomials in curvature, Chern-Weil theorem, characteristic classes, Chern class and character, Pontrjagin class, Euler form, integral quantities.
- Curvature splitting on submanifolds
- The first and second fundamental form, the orthogonal projection of the curvature, Gauss, Weingarten, and Codazzi–Mainardi equations. Extrinsic curvature for hypersurfaces. Time flow complementary to the hypersurface. Gauss's Theorema Egregium for 2-surfaces.

Vedle literatury uvedené níže jsou k dispozici texty speciálně k přednášce NTMF059:

- O. Kowalski:
*Základy Riemannovy geometrie*, skripta, Karolinum, Praha 1995 - P. Krtouš:
*Geometrické metody ve fyzice*, studijní text, WWWhttp://utf.mff.cuni.cz/vyuka/NTMF059/GeometrickeMetody/, 2006-2014

- M. Fecko:
*Differential Geometry and Lie Groups for Physicists*, Cambridge Univ. Press, 2011. - C. von Westenholz:
*Differential Forms in Mathematical Physics*, North-Holland, 1978. - T. Frankel:
*The Geometry of Physics - An Introduction*, Cambridge Univ. Press, 1999. - M. Nakahara:
*Geometry, Topology and Physics*, Taylor&Francis, 2003. - Ch. Nash, S. Sen:
*Topology and Geometry for Physicists*, Dover Publ., 2011. - Ch. J. Isham:
*Moddern Differential Geometry For Physicists*, World Scientific, 1989. - E. W. Mielke:
*Geometrodynamics of Gauge Fields*, Springer, 2017. - C. W. Misner, K. S. Thorne a J. A. Wheeler:
*Gravitation*, Freedman, 1973. - S. Kobayashi a K. Nomizu:
*Foundations of Differential Geometry I*, Interscience Publishers, 1963. - M. Spivak:
*A Comprehensive Introduction to Differential Geometry*, Publish or Perish Press, 1970-1979. - R. Penrose a W. Rindler:
*Spinors and space-time*, Cambridge Univ. Press, 1999. - P. O'Donnell:
*2-Spinors in General Relativity*, World Scientific, 2003. - V. I. Arnold:
*Mathematical methods of classical mechanics*, Graduate Texts in Math. No. 60, Springer-Verlag, 1978. - R. Abraham a J. E. Marsden:
*Foundations of Mechanics*, Addison-Wesley, 1985.