NTMF060 - Geometrical Methods of Theoretical Physics II

Annotation:

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.

Information about lectures in spring term 2026:

The course is scheduled on Thursday at 10:40–13:00 in lecture room T2.

The lectures are given in Czech. However, there is a possibility of attending the course using English recordings from previous years. Please contact the lecturer to discuss the details.

Syllabus:

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.

Studijní texty:

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

Literature:

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.
J. A. de Azcárraga, J. M. Izquierdo: Lie Groups, Lie Algebras, Cohomology and some Applications in Physics, Cambridge Univ. Press, 1995.
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.
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.
I. B. Benn a R. W. Tucker: An introduction to spinors and geometry with applications in physics, Adam Hilger, 1990.
J. Vaz, Jr. a R. da. Rocha, Jr.: An introduction to Clifford algebras and spinors, Oxford Univ. Press, 2016.
D. J. H. Garling: Clifford Algebras - An Introduction, Cambridge Univ. Press, 2009.
P. Budinich a A. Trautman: The spinorial chessboard, Springer, 1988.
R. Penrose a W. Rindler: Spinors and space-time, Cambridge Univ. Press, 1999.
P. O'Donnell: 2-Spinors in General Relativity, World Scientific, 2003.
R. U. Sexl a H. K. Urbantke: Relativity, groups, particles - special relativity and relativistic symmetry in field and particle physics Springer, 2001.