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Course, academic year 2023/2024
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Geometrical Methods of Theoretical Physics I - NTMF059
Title: Geometrické metody teoretické fyziky I
Guaranteed by: Institute of Theoretical Physics (32-UTF)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Additional information: http://utf.mff.cuni.cz/vyuka/NTMF059
Guarantor: prof. RNDr. Pavel Krtouš, Ph.D.
doc. RNDr. Robert Švarc, Ph.D.
Annotation -
Last update: doc. RNDr. Karel Houfek, Ph.D. (14.05.2021)
Foundations of topology; differentiable manifolds, tangent bundles, vector and tensor fields; maps of manifolds, diffeomorphism, induced mapping, Lie derivative; exterior calculus; covariant derivative, parallel transfer and geodesic curves, torsion and curvature, space of connections; (pseudo-)Riemann manifolds, metric derivatives, Levi-Civita derivative, Killing vectors; integrability and Frobenius theorem; integration on manifolds, integrable densities, integral theorems.
Aim of the course -
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (01.09.2014)

The goal of this lecture is to acquaint the students with differential geometry and its applications in physics.
Course completion requirements -
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (21.04.2023)

The credit is awarded for a correct solution of a problem given to the students during the term. If the solution of the problem is not satisfactory, it is returned to the student for revisions. The correct solution has to be submitted to the examiner before the beginning of the spring term. The condition for the credit cannot be fulfilled by other means.

The examination has written and oral parts. The written part contains one problem similar to those solved during the term at seminars. The oral part contains two questions on topics covered by the lectures.

Evaluation is based both on the written and oral parts.

A repeated exam contains both the written and oral parts again.
Literature -
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (23.09.2021)

C. W. Misner, K. S. Thorne, J. A. Wheeler: Gravitation, Freedman, 1973.

S. W. Hawking, G. F. R. Ellis: The Large Scale Structure of Space-Time, Cambridge Univ. Press, 1973.

R. Wald: General Relativity, Univ. of Chicago Press, 1984.

R. Penrose, W. Rindler: Spinors and space-time, Cambridge Univ. Press, 1999.

M. Fecko: Differential Geometry and Lie Groups for Physicists, Cambridge Univ. Press, 2011.

T. Frankel: The Geometry of Physics - An Introduction, Cambridge Univ. Press, 1999.

Ch. J. Isham: Modern Differential Geometry For Physicists, World Scientific, 1989.

C. von Westenholz: Differential Forms in Mathematical Physics, North-Holland, 1978.

S. Kobayashi, K. Nomizu: Foundations of Differential Geometry I, Interscience Publishers, 1963.

M. Spivak: A Comprehensive Introduction to Differential Geometry, Publish or Perish Press, 1970-1979.

J. M. Lee: Manifolds and Differential Geometry, Graduate Studies in Mathematics Vol. 107, AMS, 2009.
Teaching methods -
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (01.09.2014)

The teaching method is a lecture and a seminar.
Requirements to the exam -
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (21.04.2023)

The examination has written and oral parts. The written part contains one problem similar to those solved during the term at seminars. The oral part contains two questions on topics covered by the lectures.

Syllabus -
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (23.09.2021)
Tensors
vector space and its dual, tensor product, multi-linear tensor maps, transformation of components, tensor notation
Manifolds
basic notion of topology, differential structure, tangent spaces, vector and tensor fields, Lie brackets
Mappings of manifolds and Lie derivative
mappings of manifolds, induced map, diffeomorphism, flow, Lie derivative
Exterior calculus
wedge product, exterior derivative, exact and closed forms
Riemann and pseudo-Riemann geometry
metric, signature, length of curves and distance, Hodge dual, Levi-Civita tensor, coderivative
Covariant derivative
parallel transport, covariant derivative, covariant differential, geodesics, normal coordinates; torsion, Riemann curvature tensor, commutator of covariant derivatives for scalars and general tensors, Bianchi identities, Ricci tensor
Space of covariant derivatives
pseudo-derivative, difference of two connections and differential tensor, coordinate derivative, n-ade derivative, Ricci (spin) coefficients, metric derivatives, contorsion tensor
Levi-Civita covariant derivative
uniqueness, Christoffel symbols, Cartan structure equations, irreducibile splitting of Riemann tensor, Weyl tensor, scalar curvature, Einstein tensor
Relations between Lie, exterior and covariant derivatives
Lie and exterior derivative in terms of covariant derivative; Killing vectors and symmetries
Submanifolds and distributions
immersion and embedding, adjusted coordinates, tangent and normal spaces; distributions, integrability conditions, Frobenius theorem
Integration on manifolds
integrable densities, relation to anti-symmetric forms, integration of forms and densities; tensor of orientation, density dual, metric density; divergence of tensor densities, covariant derivative of densities, derivative annihilating density
Integral theorems
generalized Stokes' theorem for forms, normal and tangent restriction of tensor densities, Stokes and Gauss theorems
 
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