SubjectsSubjects(version: 875)
Course, academic year 2020/2021
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 [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Additional information:
Guarantor: prof. RNDr. Pavel Krtouš, Ph.D.
RNDr. Robert Švarc, Ph.D.
Annotation -
Last update: T_UTF (10.05.2012)
Elements of topology; differentiable manifolds, tangent bundles, vector and tensor fields; affine connection, parallel transfer and geodesic curves, torsion and curvature; Riemann and pseudo-Riemann manifolds, Riemann connection; Gauss theory of surfaces, Gauss formula; Lie derivative, Killing vectors; exterior calculus; integration on manifolds, integrable densities.
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. (12.10.2017)

The credit is awarded for a correct solution of a problem given to the student 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 handled to the examiner before beginning of the spring term. The condition for the credit cannot be fulfilled by other means.

Literature - Czech
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (01.09.2014)

O. Kowalski: Základy Riemannovy geometrie , skripta, 2. vydání, vydavatelství Karolinum, 2001.

P. Krtouš: Geometrické metody ve fyzice, studijní text, WWW, 2006-2014.

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

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

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

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

M. Fecko: Diferenciálna geometria a Lieove grupy pre fyzikov, IRIS, Bratislava 2004.

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

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

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

V. I. Arnold: Mathematical methods of classical mechanics, Graduate Texts in Math. No. 60, Springer-Verlag, New York 1978.

R. Abraham a J. E. Marsden: Foundations of Mechanics, Addison-Wesley, Reading 1985.

S. Kobayashi a K. Nomizu: Foundations of Differential Geomatry I, Interscience Publishers, New York 1963.

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

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. (12.10.2017)

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 part.

A repeated exam contains again both the written and oral parts.

Syllabus -
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (12.10.2017)
vector space and its dual, tensor product, multi-linear tensor maps, transformation of components, tensor notation
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, Poincaré lemma
Riemann and pseudo-Riemann geometry
metric, signature, length of curves and distance, Hodge dual, Levi-Civita tensor, coderivative, examples of Einstein spaces and maximally symmetric spaces
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, derivatives annihilating metric, contorsion tensor
Levi-Civita covariant derivative
metric derivative, Christoffel symbols, splitting of Riemann tensor, Weyl tensor, scalar curvature, Einstein tensor, Einstein spaces and spaces of maximal curvature
Relations between Lie, exterior and covariant derivatives
Lie and exterior derivative in terms of covariant derivative; Killing vectors and tensors, symmetries
Submanifolds and distributions
manifolds with boundaries, 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 and symplectic 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, integration of tensor densities over submanifold, Stokes and Gauss theorems
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