Last update: doc. RNDr. Karel Houfek, Ph.D. (04.02.2021)
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.
Last update: doc. RNDr. Karel Houfek, Ph.D. (04.02.2021)
Základy topologie; diferencovatelné variety, jejich tečné prostory, vektorová a tenzorová pole; afinní konexe,
paralelní přenos a geodetické křivky, torze a křivost, prostor konexí; Riemannovy a pseudo-Riemannovy variety,
Riemannova konexe; Gaussova teorie ploch, Gaussova formule; Lieova derivace, Killingovy vektory; vnější
kalkulus; integrování na varietách, hustoty, integrální věty.
Přednáška je určena zejména pro zájemce o teoretickou fyziku v závěru bakalářského či začátkem magisterského
studia.
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.
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (01.09.2014)
Cílem předmětu je seznámit posluchače s metodami diferenciální geometrie a jejich aplikacemi ve fyzice.
Literature - Czech
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (02.02.2021)
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.
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (01.09.2014)
Metodou výuky je přednáška a cvičení.
Syllabus -
Last update: prof. RNDr. Pavel Krtouš, Ph.D. (12.10.2017)
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, 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
Symplectic geometry
symplectic form, symplectic potencial, canonical coordinates, Hamilton flow, Hamilton equations, symplectic structure of cotangent space
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, symmetry and conserved quantities, Schouten-Nijenhuis brackets
Manifolds with boundaries and submanifolds
manifolds with boundaries, immersion and embedding, adjusted coordinates; tangent and normal spaces, intrinsic and extrinsic curvature, Gauss-Codazzi formula, curvature splitting for hypersurfaces, 2-dimensional surfaces, "Theorema Egregium"; distributions, foliations and 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
