Geometrical Methods of Theoretical Physics I

NTMF059

doc. RNDr. Robert Švarc, Ph.D.

Mgr. Ivan Kolář, Ph.D.

prof. RNDr. Pavel Krtouš, Ph.D.

fall: 2/2 Ex C

Annotation:

Differentiable manifolds, tangent structure. Fibre bundles, vector bundles. Tensor calculus. Maps of manifolds, diffeomorphism, Lie derivative. Exterior calculus. Riemann and pseudo-Riemann geometry. Covariant derivative on vector bundles, parallel transport, space of connections. Covariant derivative on tangent bundle, geodesics, torsion. Curvature. Metric derivatives, Levi-Civita derivative. Relation of Lie, exterior, and covariant derivatives, covariant exterior derivative. Integration on manifolds and submanifolds, integrable densities, integral theorems.

The lectures are aimed at students interested in theoretical physics at the end of their bachelor's or the beginning of their master's study.

This course is followed by the course NTMF060 – Geometrical Methods of Theoretical Physics II.

Information about lectures in fall 2026:

This semester, the lectures are taught by all three lecturers.

The lecture is scheduled for Tuesday mornings from 9:00 to 10:30 in lecture hall T2 and Thursday mornings from 9:00 to 10:30 in lecture hall T1.

Some lectures will be recorded. The recordings, along with those from previous years, are available on a dedicated webpage, the address of which has been sent out to enrolled students.

If anyone is interested in watching the lecture recordings without being enrolled, they should contact the lecturer by email.

The lectures are conducted in Czech. Lecture recordings in English are also available.

The exam, tutorials, and consultations can be conducted in both Czech and English.

Syllabus:

Differentiable manifolds
basics of topology, differentiable structure, tangent spaces, vector and tensor fields, Lie brackets
Fibered spaces
general fibered spaces, vector bundles, tangent tensor bundles
Tensor calculus
vector space and its dual, tensor product, multilinear mapping of tensors, transformation of components, tensor notation
Mappings of manifolds and Lie derivative
mappings of manifolds, induced mapping, diffeomorphisms, flow, Lie derivative
Exterior calculus
exterior product, exterior differentiation, exact and closed forms
Riemannian and pseudo-Riemannian geometry
metric, metric signature, curve length and distance, Hodge dual, Levi-Civita tensor, coderivative
Covariant derivative on vector bundles
parallel transport in a vector bundle, covariant differentiation, covariant differential; space of covariant derivatives, pseudoderivative, difference between two connections and the difference tensor; trivialization, connection components
Covariant derivative on the tangent bundle
geodesics, normal coordinates; torsion tensor; coordinate derivative, n-ad derivative, Ricci (spin) coefficients
Curvature
curvature operator and tensor, Bianchi identities, conservation law; Riemann tensor, commutator of covariant derivatives for a scalar and a general tensor, Ricci and Einstein tensors
Metric derivatives, Levi-Civita covariant derivative
metricity, contorsion and torsion tensor; uniqueness of the Levi-Civita derivative, Christoffel symbols, Cartan structure equations; symmetries of the curvature tensor, irreducible parts of the Riemann tensor, Weyl tensor, scalar curvature, Einstein tensor
Relationship between Lie, exterior, and covariant derivatives
expressions for Lie and exterior derivatives, covariant exterior derivative; Killing vectors and symmetries
Integration on manifolds
integrable densities, relationship to antisymmetric forms, integration of forms and densities; orientation tensor, density dual, metric volume element; divergence of tensor densities, covariant derivative of densities, derivative annihilating the volume element
Integral theorems
submanifolds, adapted coordinates, tangent and normal space, integration over submanifolds; generalized Stokes' theorem for forms; normal and tangent restriction of tensor densities to a submanifold, Gauss' theorem

Literature: