Geometrical Methods of Theoretical Physics I

NTMF059

prof. RNDr. Pavel Krtouš, Ph.D.

fall: 2/2 Ex C

Annotation:

Tensor calculus. Differentiable manifolds, tangent bundles. Maps of manifolds, diffeomorphism, Lie derivative. Exterior calculus. Riemann and pseudo-Riemann geometry. Covariant derivative, parallel transfer, geodesics. Torsion and curvature, space of connections. Metric derivatives, Levi-Civita derivative. Relation of Lie, exterior, and covariant derivatives. Submanifolds, integrability, Frobenius theorem. Integration on manifolds, 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 usually followed by the course NTMF060 – Geometrical Methods of Theoretical Physics II. However, this academic year, this course is not opened.

Information about lectures in fall 2022:

Lectures and practicals are scheduled each Tuesday at 14:50–18:00 in lecture room T1.

Both lectures and practicals are taught in person.

Recordings of lectures (from this and previous years) are available on a special page, the address of which was sent to enrolled students.

If anyone is interested in watching recordings of the lectures without enrolment, please, contact the lecturer by email.

This year, the lectures are given in Czech. However, the recordings of lectures in English are also available.

The examination can be both in Czech or English.

Syllabus:

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

Literature: