# LRI Seminar 2020/21

University of Presov, 17. novembra 1, Presov, Slovakia
Department of Physics, Mathematics and Technologies

## (1) A course on Jet Theory and Applications

(J. Brajercik, D. Krupka)
Introductory course for beginners, available for university students

Requirements: Basic knowledge of Algebra and Differential Calculus (matrices, determinants,
vector spaces, groups; derivatives of functions of several real variables)

#### Contents:

• Euclidean spaces: Structures
• Jets of mappings of Euclidean spaces, composition of jets, the chain rule
• The differential group
• Categories: Elementary concepts
• Manifolds: Elementary concepts, examples
• Jets of mappings of smooth manifolds
• The tangent bundle
• Introduction to tensor spaces (algebraic theory)
• Jet prolongations of tensor spaces
• Differential invariants
• Metric tensors on a manifold
• Example: Differential invariants of the metric tensor

#### Basic references:

1. D. Krupka, J. Janyska, Lectures on Differential Invariants, J.E. Purkyne University, Faculty of Science, Brno, Czechoslovakia, 1990, 193 pp.
2. D. Krupka, M. Krupka, Jets and contact elements, in: Proceedings of the Seminar on Differential Geometry, D. Krupka, Ed., Mathematical Publications Vol. 2, Silesian University in Opava, Opava, Czech Republic, 2000, 39-85

## (2) Variational Geometry

(D. Krupka, further contributors and guests)
Research seminar devoted to contemporary research topics.

#### Contents:

Natural Variational Principles

• Geometric foundations of natural ("generally covariant") variational principles, analysis of examples (general relativity: conservation laws, energy-momentum tensors)
• Main applications: General Relativity and Field Theory (mathematical structures, equations for extremals, conservation laws: energy-momentum tensors)

#### Basic references:

1. M.J. Gotay, J.E. Marsden, Stress-Energy-Momentum Tensors and the Belinfante-Rosenfeld Formula, Contemp. Math. (1992) 132, 367-392.
2. D. Krupka, Natural Lagrangian structures, in: Differential Geometry, Banach Center Publications 12, Diff. Geom. Semester, Warsaw, Sept.-Dec. 1979; Polish Scientific Publishers, Warsaw, 1984, 185-210.

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