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:
- D. Krupka, J. Janyska, Lectures on Differential Invariants, J.E. Purkyne University, Faculty of Science, Brno, Czechoslovakia, 1990, 193 pp.
- 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:
- M.J. Gotay, J.E. Marsden, Stress-Energy-Momentum Tensors and the Belinfante-Rosenfeld Formula, Contemp. Math. (1992) 132, 367-392.
- 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.
