Lecture series

• D. Saunders: Homogeneous variational principles and Riemann-Finsler geometry

  
  

  In this introductory series of lectures we use the problem of finding the shortest path between two points in a Riemannian manifold to motivate the use of the calculus of variations to construct the associated path space. Here, a path need not have any particular parametrization, although in the case of Finsler geometry it might have a specific orientation. We see how the paths can be specified by families of sprays (certain types of homogeneous vector fields on a tangent manifold) or of linear connections (covariant derivatives on vector bundles) and where parametrization at unit speed can select a particular spray or linear connection.

  More generally, we see how a certain type of path space (which might not arise from a variational problem, and so might not have a measure of speed) determines a single spray or linear connection on a manifold with one extra dimension. We also see that this is related to Cartan’s method of constructing paths by attaching a projective space at a point and then rolling it around the manifold.

  Finally, we extend the motivating problem to higher dimensions, including for example the problem of finding surfaces of least area. We see that the homogeneity of the variational problem, corresponding to invariance under reparametrization, means that the underlying geometrical structure can be either a velocity manifold (generalising a tangent manifold) or its quotient, a Grassmannian manifold (generalising a projective tangent manifold). We also construct the corresponding differential forms by using particular types of vector valued forms.

  1. Variations and Riemann geometry
  2. Finsler geometry
  3. Cartan geometry
  4. Generalisations in higher dimensions
  References:
  

  1. J. M. Lee, Introduction to Riemannian Manifolds, Springer (Graduate Texts in Mathematics
     176) 2018
  2. D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer
     (Graduate Texts in Mathematics 200) 2000

• N. Voicu: Introduction to Finsler geometry

  
  
  1. Axioms of a Finsler space and main geometric objects: Finsler function, metric tensor field, Cartan tensor field. The notion of anisotropic tensor. Consequences of each of the axioms. Examples and applications.
  2. The Poincare-Cartan form in mechanics. Hilbert form in Finsler geometry. Euler-Lagrange form of the arc length Lagrangian. Geodesics of a Finsler space. Canonical nonlinear connection, local adapted bases. Geodesic deviation in a Finsler space, nonlinear connection curvature. Chern affine connection, Berwald and Landsberg tensor fields.
  3. Homogeneity of tensor fields on the tangent bundle. Homogeneous anisotropic tensor fields. The positively (or oriented) projectivized tangent bundle PTM+. Manifold structure, manifold-defining local charts vs. homogeneous local coordinates. Principal bundle structure of TM over PTM+, natural fiber bundle structure of PTM+ over M. Calculus in homogeneous coordinates. Finslerian geometric objects as sections of fiber bundles sitting over PTM+.

• D. Krupka: Finsler geometry: Second-order generalizations

  
  Part 1
  1. Introduction
  2. Lepage forms in higher-order mechanics
  3. Parameter-invariant variational integrals, Hilbert forms
  Part 2
  4. Second-order velocity spaces, second-order Grassmann fibrations
  5. Second-order Lepage-Hilbert forms
  6. Second-order prolongations of vector fields
  7. Example
  References:
  1. S. S. Chern, Finsler Geometry is just Riemannian geometry without the quadratic restriction, Notices of the AMS, 1966, 959-963
  2. D. Krupka, Lepage forms in Kawaguchi spaces and the Hilbert form, Publ. Math. Debrecen 84 (2014), 147-164
  3. D. Krupka, Variational principles: Projectability onto Grassmann fibrations, J. Math. Phys. 61, 123501 (2020)
  4. Z. Urban, D. Krupka, Foundations of higher-order variational theory on Grassmann fibrations, Int. J. Geom. Meth. Mod. Phys. 11 (2014) 1460023
  5. S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry, World Scientific (Series on University Mathematics – Vol. 1) 1999

Lectures on classical and modern research problems, workshop

• M. Krššák, Gravitational Instantons in Teleparallel Gravity

  Finite Euclidean action solutions—commonly known as instantons—play an important role in understanding non-perturbative and topological aspects of quantum field theories. The best-known example is in the Yang-Mills case, where the action for self-dual BPST instantons reduces to a topological term and turns out be a global minimum. In the case of gravity, the situation is more complicated and significantly less clear. While there exist self-dual gravitational instantons (Eguchi-Hanson), their action is not related to any topological term, which significantly limits the analogy with the Yang-Mills case. In this talk, I will show how to overcome this limitation within the teleparallel formulation of general relativity byintroducing a new class of “self-excited” instantons for which the gravitational action reduces to a topological (Nieh–Yan) term.

• Z. Urban, Equilibrium stabilization of mechanical systems using variational forces

• C. K. Mishra, Curvature Inheritance Symmetry in Finsler space

Presentation of posters