The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as pseudo-Riemannian geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little was known about global properties of the geometry even if we impose a locally homogeneous structure. This theme has been developed rapidly in the last three decades.In the series of lectures, I plan to discuss two topics by the general theory and some typical examples.
1. Global geometry: Properness criterion and its quantitative estimate for the action of discrete groups of isometries on reductive homogeneous spaces, existence problem of compact manifolds modeled on homogeneous spaces, and their deformation theory.
2. Spectral analysis: Construction of periodic $L^2$ eigenfunctions for the Laplacian with indefinite signature, stability question of eigenvalues under deformation of geometric structure, and spectral decomposition on the locally homogeneous space of indefinite metric.
References for the first topic include:
References for the second topic include:
- [1a] T. Kobayashi, Proper actions on homogeneous spaces of reductive type, Math. Ann. 1989,
- [1b] --, Deformation of compact Clifford-Klein forms of indefinite- Riemannian homogeneous manifolds. Math. Ann., 1998.
- [1c] T. Kobayashi-T.Yoshino, Compact Clifford--Klein forms of symmetric spaces---revisited. Pure and Appl. Math. Quarterly, 1, 2005. (Special Issue: In Memory of Armand Borel)
- [2a] F. Kassel-T. Kobayashi, Poincaré series for non-Riemannian locally symmetric spaces. Adv. Math. 2016.
- [2b] Kobayashi. Global analysis by hidden symmetry. Mathematics 323, 2017. (Special Issue in honor of Roger Howe)
- [2-c] T.Kobayashi-F.Kassel, Spectral analysis on standard locally homogeneous spaces, preprint, 69 pages. arXiv: 1912.12601.
© Toshiyuki Kobayashi