Global Geometry and Analysis on Locally Symmetric Spaces with Indefinite-metric. Glances at Manifolds: Conference in celebration of Aleksy Tralle for his 60th birthday. the Jagiellonian University, Krakow, Poland, 2-6 July 2018.

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 preudo-Riemannian geometry, familiar to us as the spacetime of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

I plan to explain two programs:

1. (global shape) Existence problem of compact locally homogeneous spaces, and deformation theory.

2. (spectral analysis) Construction of periodic eigenfunctions for the Laplacian for indefinite-metric, and discuss the stability of eigenvalued under the deformation of the geometric structure.

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© Toshiyuki Kobayashi