## *Global Geometry and Analysis on Locally Symmetric Spaces with
Indefinite-metric*.
Analysis on
Manifolds with Symmetries and Related Structures. University of Bath, UK,
28-29 June 2016.

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.

by taking anti-de Sitter manifolds as a typical example.

© Toshiyuki Kobayashi