Introduction to Riemannian symmetric spaces and R-spaces

Data

• This is an intensive lecture for graduate students, held at Tsinghua University (清華大学, 中国), on 19 August 2011.

Abstract

• In this lecture, I will give a brief introduction to the theory of Riemannian symmetric spaces, and its application to R-spaces. A Riemannian symmetric space is a Riemannian manifold together with the symmetries at each point. Many basic examples of manifolds, such as spheres, projective spaces and Grassmannian manifolds, are Riemannian symmetric spaces. Symmetric spaces are interesting, by its beautiful theory, and also by many applications. One of the important application of symmetric spaces is R-spaces, which coincides with orbits of the linear isotropy representations. R-spaces play very important roles in submanifold theory, in particular, provide examples of homogeneous hypersurfaces in spheres. We explain these facts by describing explicit examples in terms of matrices.

詳細

• 3 時間の講義を (スライドとホワイトボードを併用して) 行いました. 内容は以下の 3 つで, それぞれ 1 時間程度で解説しました.
  • 等質多様体 (Warner の本のあらすじ紹介)
  • リーマン対称空間 (Helgason の本のあらすじ紹介)
  • R 空間
• 講義で使用したスライドのファイル: pdf (42pp, 110KB)
• 諸注意と覚書:
  • ホワイトボードでは紹介したけどスライドには書かれていない具体例がいろいろある.
  • 最初の等質多様体の章に, リーマン等質空間の話も入れといた方が良い気がする.