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Introduction to Riemannian symmetric spaces and R-spaces
This is an intensive lecture for graduate students,
held at Tsinghua University ($B@62ZBg3X(B, $BCf9q(B), on 19 August 2011.
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.
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pdf (42pp, 110KB)