A Riemannian symmetric space is a connected Riemannian manifold 
such that for every point 
there is an isometry 
fixing 
whose differential at 
is 
, where 
is the identity operator
on the tangent space at 
. Equivalently, 
reverses every geodesic through
.
The Euclidean space, sphere, and hyperbolic space are symmetric spaces. More
generally, a Riemannian symmetric space can be represented as a homogeneous
space 
,
where 
is a Lie group of isometries
and 
is the stabilizer of a point.
For example,
 |
Symmetric spaces are basic objects in differential geometry and in the study of representations of Lie groups. Harmonic maps into such spaces can be studied using loop groups and the DPW method.
