Symmetric Spaces
Symmetric spaces are classified as follows.
First, the universal cover of a symmetric space is still symmetric, so we can reduce to the case of simply connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.)
Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces).
The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each non-compact simple Lie group G, one compact and one non-compact. The non-compact one is a cover of the quotient of G by a maximal compact subgroup H, and the compact one is a cover of the quotient of the compact form of G by the same subgroup H. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.
Read more about this topic: List Of Simple Lie Groups
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