Editing Simple Lie group (section)

===Symmetric spaces===
{{Main|Symmetric space#Classification of Riemannian 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.