Abstract
An important special case is that of symmetric spaces X D KnG , where G is a connected semisimple Lie group without center and with no compact factors, and K is a maximal compact subgroup. Then X is non-positively curved, and G is the connected component of Isom.X /. In this case, X= has finite volume iff is a lattice in G . The curvature of X is non-positive if rank.G/ 2, and is negative if rank.G/D 1.
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