In Riemannian geometry, including applications to general relativity, a (Riemannian) symmetric space is a certain kind of homogeneous space in the theory of Lie groups. A geometric characterization is that it is a Riemannian manifold such that for every point there exists an isometry fixing that point and inducing minus the identity on the tangent space at that point. A Lie group characterisation is as G/H where G is a Lie group and H a subgroup that is open in the fixed set of an automorphism of G of order 2. There is a classification of such spaces, by Elie Cartan.
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In mathematics, a (Riemannian) symmetricspace in differential geometry is a certain kind of homogeneous space in the theory of Lie groups.
A Lie group characterisation of symmetricspaces is as G/H where G is a Lie group and H a compact subgroup that is open in the fixed set of an automorphism of G of order 2.
A symmetricspace is homogeneous, so can be written as G/H where G is a Lie group acting on it and H is the subgroup fixing some fixed point.
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