Symmetric spaces

In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be made more precise, in either the language of Riemannian geometry or of Lie theory. The Riemannian definition is more geometric, and plays a deep role in the theory of holonomy. The Lie-theoretic definition is more algebraic.

In Riemannian geometry, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M,g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space ${\displaystyle T_{p}M}$ of M at p by minus the identity. Any symmetric space is complete, and has a finite cover which is a simply connected symmetric space; thus these two characterizations in fact coincide up to finite covers. Both descriptions can also naturally be extended to the setting of pseudo-Riemannian manifolds.

From the point of view of Lie theory, a symmetric space is the quotient G/H of Lie group G by a Lie subgroup H, where the Lie algebra ${\displaystyle {\mathfrak {h}}}$ of H is also required to be the ${\displaystyle +1}$-eigenspace of an involution of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of G. As stated, this characterization includes pseudo-Riemannian spaces as well as a Riemannian ones; extra algebraic conditions are needed to restrict to the Riemannian case.

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