Riemannian symmetric space of noncompact type

Definition

A Riemannian symmetric space of noncompact type is a connected, simply connected, complete Riemannian manifold $X$ such that every point is an isolated fixed point of an involutive isometry, all sectional curvatures are nonpositive, and $X$ has no nontrivial Euclidean de Rham factor. Equivalently,

X\cong G/K,

where $G$ is a connected noncompact semisimple Lie group with finite center and no compact factors, and $K$ is a maximal compact subgroup. The quotient is a homogeneous space, with $G$ acting by isometries.

Infinitesimal model

At the base point $o=eK$ , the tangent space identifies with $\mathfrak p$ in the Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$ . A suitably normalized invariant bilinear form gives the $G$ -invariant metric. The relation $[\mathfrak p,\mathfrak p]\subseteq\mathfrak k$ expresses the symmetry, and curvature is determined by

R(X,Y)Z=-\bigl[ [X,Y],Z \bigr]

up to the chosen curvature-sign convention Helgason, Chapters IV–V.

Rank and flats

A maximal abelian subspace $\mathfrak a\subseteq\mathfrak p$ exponentiates to a maximal totally geodesic flat through $o$ . Its dimension is the real rank of $X$ . The associated restricted root system controls radial geometry, invariant differential operators, and the behavior of spherical functions.

Examples and boundary of the definition

Real hyperbolic space is $\mathrm{SO}_0(n,1)/\mathrm{SO}(n)$ , and the space of positive-definite determinant-one matrices is $\mathrm{SL}_n(\mathbb R)/\mathrm{SO}(n)$ . Euclidean space is symmetric and nonpositively curved but is excluded by the no-Euclidean-factor clause. Compact symmetric spaces, such as round spheres, belong to the compact dual theory rather than the noncompact type.

References

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001. DOI record. Relevant: Chapters IV–V.
S. Helgason, Groups and Geometric Analysis, American Mathematical Society, 2000. DOI record. Relevant: Chapter I on symmetric spaces and their harmonic analysis.