Questions: Symmetric Spaces

Question 1

A Riemannian manifold M is a symmetric space if at every point p there exists an isometry σp : M → M with σp(p) = p and dσp|_p = -id (it reverses all tangent vectors at p). What geometric operation does σp perform?

• A. Translation by the vector p
• B. Geodesic reflection through p — it sends each geodesic through p to itself but reverses direction
• C. Rotation by 180° about p
• D. Projection onto the tangent space at p

Answer: (B)

The isometry σp fixes p and negates all tangent vectors there, so it sends each geodesic through p to itself with reversed parameterization: σp(exp_p(tv)) = exp_p(-tv). This is 'geodesic reflection' or 'geodesic symmetry' at p. On Euclidean space, σp is the point reflection x ↦ 2p - x. On a sphere, σp sends each point to its antipodal point with respect to p (through p along the great circle). The existence of such an isometry at every point is an extremely strong symmetry condition.

Question 2

On a Riemannian symmetric space, the curvature tensor satisfies ∇R = 0 (it is covariantly constant / parallel).

• T. True
• F. False

Answer: True

The geodesic symmetries σp act as isometries, hence preserve the curvature tensor. Since dσp = -id at p, symmetry implies ∇R|_p maps to -∇R|_p under the symmetry — but as an isometry it also preserves ∇R. So ∇R|_p = -∇R|_p, forcing ∇R|_p = 0. Since p is arbitrary, ∇R = 0 everywhere. This means the curvature is 'the same everywhere' in a precise sense — parallel transport preserves the curvature tensor. Conversely, a complete simply connected Riemannian manifold with ∇R = 0 is a symmetric space.

Question 3

Name three examples of Riemannian symmetric spaces and their isometry groups.

Model answer: 1) The sphere Sⁿ = SO(n+1)/SO(n) — the isometry group is the full orthogonal group O(n+1), and the isotropy group at a point is O(n). 2) Hyperbolic space Hⁿ = SO(n,1)/SO(n) — the isometry group is the Lorentz group O(n,1). 3) The Grassmannian Gr(k,n) = O(n)/(O(k)×O(n-k)) — the space of k-dimensional subspaces of ℝⁿ. Other examples: complex projective space ℂPⁿ = SU(n+1)/S(U(1)×U(n)), and the space of positive definite n×n matrices GL(n)/O(n).

Symmetric spaces are classified into compact type (positive curvature, like spheres and Grassmannians), non-compact type (negative curvature, like hyperbolic spaces and spaces of positive-definite matrices), and Euclidean type (zero curvature, flat tori and Euclidean space). Cartan's classification gives 7 infinite families and 12 exceptional symmetric spaces.