Ergodic Theory of Frame Flows

Abstract

Let Mⁿ be a smooth n-dimensional Riemannian manifold and let T₁Mⁿ be the manifold of unit tangent vectors on Mⁿ. The geodesic flow g^t: T₁Mⁿ → T₁Mⁿ translates every vector v ∈ T₁Mⁿ by the parallel translation along the unique geodesic determined by v at distance t. The frame flow ${\text{f}}_{\text{k}}^{\text{t}}$, acts in the space St_k(Mⁿ) of orthonormal ordered k-frames w = {x, v₁,..., v_k}, where x ∈ Mⁿ, v_i ∈ T₁Mⁿ, (v_i, v_j) = δ_ij, 1 ≤ i, j ≤ k. The flow ${\text{f}}_{\text{k}}^{\text{t}}$ translates every frame w∈ St_k(Mⁿ) along the geodesic determined by the first vector of the frame at distance t. It is clear that St₁(Mⁿ) = T₁ Mⁿ. Denote by p_k, the natural projection St_k(Mⁿ) → St₁(Mⁿ), by p_{k, m}, 1 ≤ m ≤ k ≤ n, the projections St_k(Mⁿ) → St_m(Mⁿ), p_{k, m}(x, v_l,..., v_k) = (x, v₁... v _m), and by π: St₁Mⁿ→ Mⁿ the projection π(x, v) = x. If Mⁿ is oriented, the space St_n(Mⁿ) has two connected components—the sets of positively and negatively oriented n-frames, each of the components being isomorphic with the space St_n−1 (Mⁿ). The manifold St_k(Mⁿ) is a fiber bundle over St₁(Mⁿ) = T₁(Mⁿ) with projection p_k and structure group SO(n−1).

Supported by NSF Grant #MCS79-0304.