By Oswald; Whitehead, J. H. C. Veblen

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3. Invariant Isotropic Complex Structures Let f : M → M be a smooth map. Then f∗ T(M) −→ T(M) T(T(M)) ↓π π∗ ↓ π↓ f∗∗ −→ ↓ π∗ f M M −→ T(T(M)) f∗ T(M) −→ T(M) are commutative diagrams for any smooth map f : M → M. Indeed let T(T(M)) → T(M) be the projection and V ∈ T(T(M)). Then ( f∗∗ (V )) = π∗ ( f∗∗ (V )) = (d = (d ((d (V ) f∗ )V ) = f∗ ( (V )), ( f∗∗ V ) π )(d (V ) f∗ )V = (df∗ ( (V ) (π (V ) ( ◦ f∗ ))V = (d (V )) π)(d (V ) f∗ )V f ◦ π))V = f∗ (π∗ (V )). 3. 35) where K : T(T(M)) → π −1 TM is the Dombrowski map and p : π −1 TM → T(M) is the restriction to π −1 TM of the projection p : T(M) × T(M) → T(M) given by p(v, ξ ) = ξ for any v, ξ ∈ T(M).

Then the vertical tangent vector field γ X is tangent to S(M) if and only if gˆ(X, L) = 0 where gˆ = π −1 g is 28 Chapter 1 Geometry of the Tangent Bundle the Riemannian bundle metric induced by g in π −1 TM → T(M) and L is the Liouville vector field. 44). A vector field B = Bi ∂i + Bi+n ∂˙i is orthogonal to S(M) if and only if gij Ai B j + gk (Ak+n + Nik Ai )(B +n + Nj B j ) = 0. 29 Let (M, g) be a Riemannian manifold and S(M) → M its tangent sphere bundle. 47) where ωx : (π −1 TM)x → R is defined by ωx (X) = gˆx (X, Lx ) for any X ∈ (π −1 TM)x and any x ∈ M.

Also [110], p. 321–329. By a result of S. Tanno, [282], T1,0 (S(M)) is integrable if and only if (M, g) (with n > 2) has constant sectional curvature. Similarly any isotropic almost complex structure Jδ,σ , defined on an open subset A ⊆ T(M), induces an almost CR structure T1,0 (T(M))δ,σ on A ∩ S(M). It is noteworthy that Jδ,σ with 1 , δ=√ κ(2E − 1) + 1 σ = 0, induces the same almost CR structure as J1,0 yet (unlike J1,0 ) has the same integrability condition as T1,0 (S(M)). 5. THE TANGENT SPHERE BUNDLE OVER A TORUS Let d1 , d2 ∈ R2 be two linearly independent vectors and let be the lattice given by ⊂ R2 = {m d1 + n d2 : m, n ∈ Z}.