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For every globally hyperbolic spacetime M, we derive new mixed gravitational field equations embodying the smooth Geroch infinitesimal splitting T (M) = D ⊕ R∇T of M, as exhibited by Bernal and Sánchez (2005 Commun. Math. Phys. 257 43-50). We give sufficient geometric conditions (e.g. T is isoparametric and D is totally umbilical) for the existence of exact solutions −β dT ⊗ dT + g to mixed field equations in free space. We linearize and solve the mixed field equations Ric D (g) μν − ρ D (g) g μν = 0 for empty space, where ρ D (g) is the mixed scalar curvature of foliated spacetime (M, D) (due to Rovenski (2010 arXiv:1010.2986 v1[math.DG])). If g = g 0 + γ is a solution to the linearized field equations, then each leaf of D is totally geodesic in (R 4 \ R, g ) to order O( ). We derive the equations of motion of a material particle in the gravitational field g μν governed by the mixed field equations Ric D (g) μν − ρ D (g) ω μ ω ν − g μν = 2πκc −2 T μν − 1 2 T g μν . In the weak field ( 1) and low velocity ( v /c 1) limit, the motion equations are d 2 r/dt 2 = ∇φ + F, where φ = ( /2)c 2 γ 00 .

We build a variational theory of geodesics of the Tanaka-Webster connection ∇ on a strictly pseudoconvex CR manifold M . Given a contact form θ on M such that (M, θ) has nonpositive pseudohermitian sectional curvature (k θ (σ) ≤ 0) we show that (M, θ) has no horizontally conjugate points. Moreover, if (M, θ) is a Sasakian manifold such that k θ (σ) ≥ k 0 > 0 then we show that the distance between any two consecutive conjugate points on a lengthy geodesic of ∇ is at most π/(2 √ k 0 ). We obtain the first and second variation formulae for the Riemannian length of a curve in M and show that in general geodesics of ∇ admitting horizontally conjugate points do not realize the Riemannian distance.

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