The geometrical flamelet statistics in a turbulent premixed flame are represented by c ¯ , Σ f , 〈 n 〉 f , 〈 ∇ · n 〉 f and 〈 | ∇ · n | 〉 f which are respectively mean reaction progress variable, flame surface density, mean orientation vector, mean curvature and mean absolute curvature. New conditional transport equations are derived for the listed geometrical statistics of successively higher orders in terms of net normal flame motion from the equation of the reaction progress variable. They are simpler and allow easier closure than the unconditionally averaged formulations involving the tangential velocity component and the turbulent flux term involving countergradient diffusion. Balance is checked for all component terms of those transport equations in two freely propagating, constant density DNS flames in statistical steadiness. Results show that the mean orientation vector and the mean absolute curvature remain approximately uniform in local equilibrium with negligible spatial transport except near the edges of a flame brush . A simple closure strategy is suggested with the flame surface density given in terms of uniform mean absolute curvature and mean orientation vectors, 〈 n x 〉 f and 〈 n x 〉 K , given as the conditional averages weighted by Σ f ′ ( = ∂ c / ∂ n ) and ∂ 2 c / ∂ n 2 respectively. The governing equations and closure models are expected to be valid in planar freely propagating turbulent premixed flames of either constant or variable density with heat release. The suggested closure requires prior knowledge on the magnitude of the mean orientation vector and the mean absolute curvature given as the ratio of laminar and turbulent flame speeds and the inverse characteristic wrinkling scale through a flame brush. Good agreement is shown against DNS results of two constant density flames for closure models of the conditional velocities, 〈 u 〉 f and 〈 S d 〉 f , and for the predicted profiles of c ¯ and Σ f except for minor deviation near the edges. The suggested closure scheme may be extended to general turbulent premixed flames, while further work may be required for proper closure relationships under the given circumstances.