Direct numerical simulations of a turbulent premixed stoichiometric methane-oxygen flame were conducted. The chosen combustion pressure was 20 bar, to resemble conditions encountered in modern rocket combustors. The chemical reactions followed finite rate detailed mechanism integrated into the EBI-DNS solver within the OpenFOAM framework. Flame geometry was thoroughly investigated to assess its interaction with the transport of turbulent properties. The resulting flame front was remarkably thin, with high density gradients and moderate Karlovitz and Damköhler numbers. At mid-flame positions, the variable-density transport mechanisms dominated, leading to the generation of both vorticity and turbulence. A reversion of this trend towards the products was observed. For intermediate combustion progress, vorticity transport is essentially a competition between the baroclinic torque and vortex dilatation. The growth of turbulent kinetic energy is strongly correlated to this process. A geometrical analysis reveals that the generation of enstrophy and turbulence is restricted to specific topologies. Convergent and divergent flame propagation promote turbulence creation due to pressure fluctuation gradients through different physical processes. The possibility of modeling turbulence transport based on curvature is discussed along with the inherent challenges.