  Incompressible and statistically homogeneous flows obey exact kinematic
relations. The Betchov homogeneity constraints (Betchov, J. Fluid Mech., vol.
1, 1956, pp. 497-504) for the average principal invariants of the velocity
gradient are among the most well known and extensively employed homogeneity
relations. These homogeneity relations have far-reaching implications for the
coupled dynamics of strain and vorticity, as well as for the turbulent energy
cascade. Whether the Betchov homogeneity constraints are the only possible ones
or whether additional homogeneity relations exist has not been proven yet. Here
we show that the Betchov homogeneity constraints are the only homogeneity
constraints for incompressible and statistically isotropic velocity gradient
fields. We also extend our results to derive homogeneity relations involving
the velocity gradient and other dynamically relevant quantities, such as the
pressure Hessian.
