Geometric properties of passive random advection

Geometric properties of a random Gaussian short-time correlated velocity field are studied by considering the statistics of a passively advected metric tensor. That describes the universal properties of the fluctuations of tensor objects frozen into the fluid and passively advected by it. The problem of the one-point statistics of covariant and contravariant tensors is solved exactly, provided that the advected fields do not reach diffusive scales, which would break the symmetry of the problem.