  We will consider the Navier-Stokes equation on a Riemannian manifold M with
Ricci tensor bounded below, the involved Laplacian operator is De Rham-Hodge
Laplacian. The novelty of this work is to introduce a family of connections
which are related to solutions of the Navier-Stokes equation, so that vorticity
and helicity can be linked through the associated time-dependent Ricci tensor
in intrinsic way in the case where dim(M) = 3. MSC 2010: 35Q30, 58J65
