Affine distributions on Riemannian manifolds with applications to dissipative dynamics

Using a coordinate free characterization of hyperplanes intersection, we provide explicitly a set of local generators for a smooth affine distribution given by those smooth vector fields X∈X(U)X∈X(U) defined eventually on an open subset U⊆MU⊆M of a smooth Riemannian manifold (M,g)(M,g), that verifies the relations g(X,X1)=⋯=g(X,Xk)=0g(X,X1)=⋯=g(X,Xk)=0, g(X,Y1)=h1,…,g(X,Yp)=hpg(X,Y1)=h1,…,g(X,Yp)=hp, where X1,…,Xk,Y1,…,Yp∈X(U)X1,…,Xk,Y1,…,Yp∈X(U), and respectively h1,…,hp∈C∞(U,R)h1,…,hp∈C∞(U,R), are a-priori given quantities.In the case when X1,…,Xk,Y1,…,YpX1,…,Xk,Y1,…,Yp are gradient vector fields associated with some smooth functions I1,…,Ik,D1,…,Dp∈C∞(U,R)I1,…,Ik,D1,…,Dp∈C∞(U,R), i.e.,  X1=∇gI1,…,Xk=∇gIk,Y1=∇gD1,…,Yp=∇gDpX1=∇gI1,…,Xk=∇gIk,Y1=∇gD1,…,Yp=∇gDp, then we obtain a set of local generators for the smooth affine distribution of smooth vector fields which conserve the quantities I1,…,IkI1,…,Ik and dissipate the scalar quantities D1,…,DpD1,…,Dp with prescribed rates h1,…,hph1,…,hp.