Article ID Journal Published Year Pages File Type
1894710 Journal of Geometry and Physics 2015 14 Pages PDF
Abstract

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.

Related Topics
Physical Sciences and Engineering Mathematics Mathematical Physics
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