On local generators of affine distributions on Riemannian manifolds

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 $X\in\mathfrak{X}(U)$ defined eventually on an open subset $U\subseteq M$ of a smooth Riemannian manifold $(M,g)$, that verifies the relations $g(X,X_1)=\dots =g(X,X_k)=0$, $g(X,Y_1)=h_1, \dots, g(X,Y_p)=h_p$, where $X_1,\dots, X_k, Y_1, \dots, Y_p \in\mathfrak{X}(U)$, and respectively $h_1,\dots, h_p \in \mathcal{C}^{\infty}(U,\mathbb{R})$, are a-priori given quantities. In the case when $X_1,\dots, X_k, Y_1, \dots, Y_p$ are gradient vector fields associated with some smooth functions $I_1,\dots, I_k, D_1, \dots, D_p \in \mathcal{C}^{\infty}(U,\mathbb{R})$, i.e., $X_1 =\nabla_{g}I_1$, $\dots$, $X_k = \nabla_{g}I_k$, $Y_1 = \nabla_{g}D_1$, $\dots, Y_p = \nabla_{g}D_p$, then we obtain a set of local generators for the smooth affine distribution of smooth vector fields which conserve the quantities $I_1, \dots, I_k$ and dissipate the scalar quantities $D_1, \dots, D_p$ with prescribed rates $h_1, \dots, h_p$.