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 $g(X,X_1)=\cdots =g(X,X_k)=0$, $g(X,Y_1)=h_1,\dots ,g(X,Y_p)=h_p$, 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 $h_1,\dots ,h_p$.

21 pages

References in corpus (2)

• On a unified formulation of completely integrable systems
• Geometrical dissipation for dynamical systems

Cited by in corpus (5)

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• Controlling the stability of periodic orbits of completely integrable systems
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