黎曼流形上仿射分布的局部生成元
On local generators of affine distributions on Riemannian manifolds
摘要：
利用超平面相交的无坐标特征，我们明确地给出了一组光滑仿射分布的局部生成元，该分布由最终定义在光滑黎曼流形$(M,g)$的开子集$U\subseteq M$上的光滑向量场$X\in\mathfrak{X}(U)$给出，从而验证了$g(X,X_1)=\dots =g(X,X_k)=0$，$g(X,Y_1)=h_1, \dots, g(X,Y_p)=h_p$之间的关系，其中$X_1,\dots, X_k, Y_1, \dots, Y_p \in\mathfrak{X}(U)$和$h_1,\dots, h_p \in \mathcal{C}^{\infty}(U,\mathbb{R})$分别是先验给定量.在$X_1,\dots, X_k, Y_1, \dots, Y_p$是与某些平滑函数$I_1,\dots, I_k, D_1, \dots, D_p \in \mathcal{C}^{\infty}(U,\mathbb{R})$相关联的梯度向量场的情况下，即，$X_1 =\nabla_{g}I_1$，$\dots$，$X_k = \nabla_{g}I_k$，$Y_1 = \nabla_{g}D_1$，$\dots, Y_p = \nabla_{g}D_p$，得到了光滑向量场的光滑仿射分布的一组局部生成元，这些局部生成元使量$I_1, \dots, I_k$守恒，标量量$D_1, \dots, D_p$以给定的速率$h_1, \dots, h_p$耗散.
英文摘要：
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 $h_1, \dots, h_p$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
