We give a global geometric decomposition of continuously differentiable vector fields on \(\mathbb{R}^{n}\). More precisely, given a vector field of class \(\mathcal{C}^{1}\) on \(\mathbb{R}^{n}\), and a geometric structure on \(\mathbb{R}^{n}\), we provide a unique global decomposition of the vector field as the sum of a left (right) gradient-like vector field (naturally associated to the geometric structure) with potential function vanishing at the origin, and a vector field which is left (right) orthogonal to the Euler vector field, with respect to the geometric structure. As application, we provide a criterion to decide topological conjugacy of complete vector fields of class \(\mathcal{C} ^{1}\) on \(\mathbb{R}^{n}\) based on topological conjugacy of the corresponding parts given by the associated geometric decompositions.

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向量场的全局几何分解及其在拓扑共轭中的应用

我们给出\（\ mathbb {R} ^ {n} \）上连续可微向量场的全局几何分解。更精确地，给定类的矢量场\（\ mathcal {C} ^ {1} \）上\（\ mathbb {R} ^ {N} \），和在几何结构\（\ mathbb {R} ^ {n} \），我们提供矢量场的唯一全局分解，作为左（右）渐变状矢量场（与几何结构自然相关）的和，该矢量在原点处消失，并且矢量场相对于几何结构，它与Euler矢量场正交（左）。作为应用，我们提供了一个标准来决定类的完整的矢量场的拓扑共轭\（\ mathcal {C} ^ {1} \）上\（\ mathbb {R} ^ {n} \）基于相关几何分解给出的相应部分的拓扑共轭。