Poincaré-Hopf定理的推广
A generalisation of the Poincaré-Hopf Theorem
摘要：
庞加莱-霍普夫定理（Poincaré-Hopf Theorem）是实解析向量场的一个守恒律，它切向于一个闭曲面（如环面或球面）。该定理也适用于实解析向量场，它与光滑边界的曲面相切;在这些情况下，向量场必须指向沿边界向外的法线方向。在本文中，我将推广的庞加莱-霍普夫定理的实解析向量场是切表面与分片光滑边界，而不是平行于外法线的边界。
英文摘要：
The Poincaré-Hopf Theorem is a conservation law for real-analytic vector fields, which are tangential to a closed surface (such as a torus or a sphere). The theorem also governs real-analytic vector fields, which are tangential to surfaces with smooth boundaries; in these cases, the vector field must be pointing in the outward normal direction along the boundary. In this paper, I will generalise the Poincaré-Hopf Theorem for real-analytic vector fields that are tangential to surfaces with piecewise smooth boundaries, and not parallel to the outward normal of the boundary.
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