标题： 关于三角形庞塞罗抛物线中的凯莱条件的光滑性 显示英文标题

标题： On The Smoothness of Cayley Conditions in Poncelet Porism for Triangles

摘要： 庞塞莱闭合定理涉及同时内接于一个圆锥曲线且外切于另一个圆锥曲线的多边形的存在性。凯莱定理通过某个行列式展开系数提供了此类多边形存在的代数条件。本文集中讨论三角形的情况，并就凯莱条件的几何性质建立了三个主要结果。首先，我们推导出一个明确的代数方程来定义凯莱集——即允许庞塞莱三角形存在的圆锥曲线对的轨迹——并证明这个集合是圆锥曲线对空间中的一个可构造集合。其次，我们通过展示凯莱集是具有二次超曲面纤维的平滑平凡纤维丛的开子集，证明凯莱集是一个光滑流形。最后，我们将庞塞莱对应关系表述为一个以复环面（椭圆曲线）为纤维的主丛，证明来自凯莱集内圆锥曲线对的所有椭圆曲线都是同构的。这种微分几何方法不仅阐明了庞塞莱三角形解空间的结构。 显示英文摘要

摘要： Poncelet's porism concerns the existence of polygons simultaneously inscribed in one conic and circumscribed about another. Cayley's theorem provides algebraic conditions for such polygons to exist, expressed through the coefficients of a certain determinantal expansion. In this paper, we focus on the case of triangles and establish three main results concerning the geometry of the Cayley condition. First, we derive an explicit algebraic equation that defines the Cayley set - the locus of conic pairs admitting Poncelet triangles - and prove that this set is a constructible set in the space of conic pairs. Second, we demonstrate that the Cayley set is a smooth manifold by showing it is an open subset of the total space of a smooth trivial fiber bundle with fibers isomorphic to a quadric hypersurface. Finally, we formulate the Poncelet correspondence as a principal bundle whose fibers are complex tori (elliptic curves), proving that all elliptic curves arising from conic pairs in the Cayley set are isomorphic. This differential-geometric approach not only clarifies the structure of the solution space for Poncelet triangles.