标题： 关于三角形庞斯列小瓣摆线定理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： 彭赛列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问题。凯莱定理通过特定行列式展开的系数给出了这类多边形存在的代数条件。本文专注于三角形的情形，并建立了关于凯莱条件几何性质的三个主要结果。首先，我们推导出一个明确的代数方程来定义凯莱集——即允许彭赛列三角形存在的圆锥曲线对的轨迹——并证明该集合是在圆锥曲线对空间中的可构造集合。其次，我们通过展示凯莱集是具有二次超曲面同构纤维的平滑平凡纤维丛的总空间的一个开子集，证明凯莱集是一个光滑流形。最后，我们将彭赛列对应关系表述为主丛，其纤维为复环面（椭圆曲线），并证明凯莱集中由圆锥曲线对产生的所有椭圆曲线彼此同构。这种微分几何方法不仅阐明了彭赛列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.