On The Geometry and Topology of Cayley Condition in Poncelet Porism for Triangles

Abstract

This paper investigates the differential-geometric and topological properties of the Cayley condition in Poncelet porism for triangles, defined as the locus of pairs of non-degenerate conics that admit a Poncelet triangle. While the algebraic condition for this porism, established by Cayley, is classical, the geometric nature of the set of solutions has remained largely unexplored. We demonstrate that this Cayley set is a smooth, connected, 9-dimensional complex manifold. This is proven by showing it is an open subset of a smooth algebraic variety endowed with a trivial fiber bundle structure over the space of non-degenerate conics. To further analyze its structure, we construct the moduli space of transversely intersecting conic pairs under the action of $\mathbb{P}GL_3(\mathbb{C})$ and identify it with an open subset of $\mathbb{CP}^2/S_3$. We compute the fundamental group of a generic orbit. The elliptic j-invariant is then introduced as a holomorphic map on this space of conics, which factors through this moduli space. We analyze the subset of the Cayley set where this map is a submersion - its regular part - which corresponds to excluding points whose j-invariant is one of the critical values $0$ or $1728$. We prove that this regular part is the total space of a fiber bundle over $\mathbb{C} \setminus \{0,1728\}$. This structure allows for the computation of the fundamental group via the long exact sequence of homotopy. Finally, we provide a principal bundle formulation for the Poncelet correspondence itself over orbits of conic pairs.