A cone structure on a complex manifold M is a closed submanifold \(\mathcal {C}\subset \mathbb {P}TM\) of the projectivized tangent bundle which is submersive over M. A conic connection on \(\mathcal {C}\) specifies a distinguished family of curves on M in the directions specified by \(\mathcal {C}\). There are two common sources of cone structures and conic connections, one in differential geometry and another in algebraic geometry. In differential geometry, we have cone structures induced by the geometric structures underlying holomorphic parabolic geometries, a classical example of which is the null cone bundle of a holomorphic conformal structure. In algebraic geometry, we have the cone structures consisting of varieties of minimal rational tangents (VMRT) given by minimal rational curves on uniruled projective manifolds. The local invariants of the cone structures in parabolic geometries are given by the curvature of the parabolic geometries, the nature of which depend on the type of the parabolic geometry, i.e., the type of the fibers of \(\mathcal {C}\rightarrow M\). For the VMRT-structures, more intrinsic invariants of the conic connections which do not depend on the type of the fiber play important roles. We study the relation between these two different aspects for the cone structures induced by parabolic geometries associated with a long simple root of a complex simple Lie algebra. As an application, we obtain a local differential-geometric version of the global algebraic-geometric recognition theorem due to Mok and Hong–Hwang. In our local version, the role of rational curves is completely replaced by appropriate torsion conditions on the conic connection.

复流形M上的锥结构是投影化切丛的封闭子流形\(\mathcal {C}\subset \mathbb {P}TM\)，它浸没在M 上。上的一个锥形连接\（\ mathcal {C} \）指定的曲线的大的家族上中号中所指定的方向\（\ mathcal {C} \）. 圆锥结构和圆锥连接有两种常见的来源，一种是微分几何，另一种是代数几何。在微分几何中，我们有由全纯抛物线几何基础的几何结构引起的锥结构，其经典示例是全纯保形结构的零锥丛。在代数几何中，我们有锥体结构，它由由单规则射影流形上的最小有理曲线给出的各种最小有理切线 (VMRT) 组成。抛物线几何中锥体结构的局部不变量由抛物线几何的曲率给出，其性质取决于抛物线几何的类型，即\(\mathcal {C}\rightarrow M\). 对于 VMRT 结构，不依赖于纤维类型的圆锥连接的更多内在不变量起着重要作用。我们研究了由与复杂简单李代数的长简单根相关联的抛物线几何引起的锥体结构的这两个不同方面之间的关系。作为应用，由于 Mok 和 Hong-Hwang，我们获得了全局代数几何识别定理的局部微分几何版本。在我们的本地版本中，有理曲线的作用完全被圆锥连接上的适当扭转条件所取代。