发表机构
Departamento de Matemáticas, Universidad de León; Departamento de Matemática Aplicada, Universidad de Málaga（莱昂大学数学系; 马拉加大学应用数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨黎曼几何中对应空间导出的嘉当几何，证明(Euc(n),O(n-1))型嘉当几何与底流形的近接触度量结构等数据一一对应，且n≥3时存在正则嘉当联络。
AI 中文摘要
嘉当几何是齐性空间的弯曲类比，对应空间构造可从一种类型的嘉当几何在更大的底流形上生成另一种不同类型的嘉当几何。黎曼流形的单位切丛按此表述是通过该方式得到的(Euc(n),O(n-1))型嘉当几何；我们研究这类一般类型的嘉当几何，无论其是否由对应空间生成。我们证明每一个这类几何都会在其底流形上诱导出近接触度量结构、切丛的正交分解及相容线性联络；反之，这些数据可确定该几何，故对应关系为双射。当n≥3时，我们通过斯宾塞微分表述的挠率条件选出一个特殊的嘉当联络，称之为正则嘉当联络，该正则化由对应空间构造决定：黎曼流形单位切丛上诱导的嘉当联络是正则的。
英文摘要
Cartan geometries are curved analogues of homogeneous spaces, and the correspondence space construction produces, from a Cartan geometry of one type, another of a different type over a larger base manifold. The unit tangent bundle of a Riemannian manifold is, in this language, a Cartan geometry of type $(\operatorname{Euc}(n),O(n-1))$ obtained in this way; we study Cartan geometries of this type in general, whether or not they arise as correspondence spaces. We show that every such geometry induces on its base manifold an almost contact metric structure, together with an orthogonal splitting of the tangent bundle and a compatible linear connection; conversely, these data determine the geometry, so that the correspondence is a bijection. When $n\ge3$, we single out a distinguished Cartan connection, which we call normal, by a condition on its torsion formulated in terms of the Spencer differential. This normalization is dictated by the correspondence space construction: the Cartan connection induced on the unit tangent bundle of a Riemannian manifold is normal.
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