In this paper, we study germs of smooth CR mappings sending a closed orbit of $SU(\ell, m)$ into a closed orbit of $SU(\ell^\prime , m^\prime)$ in Grassmannian manifolds. We show that if the signature difference of the Levi forms of two orbits is not too large, then the mapping can be factored into a simple form and one of the factors extends to a totally geodesic embedding of the ambient Grassmannian with respect to the standard metric. As an application, we give a sufficient condition for a proper holomorphic mapping between type I bounded symmeric domains to be the product of trivial embedding and a holomorphic mapping into a subdomain.

中文翻译：

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格拉斯曼流形中闭合 $SU(\ell, m)$-轨道之间的全纯映射

在本文中，我们研究了在格拉斯曼流形中将 $SU(\ell, m)$ 的闭合轨道发送到 $SU(\ell^\prime , m^\prime)$ 的闭合轨道的平滑 CR 映射的萌芽。我们表明，如果两个轨道的 Levi 形式的签名差异不是太大，那么映射可以分解为一个简单的形式，其中一个因素扩展到环境 Grassmannian 相对于标准度量的完全测地线嵌入. 作为一个应用，我们给出了一个充分条件，使 I 型有界对称域之间的适当全纯映射是平凡嵌入和全纯映射到子域的产物。