Stiefel流形和Grassmann流形上的极值曲线
Extremal curves on Stiefel and Grassmann manifolds
摘要：
本文揭示了李群上的一类左不变的子黎曼系统，它们具有某些性质的显式解，并为Stiefel流形上的一类重要曲线（称为拟测地线）提供了几何起源，这些曲线作为黎曼测地线投影到Grassmann流形上.我们表明，准测地线是投影的亚黎曼测地线所产生的某些左不变的分布在李群上的每个Stiefel流形$St_k^n(V)$传递作用。这个结果不仅对$V=\mathbb R^n$中的实Stiefel流形有效，而且对Hermite空间$V=\mathbb C^n$和四元数空间$V=\mathbb H^n$中的Stiefel流形也有效.
英文摘要：
This paper uncovers a large class of left-invariant sub-Rie\-mannian systems on Lie groups that admit explicit solutions with certain properties, and provides geometric origins for a class of important curves on Stiefel manifolds, called quasi-geodesics, that project on Grassmann manifolds as Riemannian geodesics. We show that quasi-geodesics are the projections of sub-Riemannian geodesics generated by certain left-invariant distributions on Lie groups that act transitively on each Stiefel manifold $St_k^n(V)$. This result is valid not only for the real Stiefel manifolds in $V=\mathbb R^n$, but also for the Stiefels in the Hermitian space $V=\mathbb C^n$ and the quaternion space $V=\mathbb H^n$.
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