The Grassmann manifold\(Gr_m({\mathbb {R}}^n)\) of all m-dimensional subspaces of the n-dimensional space \({\mathbb {R}}^n\)\((m<n)\) is widely used in image analysis, statistics and optimization. Motivated by interpolation in the manifold \(Gr_2({\mathbb {R}}^4)\), we first formulate the differential equation for desired interpolation curves called Riemannian cubics in symmetric spaces by the Pontryagin maximum principle (PMP) and then narrow down to it in \(Gr_2({\mathbb {R}}^4)\). Although computation on this low-dimensional manifold may not occur heavy burden for modern machines, theoretical analysis for Riemannian cubics is very limited in references due to its highly nonlinearity. This paper focuses on presenting analytical and geometrical structures for the so-called Lie quadratics associated with Riemannian cubics. By analysing asymptotics of Lie quadratics, we find asymptotics of Riemannian cubics in \(Gr_2({\mathbb {R}}^4)\). Finally, we illustrate our results by numerical simulations.

n维空间\（{\ mathbb {R}} ^ n \）所有m维子空间的Grassmann流形\（Gr_m（{\ mathbb {R}} ^ n）\）\（（m <n ）\）广泛用于图像分析，统计和优化。通过在流形\（Gr_2（{\ mathbb {R}} ^ 4）\）中进行插值，我们首先通过Pontryagin最大原理（PMP）公式化了对称空间中所需插值曲线的黎曼三次方程的微分方程，向下到\（Gr_2（{\ mathbb {R}} ^ 4）\）。尽管在此低维流形上的计算可能不会给现代机器带来沉重的负担，但由于黎曼三次方程具有高度的非线性，因此其理论分析在参考文献中非常有限。本文重点介绍与黎曼三次方相关的所谓李二次方程的解析和几何结构。通过分析李二次方程的渐近性，我们在\（Gr_2（{\ mathbb {R}} ^ 4）\）中找到黎曼三次立方的渐近性。最后，我们通过数值模拟说明了我们的结果。