On Jordan angles and triangle inequality in Grassmannian

Vector bundles over Grassmannians and the skew-symmetric curvature operator

A pseudo-Riemannian manifold is said to be spacelike Jordan IP if the Jordan normal form of the skew-symmetric curvature operator depends upon the point of the manifold, but not upon the particular spacelike 2-plane in the tangent bundle at that point. We use methods of algebraic topology to classify connected spacelike Jordan IP pseudo-Riemannian manifolds of signature $(p,q)$, where $q\ge 11$, $p\le \frac{q-6}{4}$ and where the set $\{q, . . ., q+p\}$ does not contain a power of 2.

Spherical Geometry and the Least Symmetric Triangle

We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar $n$-gons and points in the Grassmannian of 2-planes in real $n$-space introduced by Hausmann and Knutson, this corresponds to finding the point in the fundamental domain of the hyperoctahedral group action on the Grassmannian which is furthest from the boundary, which we compute exactly. We also determine the least symmetric obtuse and acute triangles. These calculations provide prototypes for comp…

Grassmann流形上的几何性质与距离不等式

Geometric Properties and Distance Inequalities on Grassmannians

本文研究了Grassmann流形上元素的几何平均不等式，揭示了其作为黎曼流形的椭圆几何性质，包含半平行四边形定律、余弦定律和测地三角形不等式。

与若尔当代数相关的紧对称空间的三明治算子与爱因斯坦形变

Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras

本研究通过构造紧李代数的三明治算子、结合若尔当代数分析，解决了两类对称空间的爱因斯坦度量二阶形变问题，证明了多数可无穷小形变的不可约紧对称空间的非线性不稳定性。

Random Triangles and Polygons in the Plane

We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. In these terms, we answer Caroll's question. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space …

A spherical Bernstein theorem for minimal submanifolds of higher codimension

Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed region of a Grassmannian manifold. The proof depends on the subharmoncity of an auxiliary function, the Codazzi equations and geometric measure theory.