Geometric Properties and Distance Inequalities on Grassmannians

TL;DR

This paper explores geometric inequalities on Grassmannians, revealing how their elliptic geometry influences properties like the law of cosines and triangle inequalities in a Riemannian setting.

Contribution

It introduces new inequalities for the geometric mean on Grassmannians, highlighting their elliptic Riemannian geometry and extending classical geometric laws.

Findings

Established semi-parallelogram law on Grassmannians

Derived law of cosines for Grassmannian geodesic triangles

Proved new distance inequalities reflecting elliptic geometry

Abstract

In this paper we obtain inequalities for the geometric mean of elements in the Grassmannians. These inequalities reflect the elliptic geometry of the Grassmannians as Riemannian manifolds. These include Semi-Parallelogram Law, Law of Cosines and geodesic triangle inequalities.