Flag Manifolds and Grassmannians

TL;DR

This paper explores the geometric structures of flag manifolds and Grassmannians, revealing their roles in describing point configurations and their interactions within spacetime, along with deriving related curvature tensors.

Contribution

It establishes the relationship between flag manifolds, Grassmannians, and spacetime point configurations, and derives curvature tensors for these geometric structures.

Findings

Flag manifolds describe relations between point configurations in spacetime.

Grassmannians are special cases of flag manifolds with two subsets.

Curvature tensors for flag manifolds are derived, linking geometry to point interactions.

Abstract

Flag manifolds are shown to describe the relations between configurations of distinguished points (topologically equivalent to punctures) embedded in a general spacetime manifold. Grassmannians are flag manifolds with just two subsets of points selected out from a set of N points. The geometry of Grassmannians is determined by a group acting by linear fractional transformations, and the associated Lie algebra induces transitions between subspaces. Curvature tensors are derived for a general flag manifold, showing that interactions between a subset of k points and the remaining N-k points in the configuration is determined by the coordinates in the flag manifold.