How to understand grassmannians?

Abstract

Grassmannians or Grassmann manifolds are very important manifolds in modern mathematics. They naturally appear in algebraic topology, differential geometry, analysis, combinatorics, mathematical physics, etc. Grassmannians have very rich geometrical, combinatorial and topological structure, so understanding them has been one of the central research themes in mathematics. They occur in many important constructions such as universal bundles, flag manifolds and others, hence studying their properties and finding their topological and geometrical invariants is still a very attractive question. In this article we offer a quick introduction into the geometry of Grassmannians suitable for readers without any previous exposure to these concepts.