Relationship between Cross-cap and Grassman Manifold

Question

My professor, said that https://en.wikipedia.org/wiki/Cross-cap#/media/File:CrossCapTwoViews.PNG is a visualization of Grassmann Manifold of n=3,d=1. Can anyone help me understand this please.

Answer 1

$G_{n,k} $ is defined to be the collection of $k $-dimensional vector subspaces of $\mathbb R^n $. The case $G {3,1} $ is lines in $\mathbb R^3$. There must be a diffeomorphism between $G {3,1} $ and the cross-cap.

The space is in fact $\mathbb RP^2$,the real projective plane. See Möbius strip description . ..

(It's the sphere with antipodal points identified. ..)

Here's a reference I came across.