Relationship between Cross-cap and Grassman Manifold My professor, said that < is a visualization of Grassmann Manifold of n=3,d=1. Can anyone help me understand this please.

$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.