Timeline for Canonic identification of the tangent space of the Grassmannian

8 events

when toggle format	what		by	license	comment
Jan 19, 2019 at 20:20	comment	added	Deane Yang		It's the topology, where $\Phi_L$ is continuous (in fact, smooth) for every $K$ and $L$.
Jan 19, 2019 at 13:34	comment	added	rmdmc89		@DeaneYang, when you talk about an "open neighbourhood of $K$", you're assuming a topology on $G(k,V)$. What topology is it?
Nov 15, 2017 at 22:09	comment	added	Deane Yang		Given $K \in G(k,V)$ and a transversal subspace $L$, there is a natural map $\Phi_L: Hom(K,L) \rightarrow G(k,V)$ by $\Phi(\phi) = \{ k + \phi(k)\ :\ k \in K\}$. Check that 1) this is a diffeomorphism onto an open neighborhood of $K$ and 2) $\Phi_{L'} = \Phi_L$, if $[L] = [L'] \in V/K$. Therefore, $\Phi_L$ can be viewed as a map $\Phi: Hom(K,V/K) \rightarrow G(k,V)$. It follows from this that $T_KG(k,V) = Hom(K,V/K)$.
Nov 15, 2017 at 19:15	answer	added	Ning JIANG		timeline score: 1
Jul 6, 2013 at 8:24	answer	added	Neil Strickland		timeline score: 21
Jul 6, 2013 at 4:40	comment	added	Deane Yang		It seems to me that the easiest way to understand the Grassmannian as a manifold and its tangent bundle is by viewing it as a homogeneous space, i.e, the quotient of $GL(v)$ by the appropriate subgroup. And it's worth working everything out for projective space first.
Jul 6, 2013 at 0:53	review	First posts
Jul 6, 2013 at 1:42
Jul 6, 2013 at 0:36	history	asked	Tom Fellmann	CC BY-SA 3.0