Formalizing the equivalence $T_WG(k,n)simeq text{Lin}(W,mathbb{R}^n/W)$ in Grassmannians
$begingroup$
Let $G(k,n):={Wsubsetmathbb{R}^nmid Wtext{ is a } ktext{-dimensional subspace}}$.
I'm trying to understand the usual identification $T_WG(k,n)simeq text{Lin}(W,mathbb{R}^n/W)$.
In more than one place, I've read the following claim (without much explanation):
Take a curve $W(t)$ in $G(k,n)$ with $W(0)=W$. If $w_1(t),...,w_k(t)$ are curves in $mathbb{R}^n$ such that $W(t)=text{span}(w_1(t),...,w_k(t))$, we may idenfify $W'(0)$ with the map $Wtomathbb{R}^n/W$ defined by $w_i(0)mapsto w_i'(0)+W$.
After drawing a couple sketches, I was convinced this makes sense intuitively: the equivalent class $w_i'(0)+W$ basically measures how fast $w_i(t)$ is moving away from $W$ at $t=0$. If I know this behavior for all $w_1(t),...,w_k(t)$, then I know exactely how $W(t)$ moves away from $W$ at $t=0$.
But I'm having trouble formalizing this equivalence. How do I find the bijection $T_WG(k,n)to text{Lin}(W,mathbb{R}^n/W)$, explicitly?
I guess my problem is that I don't know how to treat elements in $T_WG(k,n)$, because they seem so abstract.
Any suggestions?
smooth-manifolds grassmannian tangent-bundle
asked Jan 23 at 1:32
rmdmc89rmdmc89
2,2271923
$endgroup$
add a comment |
$begingroup$
Let $G(k,n):={Wsubsetmathbb{R}^nmid Wtext{ is a } ktext{-dimensional subspace}}$.
I'm trying to understand the usual identification $T_WG(k,n)simeq text{Lin}(W,mathbb{R}^n/W)$.
In more than one place, I've read the following claim (without much explanation):
Take a curve $W(t)$ in $G(k,n)$ with $W(0)=W$. If $w_1(t),...,w_k(t)$ are curves in $mathbb{R}^n$ such that $W(t)=text{span}(w_1(t),...,w_k(t))$, we may idenfify $W'(0)$ with the map $Wtomathbb{R}^n/W$ defined by $w_i(0)mapsto w_i'(0)+W$.
After drawing a couple sketches, I was convinced this makes sense intuitively: the equivalent class $w_i'(0)+W$ basically measures how fast $w_i(t)$ is moving away from $W$ at $t=0$. If I know this behavior for all $w_1(t),...,w_k(t)$, then I know exactely how $W(t)$ moves away from $W$ at $t=0$.
But I'm having trouble formalizing this equivalence. How do I find the bijection $T_WG(k,n)to text{Lin}(W,mathbb{R}^n/W)$, explicitly?
I guess my problem is that I don't know how to treat elements in $T_WG(k,n)$, because they seem so abstract.
Any suggestions?
smooth-manifolds grassmannian tangent-bundle
asked Jan 23 at 1:32
rmdmc89rmdmc89
2,2271923
$endgroup$
add a comment |
$begingroup$
Let $G(k,n):={Wsubsetmathbb{R}^nmid Wtext{ is a } ktext{-dimensional subspace}}$.
I'm trying to understand the usual identification $T_WG(k,n)simeq text{Lin}(W,mathbb{R}^n/W)$.
In more than one place, I've read the following claim (without much explanation):
Take a curve $W(t)$ in $G(k,n)$ with $W(0)=W$. If $w_1(t),...,w_k(t)$ are curves in $mathbb{R}^n$ such that $W(t)=text{span}(w_1(t),...,w_k(t))$, we may idenfify $W'(0)$ with the map $Wtomathbb{R}^n/W$ defined by $w_i(0)mapsto w_i'(0)+W$.
After drawing a couple sketches, I was convinced this makes sense intuitively: the equivalent class $w_i'(0)+W$ basically measures how fast $w_i(t)$ is moving away from $W$ at $t=0$. If I know this behavior for all $w_1(t),...,w_k(t)$, then I know exactely how $W(t)$ moves away from $W$ at $t=0$.
But I'm having trouble formalizing this equivalence. How do I find the bijection $T_WG(k,n)to text{Lin}(W,mathbb{R}^n/W)$, explicitly?
I guess my problem is that I don't know how to treat elements in $T_WG(k,n)$, because they seem so abstract.
Any suggestions?
smooth-manifolds grassmannian tangent-bundle
asked Jan 23 at 1:32
rmdmc89rmdmc89
2,2271923
$endgroup$
Let $G(k,n):={Wsubsetmathbb{R}^nmid Wtext{ is a } ktext{-dimensional subspace}}$.
I'm trying to understand the usual identification $T_WG(k,n)simeq text{Lin}(W,mathbb{R}^n/W)$.
In more than one place, I've read the following claim (without much explanation):
Take a curve $W(t)$ in $G(k,n)$ with $W(0)=W$. If $w_1(t),...,w_k(t)$ are curves in $mathbb{R}^n$ such that $W(t)=text{span}(w_1(t),...,w_k(t))$, we may idenfify $W'(0)$ with the map $Wtomathbb{R}^n/W$ defined by $w_i(0)mapsto w_i'(0)+W$.
After drawing a couple sketches, I was convinced this makes sense intuitively: the equivalent class $w_i'(0)+W$ basically measures how fast $w_i(t)$ is moving away from $W$ at $t=0$. If I know this behavior for all $w_1(t),...,w_k(t)$, then I know exactely how $W(t)$ moves away from $W$ at $t=0$.
But I'm having trouble formalizing this equivalence. How do I find the bijection $T_WG(k,n)to text{Lin}(W,mathbb{R}^n/W)$, explicitly?
I guess my problem is that I don't know how to treat elements in $T_WG(k,n)$, because they seem so abstract.
Any suggestions?
smooth-manifolds grassmannian tangent-bundle
smooth-manifolds grassmannian tangent-bundle
asked Jan 23 at 1:32
rmdmc89rmdmc89
2,2271923
asked Jan 23 at 1:32
rmdmc89rmdmc89
2,2271923
edited Jan 23 at 12:20
rmdmc89
asked Jan 23 at 1:32
rmdmc89rmdmc89
2,2271923
asked Jan 23 at 1:32
rmdmc89rmdmc89
2,2271923
asked Jan 23 at 1:32
rmdmc89rmdmc89
2,2271923
2,2271923
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