Flat connection: holonomy is invariant under homotopy of loops
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I am looking for an elementary proof of the fact (and probably not any broader results) that when the connection is flat, holonomy does not change under homotopy of the loop. Although this is a first step of many further results, I have failed to find a standalone proof. Any reference would be appreciated.
Let $pi: Pmapsto M$ be a principal $G$-bundle with connection $A$. Let $x_0in M$. For a loop $gamma$ in $M$ based at $x_0$, define its holonomy as the map $h_{A,gamma}: P|_{x0}mapsto G$ such that the horizontal lift of $gamma$ that starts at $p$ ends at $ph_{A,gamma}(p)$.
(It is clear to me that $h_{A,gamma}(pg^{-1})=gh_{A,gamma}(p)g^{-1}$ $forall gin G$).
I found the following argument in Taubes' Differential Geometry, with I was not able to work out. Any help would be appreciated. (It is not necessary, but for notational convenience assume G is a matrix group.)
Let $f:[-1,1]times[0,2pi]mapsto M$ be a smooth map such that each $f(s,cdot)$ is a loop in $M$ based at $x_0$. For $pin P|_{x_0}$, let $f_{A,p}:[-1,1]times[0,2pi]mapsto P$ be such that $f_{A,p}(cdot,0)=p$ and each $f_{A,p}(s,cdot)$ is the horizontal lift of $f(s,cdot)$. We want to prove that $f_{A,p}(cdot,1)$ is constant when A is a flat connection.
Let $Fin C^{infty}(M,(Ptimes_{Ad}lie(G))otimeswedge^2T^*M))$ be the curvature of the connection. For any $pin P$, $F|_p$ is understood as a map from $wedge^2T_{pi(p)}M$ to $lie(G)$ such that $F|_{pg^{-1}}=Ad_g(F|_p)$.
The author claims that $h_{A,f(s,cdot)}(p)^{-1}partial_sh_{A,f(s,cdot)}(p)=int_0limits^{2pi}F|_{f_{A,p}(s,t)}(f_*partial_t,f_*partial_s))dt$ for any $A$, thus vanishes when $A$ is flat.
In particular, in a subset $U$ of $[-1,1]times[0,2pi]$ where $P|_{f(U)}$ is trivializable, write $P|_{f(U)}=f(U)times G$, $A=g^{-1}dg+Ad_{g^{-1}}a$ where $a$ is a $lie(G)$ valued 1-form on $f(U)$. $F=da+awedge a$ is a $lie(G)$ valued 2-form on $f(U)$.
The horizontal lift satisfies $partial_tf_{A,p}(s,t)=-a(partial_t)f_{A,p}(s,cdot)$ where now $f_{A,p}(s,t)$ is understood as a map to $G$.
I understand that we need to calculate something like $partial_t(f_{A,p}(s,t)^{-1}partial_sf_{A,p}(s,t))$, which should yield $F(partial_t,partial_s)$ (possibly plus something that integrates to zero over $t$), in order to integrate to get the desired result. However, even though I have explicit forms of $F$ and the PDE of $f_{A,p}(s,t)$, I simply could not seem to get it to work.
differential-geometry curvature connections holonomy
asked Jan 13 at 17:10
ssssss
12
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add a comment |
$begingroup$
I am looking for an elementary proof of the fact (and probably not any broader results) that when the connection is flat, holonomy does not change under homotopy of the loop. Although this is a first step of many further results, I have failed to find a standalone proof. Any reference would be appreciated.
Let $pi: Pmapsto M$ be a principal $G$-bundle with connection $A$. Let $x_0in M$. For a loop $gamma$ in $M$ based at $x_0$, define its holonomy as the map $h_{A,gamma}: P|_{x0}mapsto G$ such that the horizontal lift of $gamma$ that starts at $p$ ends at $ph_{A,gamma}(p)$.
(It is clear to me that $h_{A,gamma}(pg^{-1})=gh_{A,gamma}(p)g^{-1}$ $forall gin G$).
I found the following argument in Taubes' Differential Geometry, with I was not able to work out. Any help would be appreciated. (It is not necessary, but for notational convenience assume G is a matrix group.)
Let $f:[-1,1]times[0,2pi]mapsto M$ be a smooth map such that each $f(s,cdot)$ is a loop in $M$ based at $x_0$. For $pin P|_{x_0}$, let $f_{A,p}:[-1,1]times[0,2pi]mapsto P$ be such that $f_{A,p}(cdot,0)=p$ and each $f_{A,p}(s,cdot)$ is the horizontal lift of $f(s,cdot)$. We want to prove that $f_{A,p}(cdot,1)$ is constant when A is a flat connection.
Let $Fin C^{infty}(M,(Ptimes_{Ad}lie(G))otimeswedge^2T^*M))$ be the curvature of the connection. For any $pin P$, $F|_p$ is understood as a map from $wedge^2T_{pi(p)}M$ to $lie(G)$ such that $F|_{pg^{-1}}=Ad_g(F|_p)$.
The author claims that $h_{A,f(s,cdot)}(p)^{-1}partial_sh_{A,f(s,cdot)}(p)=int_0limits^{2pi}F|_{f_{A,p}(s,t)}(f_*partial_t,f_*partial_s))dt$ for any $A$, thus vanishes when $A$ is flat.
In particular, in a subset $U$ of $[-1,1]times[0,2pi]$ where $P|_{f(U)}$ is trivializable, write $P|_{f(U)}=f(U)times G$, $A=g^{-1}dg+Ad_{g^{-1}}a$ where $a$ is a $lie(G)$ valued 1-form on $f(U)$. $F=da+awedge a$ is a $lie(G)$ valued 2-form on $f(U)$.
The horizontal lift satisfies $partial_tf_{A,p}(s,t)=-a(partial_t)f_{A,p}(s,cdot)$ where now $f_{A,p}(s,t)$ is understood as a map to $G$.
I understand that we need to calculate something like $partial_t(f_{A,p}(s,t)^{-1}partial_sf_{A,p}(s,t))$, which should yield $F(partial_t,partial_s)$ (possibly plus something that integrates to zero over $t$), in order to integrate to get the desired result. However, even though I have explicit forms of $F$ and the PDE of $f_{A,p}(s,t)$, I simply could not seem to get it to work.
differential-geometry curvature connections holonomy
asked Jan 13 at 17:10
ssssss
12
$endgroup$
add a comment |
$begingroup$
I am looking for an elementary proof of the fact (and probably not any broader results) that when the connection is flat, holonomy does not change under homotopy of the loop. Although this is a first step of many further results, I have failed to find a standalone proof. Any reference would be appreciated.
Let $pi: Pmapsto M$ be a principal $G$-bundle with connection $A$. Let $x_0in M$. For a loop $gamma$ in $M$ based at $x_0$, define its holonomy as the map $h_{A,gamma}: P|_{x0}mapsto G$ such that the horizontal lift of $gamma$ that starts at $p$ ends at $ph_{A,gamma}(p)$.
(It is clear to me that $h_{A,gamma}(pg^{-1})=gh_{A,gamma}(p)g^{-1}$ $forall gin G$).
I found the following argument in Taubes' Differential Geometry, with I was not able to work out. Any help would be appreciated. (It is not necessary, but for notational convenience assume G is a matrix group.)
Let $f:[-1,1]times[0,2pi]mapsto M$ be a smooth map such that each $f(s,cdot)$ is a loop in $M$ based at $x_0$. For $pin P|_{x_0}$, let $f_{A,p}:[-1,1]times[0,2pi]mapsto P$ be such that $f_{A,p}(cdot,0)=p$ and each $f_{A,p}(s,cdot)$ is the horizontal lift of $f(s,cdot)$. We want to prove that $f_{A,p}(cdot,1)$ is constant when A is a flat connection.
Let $Fin C^{infty}(M,(Ptimes_{Ad}lie(G))otimeswedge^2T^*M))$ be the curvature of the connection. For any $pin P$, $F|_p$ is understood as a map from $wedge^2T_{pi(p)}M$ to $lie(G)$ such that $F|_{pg^{-1}}=Ad_g(F|_p)$.
The author claims that $h_{A,f(s,cdot)}(p)^{-1}partial_sh_{A,f(s,cdot)}(p)=int_0limits^{2pi}F|_{f_{A,p}(s,t)}(f_*partial_t,f_*partial_s))dt$ for any $A$, thus vanishes when $A$ is flat.
In particular, in a subset $U$ of $[-1,1]times[0,2pi]$ where $P|_{f(U)}$ is trivializable, write $P|_{f(U)}=f(U)times G$, $A=g^{-1}dg+Ad_{g^{-1}}a$ where $a$ is a $lie(G)$ valued 1-form on $f(U)$. $F=da+awedge a$ is a $lie(G)$ valued 2-form on $f(U)$.
The horizontal lift satisfies $partial_tf_{A,p}(s,t)=-a(partial_t)f_{A,p}(s,cdot)$ where now $f_{A,p}(s,t)$ is understood as a map to $G$.
I understand that we need to calculate something like $partial_t(f_{A,p}(s,t)^{-1}partial_sf_{A,p}(s,t))$, which should yield $F(partial_t,partial_s)$ (possibly plus something that integrates to zero over $t$), in order to integrate to get the desired result. However, even though I have explicit forms of $F$ and the PDE of $f_{A,p}(s,t)$, I simply could not seem to get it to work.
differential-geometry curvature connections holonomy
asked Jan 13 at 17:10
ssssss
12
$endgroup$
I am looking for an elementary proof of the fact (and probably not any broader results) that when the connection is flat, holonomy does not change under homotopy of the loop. Although this is a first step of many further results, I have failed to find a standalone proof. Any reference would be appreciated.
Let $pi: Pmapsto M$ be a principal $G$-bundle with connection $A$. Let $x_0in M$. For a loop $gamma$ in $M$ based at $x_0$, define its holonomy as the map $h_{A,gamma}: P|_{x0}mapsto G$ such that the horizontal lift of $gamma$ that starts at $p$ ends at $ph_{A,gamma}(p)$.
(It is clear to me that $h_{A,gamma}(pg^{-1})=gh_{A,gamma}(p)g^{-1}$ $forall gin G$).
I found the following argument in Taubes' Differential Geometry, with I was not able to work out. Any help would be appreciated. (It is not necessary, but for notational convenience assume G is a matrix group.)
Let $f:[-1,1]times[0,2pi]mapsto M$ be a smooth map such that each $f(s,cdot)$ is a loop in $M$ based at $x_0$. For $pin P|_{x_0}$, let $f_{A,p}:[-1,1]times[0,2pi]mapsto P$ be such that $f_{A,p}(cdot,0)=p$ and each $f_{A,p}(s,cdot)$ is the horizontal lift of $f(s,cdot)$. We want to prove that $f_{A,p}(cdot,1)$ is constant when A is a flat connection.
Let $Fin C^{infty}(M,(Ptimes_{Ad}lie(G))otimeswedge^2T^*M))$ be the curvature of the connection. For any $pin P$, $F|_p$ is understood as a map from $wedge^2T_{pi(p)}M$ to $lie(G)$ such that $F|_{pg^{-1}}=Ad_g(F|_p)$.
The author claims that $h_{A,f(s,cdot)}(p)^{-1}partial_sh_{A,f(s,cdot)}(p)=int_0limits^{2pi}F|_{f_{A,p}(s,t)}(f_*partial_t,f_*partial_s))dt$ for any $A$, thus vanishes when $A$ is flat.
In particular, in a subset $U$ of $[-1,1]times[0,2pi]$ where $P|_{f(U)}$ is trivializable, write $P|_{f(U)}=f(U)times G$, $A=g^{-1}dg+Ad_{g^{-1}}a$ where $a$ is a $lie(G)$ valued 1-form on $f(U)$. $F=da+awedge a$ is a $lie(G)$ valued 2-form on $f(U)$.
The horizontal lift satisfies $partial_tf_{A,p}(s,t)=-a(partial_t)f_{A,p}(s,cdot)$ where now $f_{A,p}(s,t)$ is understood as a map to $G$.
I understand that we need to calculate something like $partial_t(f_{A,p}(s,t)^{-1}partial_sf_{A,p}(s,t))$, which should yield $F(partial_t,partial_s)$ (possibly plus something that integrates to zero over $t$), in order to integrate to get the desired result. However, even though I have explicit forms of $F$ and the PDE of $f_{A,p}(s,t)$, I simply could not seem to get it to work.
differential-geometry curvature connections holonomy
differential-geometry curvature connections holonomy
asked Jan 13 at 17:10
ssssss
12
asked Jan 13 at 17:10
ssssss
12
asked Jan 13 at 17:10
ssssss
12
asked Jan 13 at 17:10
ssssss
12
asked Jan 13 at 17:10
ssssss
12
12
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