Given a map $X to text{GL}_2(mathbb{R})$ how do I determine a flat connection on this Riemann surface?
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I need help determining the Euler class of this vector bundle $phi:Eto X$.
The base space is the torus $X = mathbb{R}^2/mathbb{Z}^2$ and the fiber over each point, $f^{-1}(x) simeq mathbb{R}^2$. So it is a vector bundle of rank $2$.
The fundamental group of this torus is the $pi_0(X) simeq mathbb{Z}^2$ and consider the map from $pi_0(X) to text{GL}_2(mathbb{R})$
- $(1,0) to frac{1}{sqrt{5}}left[ begin{array}{rr} 2 & -1 \ 1 & 2 end{array} right]$
- $(0,1) to ;;,frac{1}{5}left[ begin{array}{rr} 3 & -4 \ 4 & 3 end{array} right]$
These are commuting matrices so this is a well-defined map. There should be a flat vector bundle with these properties.
How can I find a chart for this vector bundles? Can I just take a chart from the torus and lift it?
What is a formula for the flat conneciton $nabla$ ?
vector-bundles connections k-theory
asked Dec 22 '18 at 4:38
cactus314cactus314
15.4k42269
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add a comment |
$begingroup$
I need help determining the Euler class of this vector bundle $phi:Eto X$.
The base space is the torus $X = mathbb{R}^2/mathbb{Z}^2$ and the fiber over each point, $f^{-1}(x) simeq mathbb{R}^2$. So it is a vector bundle of rank $2$.
The fundamental group of this torus is the $pi_0(X) simeq mathbb{Z}^2$ and consider the map from $pi_0(X) to text{GL}_2(mathbb{R})$
- $(1,0) to frac{1}{sqrt{5}}left[ begin{array}{rr} 2 & -1 \ 1 & 2 end{array} right]$
- $(0,1) to ;;,frac{1}{5}left[ begin{array}{rr} 3 & -4 \ 4 & 3 end{array} right]$
These are commuting matrices so this is a well-defined map. There should be a flat vector bundle with these properties.
How can I find a chart for this vector bundles? Can I just take a chart from the torus and lift it?
What is a formula for the flat conneciton $nabla$ ?
vector-bundles connections k-theory
asked Dec 22 '18 at 4:38
cactus314cactus314
15.4k42269
$endgroup$
add a comment |
$begingroup$
I need help determining the Euler class of this vector bundle $phi:Eto X$.
The base space is the torus $X = mathbb{R}^2/mathbb{Z}^2$ and the fiber over each point, $f^{-1}(x) simeq mathbb{R}^2$. So it is a vector bundle of rank $2$.
The fundamental group of this torus is the $pi_0(X) simeq mathbb{Z}^2$ and consider the map from $pi_0(X) to text{GL}_2(mathbb{R})$
- $(1,0) to frac{1}{sqrt{5}}left[ begin{array}{rr} 2 & -1 \ 1 & 2 end{array} right]$
- $(0,1) to ;;,frac{1}{5}left[ begin{array}{rr} 3 & -4 \ 4 & 3 end{array} right]$
These are commuting matrices so this is a well-defined map. There should be a flat vector bundle with these properties.
How can I find a chart for this vector bundles? Can I just take a chart from the torus and lift it?
What is a formula for the flat conneciton $nabla$ ?
vector-bundles connections k-theory
asked Dec 22 '18 at 4:38
cactus314cactus314
15.4k42269
$endgroup$
I need help determining the Euler class of this vector bundle $phi:Eto X$.
The base space is the torus $X = mathbb{R}^2/mathbb{Z}^2$ and the fiber over each point, $f^{-1}(x) simeq mathbb{R}^2$. So it is a vector bundle of rank $2$.
The fundamental group of this torus is the $pi_0(X) simeq mathbb{Z}^2$ and consider the map from $pi_0(X) to text{GL}_2(mathbb{R})$
- $(1,0) to frac{1}{sqrt{5}}left[ begin{array}{rr} 2 & -1 \ 1 & 2 end{array} right]$
- $(0,1) to ;;,frac{1}{5}left[ begin{array}{rr} 3 & -4 \ 4 & 3 end{array} right]$
These are commuting matrices so this is a well-defined map. There should be a flat vector bundle with these properties.
How can I find a chart for this vector bundles? Can I just take a chart from the torus and lift it?
What is a formula for the flat conneciton $nabla$ ?
vector-bundles connections k-theory
vector-bundles connections k-theory
asked Dec 22 '18 at 4:38
cactus314cactus314
15.4k42269
asked Dec 22 '18 at 4:38
cactus314cactus314
15.4k42269
asked Dec 22 '18 at 4:38
cactus314cactus314
15.4k42269
asked Dec 22 '18 at 4:38
cactus314cactus314
15.4k42269
asked Dec 22 '18 at 4:38
cactus314cactus314
15.4k42269
15.4k42269
add a comment |
add a comment |
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