Local Gauss-Bonnet for a rectangle
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I'm trying to satisfy the local Gauss-Bonnet equation for the surface $R$ bounded by $u=A,u=B,v=a,v=b$ in the hyperbolic plane. However I am stuck.
I know that the equation of local Gauss-Bonnet is $int_{partial R} kappa_gds +int int _R KdA + sum epsilon_j = 2pi$ where $epsilon$ re presents the interior angles.
Now I know my shape is a square and in the hyperbolic plane hence $dA=sqrt{EG-F}= frac{1}{v^2}$ and $K=-1$.
So $int int _R KdA = -int_a^bint_A^Bfrac{1}{v^2}dudv = (B-A)(frac{1}{b}-frac{1}{a})$.
And $kappa_g$ for the lines $v=a,v=b$ is $1$ and $-1$ respectively by direction. Therefore we would have that $int_{partial R} kappa_gds = int_A^Bds-int_A^Bds =0$.
However then $sum epsilon_j= 4frac{pi}{2}=2pi$. However this doesnt satisfy local Gauss-Bonnet since $int_{partial R} kappa_gds +int int _R KdA + sum epsilon_j = (B-A)(frac{1}{b}-frac{1}{a}) +2pi$ so I'm not sure where went wrong.
geometry differential-geometry hyperbolic-geometry
asked Nov 28 '18 at 18:20
SashaSasha
537
$endgroup$
add a comment |
$begingroup$
I'm trying to satisfy the local Gauss-Bonnet equation for the surface $R$ bounded by $u=A,u=B,v=a,v=b$ in the hyperbolic plane. However I am stuck.
I know that the equation of local Gauss-Bonnet is $int_{partial R} kappa_gds +int int _R KdA + sum epsilon_j = 2pi$ where $epsilon$ re presents the interior angles.
Now I know my shape is a square and in the hyperbolic plane hence $dA=sqrt{EG-F}= frac{1}{v^2}$ and $K=-1$.
So $int int _R KdA = -int_a^bint_A^Bfrac{1}{v^2}dudv = (B-A)(frac{1}{b}-frac{1}{a})$.
And $kappa_g$ for the lines $v=a,v=b$ is $1$ and $-1$ respectively by direction. Therefore we would have that $int_{partial R} kappa_gds = int_A^Bds-int_A^Bds =0$.
However then $sum epsilon_j= 4frac{pi}{2}=2pi$. However this doesnt satisfy local Gauss-Bonnet since $int_{partial R} kappa_gds +int int _R KdA + sum epsilon_j = (B-A)(frac{1}{b}-frac{1}{a}) +2pi$ so I'm not sure where went wrong.
geometry differential-geometry hyperbolic-geometry
asked Nov 28 '18 at 18:20
SashaSasha
537
$endgroup$
$begingroup$
Hmm, this question looks familiar. You have $ds$ wrong when you're doing the $int kappa_g,ds$ computations.
$endgroup$
– Ted Shifrin
Nov 29 '18 at 0:25
add a comment |
$begingroup$
I'm trying to satisfy the local Gauss-Bonnet equation for the surface $R$ bounded by $u=A,u=B,v=a,v=b$ in the hyperbolic plane. However I am stuck.
I know that the equation of local Gauss-Bonnet is $int_{partial R} kappa_gds +int int _R KdA + sum epsilon_j = 2pi$ where $epsilon$ re presents the interior angles.
Now I know my shape is a square and in the hyperbolic plane hence $dA=sqrt{EG-F}= frac{1}{v^2}$ and $K=-1$.
So $int int _R KdA = -int_a^bint_A^Bfrac{1}{v^2}dudv = (B-A)(frac{1}{b}-frac{1}{a})$.
And $kappa_g$ for the lines $v=a,v=b$ is $1$ and $-1$ respectively by direction. Therefore we would have that $int_{partial R} kappa_gds = int_A^Bds-int_A^Bds =0$.
However then $sum epsilon_j= 4frac{pi}{2}=2pi$. However this doesnt satisfy local Gauss-Bonnet since $int_{partial R} kappa_gds +int int _R KdA + sum epsilon_j = (B-A)(frac{1}{b}-frac{1}{a}) +2pi$ so I'm not sure where went wrong.
geometry differential-geometry hyperbolic-geometry
asked Nov 28 '18 at 18:20
SashaSasha
537
$endgroup$
I'm trying to satisfy the local Gauss-Bonnet equation for the surface $R$ bounded by $u=A,u=B,v=a,v=b$ in the hyperbolic plane. However I am stuck.
I know that the equation of local Gauss-Bonnet is $int_{partial R} kappa_gds +int int _R KdA + sum epsilon_j = 2pi$ where $epsilon$ re presents the interior angles.
Now I know my shape is a square and in the hyperbolic plane hence $dA=sqrt{EG-F}= frac{1}{v^2}$ and $K=-1$.
So $int int _R KdA = -int_a^bint_A^Bfrac{1}{v^2}dudv = (B-A)(frac{1}{b}-frac{1}{a})$.
And $kappa_g$ for the lines $v=a,v=b$ is $1$ and $-1$ respectively by direction. Therefore we would have that $int_{partial R} kappa_gds = int_A^Bds-int_A^Bds =0$.
However then $sum epsilon_j= 4frac{pi}{2}=2pi$. However this doesnt satisfy local Gauss-Bonnet since $int_{partial R} kappa_gds +int int _R KdA + sum epsilon_j = (B-A)(frac{1}{b}-frac{1}{a}) +2pi$ so I'm not sure where went wrong.
geometry differential-geometry hyperbolic-geometry
geometry differential-geometry hyperbolic-geometry
asked Nov 28 '18 at 18:20
SashaSasha
537
asked Nov 28 '18 at 18:20
SashaSasha
537
asked Nov 28 '18 at 18:20
SashaSasha
537
asked Nov 28 '18 at 18:20
SashaSasha
537
asked Nov 28 '18 at 18:20
SashaSasha
537
537
$begingroup$
Hmm, this question looks familiar. You have $ds$ wrong when you're doing the $int kappa_g,ds$ computations.
$endgroup$
– Ted Shifrin
Nov 29 '18 at 0:25
add a comment |
$begingroup$
Hmm, this question looks familiar. You have $ds$ wrong when you're doing the $int kappa_g,ds$ computations.
$endgroup$
– Ted Shifrin
Nov 29 '18 at 0:25
$begingroup$
Hmm, this question looks familiar. You have $ds$ wrong when you're doing the $int kappa_g,ds$ computations.
$endgroup$
– Ted Shifrin
Nov 29 '18 at 0:25
$begingroup$
Hmm, this question looks familiar. You have $ds$ wrong when you're doing the $int kappa_g,ds$ computations.
$endgroup$
– Ted Shifrin
Nov 29 '18 at 0:25
add a comment |
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$begingroup$
Hmm, this question looks familiar. You have $ds$ wrong when you're doing the $int kappa_g,ds$ computations.
$endgroup$
– Ted Shifrin
Nov 29 '18 at 0:25