Computing the Fubini-Study metric
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I would like to compute the Fubini-Study metric $g_FS$ for $mathbbCP^n$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration.
I tried to compute on homogenous coordinates $varphi_0:U_0rightarrow mathbbC^n$, where $U_0=lbrace[z_0:dots:z_n] :z_0neq0rbrace$ and $varphi_0[z_0:dots:z_n] =(fracz_1z_0,dots,fracz_nz_0)$.
For this i factored the map $varphi_0^-1=picircpsi$, where $pi$ is the projection $pi:S^2n+1rightarrow mathbbCP^n$ and $psi:mathbbC^n rightarrow S^2n+1$ given by $psi(z)=frac(1,z)sqrt1+$ (the image of $psi$ is a submanifold N of $S^2n+1$).
Then $(varphi_0^-1)^astg_FS=(psi^astcircpi^ast)g_FS$, now $pi^astg_FS$ is the hermitian metric of $mathbbC^n+1$ induced on the submanifold N.
So i just have to compute $psi^ast(dz_aotimes dbarz^a)$, and the result i get is
beginequation
sum_a,bfrac(1+(1+dz_aotimes dbarz_b,
endequation
but the factor $frac34$ should be $1$. So i ask what am i doing wrong?
differential-geometry riemannian-geometry complex-geometry
asked Sep 23 '14 at 3:08
Pedro
54028
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up vote
6
down vote
favorite
I would like to compute the Fubini-Study metric $g_FS$ for $mathbbCP^n$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration.
I tried to compute on homogenous coordinates $varphi_0:U_0rightarrow mathbbC^n$, where $U_0=lbrace[z_0:dots:z_n] :z_0neq0rbrace$ and $varphi_0[z_0:dots:z_n] =(fracz_1z_0,dots,fracz_nz_0)$.
For this i factored the map $varphi_0^-1=picircpsi$, where $pi$ is the projection $pi:S^2n+1rightarrow mathbbCP^n$ and $psi:mathbbC^n rightarrow S^2n+1$ given by $psi(z)=frac(1,z)sqrt1+$ (the image of $psi$ is a submanifold N of $S^2n+1$).
Then $(varphi_0^-1)^astg_FS=(psi^astcircpi^ast)g_FS$, now $pi^astg_FS$ is the hermitian metric of $mathbbC^n+1$ induced on the submanifold N.
So i just have to compute $psi^ast(dz_aotimes dbarz^a)$, and the result i get is
beginequation
sum_a,bfrac(1+(1+dz_aotimes dbarz_b,
endequation
but the factor $frac34$ should be $1$. So i ask what am i doing wrong?
differential-geometry riemannian-geometry complex-geometry
asked Sep 23 '14 at 3:08
Pedro
54028
Are you writing the metric on $S^2n+1$ as $(dZ,dZ)$, where $ZinBbb C^n+1$ and $(,)$ is the standard hermitian inner product?
– Ted Shifrin
Sep 23 '14 at 3:14
I tried using the standard hermitian metric on $S^n$ and the riemannian metric as subset of $mathbbR^2n+2$ and then turning it hermitian (i think it is the same thing as it gives the same result).
– Pedro
Sep 23 '14 at 13:21
There is no hermitian metric on $S^n$. I recommend you think of it the way I suggested and compute $(dZ,dZ)$ where $Z=(z,1)/sqrt1+$. I will comment, however, that the Fubini-Study metric on $Bbb CP^n$ really looks nicer in homogeneous coordinates.
– Ted Shifrin
Sep 23 '14 at 21:36
Sorry, i meant $S^2n+1$ as you said. I computed $(dZ,dZ)$ and obtained $(partial_i,barpartial_j)=frac2(1+(1+$.
– Pedro
Sep 24 '14 at 0:35
After your comment i realised that what i am trying to accomplish is not what i am doing. I thought that the computation result would give the metric in homogeneous coordinates, that is my mistake. So now the question is how to compute the metric in homogeneous coordinates.
– Pedro
Sep 24 '14 at 1:18
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
I would like to compute the Fubini-Study metric $g_FS$ for $mathbbCP^n$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration.
I tried to compute on homogenous coordinates $varphi_0:U_0rightarrow mathbbC^n$, where $U_0=lbrace[z_0:dots:z_n] :z_0neq0rbrace$ and $varphi_0[z_0:dots:z_n] =(fracz_1z_0,dots,fracz_nz_0)$.
For this i factored the map $varphi_0^-1=picircpsi$, where $pi$ is the projection $pi:S^2n+1rightarrow mathbbCP^n$ and $psi:mathbbC^n rightarrow S^2n+1$ given by $psi(z)=frac(1,z)sqrt1+$ (the image of $psi$ is a submanifold N of $S^2n+1$).
Then $(varphi_0^-1)^astg_FS=(psi^astcircpi^ast)g_FS$, now $pi^astg_FS$ is the hermitian metric of $mathbbC^n+1$ induced on the submanifold N.
So i just have to compute $psi^ast(dz_aotimes dbarz^a)$, and the result i get is
beginequation
sum_a,bfrac(1+(1+dz_aotimes dbarz_b,
endequation
but the factor $frac34$ should be $1$. So i ask what am i doing wrong?
differential-geometry riemannian-geometry complex-geometry
asked Sep 23 '14 at 3:08
Pedro
54028
I would like to compute the Fubini-Study metric $g_FS$ for $mathbbCP^n$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration.
I tried to compute on homogenous coordinates $varphi_0:U_0rightarrow mathbbC^n$, where $U_0=lbrace[z_0:dots:z_n] :z_0neq0rbrace$ and $varphi_0[z_0:dots:z_n] =(fracz_1z_0,dots,fracz_nz_0)$.
For this i factored the map $varphi_0^-1=picircpsi$, where $pi$ is the projection $pi:S^2n+1rightarrow mathbbCP^n$ and $psi:mathbbC^n rightarrow S^2n+1$ given by $psi(z)=frac(1,z)sqrt1+$ (the image of $psi$ is a submanifold N of $S^2n+1$).
Then $(varphi_0^-1)^astg_FS=(psi^astcircpi^ast)g_FS$, now $pi^astg_FS$ is the hermitian metric of $mathbbC^n+1$ induced on the submanifold N.
So i just have to compute $psi^ast(dz_aotimes dbarz^a)$, and the result i get is
beginequation
sum_a,bfrac(1+(1+dz_aotimes dbarz_b,
endequation
but the factor $frac34$ should be $1$. So i ask what am i doing wrong?
differential-geometry riemannian-geometry complex-geometry
differential-geometry riemannian-geometry complex-geometry
asked Sep 23 '14 at 3:08
Pedro
54028
asked Sep 23 '14 at 3:08
Pedro
54028
edited Sep 25 '14 at 16:48
asked Sep 23 '14 at 3:08
Pedro
54028
asked Sep 23 '14 at 3:08
Pedro
54028
asked Sep 23 '14 at 3:08
Pedro
54028
54028
Are you writing the metric on $S^2n+1$ as $(dZ,dZ)$, where $ZinBbb C^n+1$ and $(,)$ is the standard hermitian inner product?
– Ted Shifrin
Sep 23 '14 at 3:14
I tried using the standard hermitian metric on $S^n$ and the riemannian metric as subset of $mathbbR^2n+2$ and then turning it hermitian (i think it is the same thing as it gives the same result).
– Pedro
Sep 23 '14 at 13:21
There is no hermitian metric on $S^n$. I recommend you think of it the way I suggested and compute $(dZ,dZ)$ where $Z=(z,1)/sqrt1+$. I will comment, however, that the Fubini-Study metric on $Bbb CP^n$ really looks nicer in homogeneous coordinates.
– Ted Shifrin
Sep 23 '14 at 21:36
Sorry, i meant $S^2n+1$ as you said. I computed $(dZ,dZ)$ and obtained $(partial_i,barpartial_j)=frac2(1+(1+$.
– Pedro
Sep 24 '14 at 0:35
After your comment i realised that what i am trying to accomplish is not what i am doing. I thought that the computation result would give the metric in homogeneous coordinates, that is my mistake. So now the question is how to compute the metric in homogeneous coordinates.
– Pedro
Sep 24 '14 at 1:18
add a comment |Â
Are you writing the metric on $S^2n+1$ as $(dZ,dZ)$, where $ZinBbb C^n+1$ and $(,)$ is the standard hermitian inner product?
– Ted Shifrin
Sep 23 '14 at 3:14
I tried using the standard hermitian metric on $S^n$ and the riemannian metric as subset of $mathbbR^2n+2$ and then turning it hermitian (i think it is the same thing as it gives the same result).
– Pedro
Sep 23 '14 at 13:21
There is no hermitian metric on $S^n$. I recommend you think of it the way I suggested and compute $(dZ,dZ)$ where $Z=(z,1)/sqrt1+$. I will comment, however, that the Fubini-Study metric on $Bbb CP^n$ really looks nicer in homogeneous coordinates.
– Ted Shifrin
Sep 23 '14 at 21:36
Sorry, i meant $S^2n+1$ as you said. I computed $(dZ,dZ)$ and obtained $(partial_i,barpartial_j)=frac2(1+(1+$.
– Pedro
Sep 24 '14 at 0:35
After your comment i realised that what i am trying to accomplish is not what i am doing. I thought that the computation result would give the metric in homogeneous coordinates, that is my mistake. So now the question is how to compute the metric in homogeneous coordinates.
– Pedro
Sep 24 '14 at 1:18
Are you writing the metric on $S^2n+1$ as $(dZ,dZ)$, where $ZinBbb C^n+1$ and $(,)$ is the standard hermitian inner product?
– Ted Shifrin
Sep 23 '14 at 3:14
Are you writing the metric on $S^2n+1$ as $(dZ,dZ)$, where $ZinBbb C^n+1$ and $(,)$ is the standard hermitian inner product?
– Ted Shifrin
Sep 23 '14 at 3:14
I tried using the standard hermitian metric on $S^n$ and the riemannian metric as subset of $mathbbR^2n+2$ and then turning it hermitian (i think it is the same thing as it gives the same result).
– Pedro
Sep 23 '14 at 13:21
I tried using the standard hermitian metric on $S^n$ and the riemannian metric as subset of $mathbbR^2n+2$ and then turning it hermitian (i think it is the same thing as it gives the same result).
– Pedro
Sep 23 '14 at 13:21
There is no hermitian metric on $S^n$. I recommend you think of it the way I suggested and compute $(dZ,dZ)$ where $Z=(z,1)/sqrt1+$. I will comment, however, that the Fubini-Study metric on $Bbb CP^n$ really looks nicer in homogeneous coordinates.
– Ted Shifrin
Sep 23 '14 at 21:36
There is no hermitian metric on $S^n$. I recommend you think of it the way I suggested and compute $(dZ,dZ)$ where $Z=(z,1)/sqrt1+$. I will comment, however, that the Fubini-Study metric on $Bbb CP^n$ really looks nicer in homogeneous coordinates.
– Ted Shifrin
Sep 23 '14 at 21:36
Sorry, i meant $S^2n+1$ as you said. I computed $(dZ,dZ)$ and obtained $(partial_i,barpartial_j)=frac2(1+(1+$.
– Pedro
Sep 24 '14 at 0:35
Sorry, i meant $S^2n+1$ as you said. I computed $(dZ,dZ)$ and obtained $(partial_i,barpartial_j)=frac2(1+(1+$.
– Pedro
Sep 24 '14 at 0:35
After your comment i realised that what i am trying to accomplish is not what i am doing. I thought that the computation result would give the metric in homogeneous coordinates, that is my mistake. So now the question is how to compute the metric in homogeneous coordinates.
– Pedro
Sep 24 '14 at 1:18
After your comment i realised that what i am trying to accomplish is not what i am doing. I thought that the computation result would give the metric in homogeneous coordinates, that is my mistake. So now the question is how to compute the metric in homogeneous coordinates.
– Pedro
Sep 24 '14 at 1:18
add a comment |Â
1 Answer
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The Fubini Study metric on $Cmathbb P^n$
in normal coordinates $(t, θ)$, $t$ positive
real and $θ ∈ S^2n−1(1) ⊂ T_pCmathbb P^n$
is given by
$$ds^2=dt^2+frac14sin^2(2t)dtheta^2|_F+sin^2(t)dtheta^2|_F^perp$$ where where $F$ is tangent to the Hopf fibration $S^2n−1(1) → Cmathbb P^n−1$ in the tangent space and $F^⊥$ orthogonal to it
For proof see
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
The Fubini Study metric on $Cmathbb P^n$
in normal coordinates $(t, θ)$, $t$ positive
real and $θ ∈ S^2n−1(1) ⊂ T_pCmathbb P^n$
is given by
$$ds^2=dt^2+frac14sin^2(2t)dtheta^2|_F+sin^2(t)dtheta^2|_F^perp$$ where where $F$ is tangent to the Hopf fibration $S^2n−1(1) → Cmathbb P^n−1$ in the tangent space and $F^⊥$ orthogonal to it
For proof see
add a comment |Â
up vote
1
down vote
The Fubini Study metric on $Cmathbb P^n$
in normal coordinates $(t, θ)$, $t$ positive
real and $θ ∈ S^2n−1(1) ⊂ T_pCmathbb P^n$
is given by
$$ds^2=dt^2+frac14sin^2(2t)dtheta^2|_F+sin^2(t)dtheta^2|_F^perp$$ where where $F$ is tangent to the Hopf fibration $S^2n−1(1) → Cmathbb P^n−1$ in the tangent space and $F^⊥$ orthogonal to it
For proof see
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The Fubini Study metric on $Cmathbb P^n$
in normal coordinates $(t, θ)$, $t$ positive
real and $θ ∈ S^2n−1(1) ⊂ T_pCmathbb P^n$
is given by
$$ds^2=dt^2+frac14sin^2(2t)dtheta^2|_F+sin^2(t)dtheta^2|_F^perp$$ where where $F$ is tangent to the Hopf fibration $S^2n−1(1) → Cmathbb P^n−1$ in the tangent space and $F^⊥$ orthogonal to it
For proof see
The Fubini Study metric on $Cmathbb P^n$
in normal coordinates $(t, θ)$, $t$ positive
real and $θ ∈ S^2n−1(1) ⊂ T_pCmathbb P^n$
is given by
$$ds^2=dt^2+frac14sin^2(2t)dtheta^2|_F+sin^2(t)dtheta^2|_F^perp$$ where where $F$ is tangent to the Hopf fibration $S^2n−1(1) → Cmathbb P^n−1$ in the tangent space and $F^⊥$ orthogonal to it
For proof see
edited Nov 1 '17 at 6:16
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Are you writing the metric on $S^2n+1$ as $(dZ,dZ)$, where $ZinBbb C^n+1$ and $(,)$ is the standard hermitian inner product?
– Ted Shifrin
Sep 23 '14 at 3:14
I tried using the standard hermitian metric on $S^n$ and the riemannian metric as subset of $mathbbR^2n+2$ and then turning it hermitian (i think it is the same thing as it gives the same result).
– Pedro
Sep 23 '14 at 13:21
There is no hermitian metric on $S^n$. I recommend you think of it the way I suggested and compute $(dZ,dZ)$ where $Z=(z,1)/sqrt1+$. I will comment, however, that the Fubini-Study metric on $Bbb CP^n$ really looks nicer in homogeneous coordinates.
– Ted Shifrin
Sep 23 '14 at 21:36
Sorry, i meant $S^2n+1$ as you said. I computed $(dZ,dZ)$ and obtained $(partial_i,barpartial_j)=frac2(1+(1+$.
– Pedro
Sep 24 '14 at 0:35
After your comment i realised that what i am trying to accomplish is not what i am doing. I thought that the computation result would give the metric in homogeneous coordinates, that is my mistake. So now the question is how to compute the metric in homogeneous coordinates.
– Pedro
Sep 24 '14 at 1:18