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Thanks for reporting this issue. This is a bug, which is now fixed in the ticket #28462. Hopefully, this ticket will be merged in Sage 8.9 (to come out soon).

With #28462, your code leads to

sage: K = Q.extrinsic_curvature()
sage: K.display_comp()
K_uu = -4*(2*u*y^2 + 2*u*z^2 - (u^2 + v^2 + 1)*y - (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_uv = -4*(2*u*y^2 + 2*u*z^2 - (u^2 + v^2 + 1)*y + (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_vu = -4*(2*u*y^2 + 2*u*z^2 - (u^2 + v^2 + 1)*y + (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_vv = -4*(2*v*y^2 + 2*v*z^2 - (u^2 + v^2 + 1)*y - (u^2 + v^2 + 1)*z + 2*v)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2)))

Side note: your declaration of phi_inv is not correct: with your code, we have

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (u, v)

As you can see, the output is ill-formed, because (u, v) should be a function of (x, y, z).

The correct declaration should be

sage: phi_inv = P.diff_map(Q, {(CP, CQ) : list(CP[1:])})

which leads to

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (y, z)

Forturnately, phi_inv plays no role in the computation of the extrinsic curvature.

Thanks for reporting this issue. This is a bug, which is now fixed in the ticket #28462. (this is a 1-line fix, as you can see here !). Hopefully, this ticket will be merged in Sage 8.9 (to come out soon).

With #28462, your code leads to

sage: K = Q.extrinsic_curvature()
sage: K.display_comp()
K_uu = -4*(2*u*y^2 + 2*u*z^2 - (u^2 + v^2 + 1)*y - (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_uv = -4*(2*u*y^2 + 2*u*z^2 - (u^2 + v^2 + 1)*y + (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_vu = -4*(2*u*y^2 + 2*u*z^2 - (u^2 + v^2 + 1)*y + (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_vv = -4*(2*v*y^2 + 2*v*z^2 - (u^2 + v^2 + 1)*y - (u^2 + v^2 + 1)*z + 2*v)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2)))

Side note: your declaration of phi_inv is not correct: with your code, we have

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (u, v)

As you can see, the output is ill-formed, because (u, v) should be a function of (x, y, z).

The correct declaration should be

sage: phi_inv = P.diff_map(Q, {(CP, CQ) : list(CP[1:])})

which leads to

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (y, z)

Forturnately, phi_inv plays no role in the computation of the extrinsic curvature.

Thanks for reporting this issue. This is a bug, which is now fixed in the ticket #28462 (this is a 1-line fix, as you can see here !). Hopefully, this ticket will be merged in Sage 8.9 (to come out soon).

With #28462, your code leads to

sage: K = Q.extrinsic_curvature()
sage: K.display_comp()
K_uu = -4*(2*u*y^2 + 2*u*z^2 -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u - (u^2 + v^2 + 1)*y v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_uv = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u + v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_vu = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u + v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_vv = 4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u - (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_uv = -4*(2*u*y^2 + 2*u*z^2 - (u^2 + v^2 + 1)*y + (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_vu = -4*(2*u*y^2 + 2*u*z^2 - (u^2 + v^2 + 1)*y + (u^2 + v^2 + 1)*z + 2*u)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))) 
K_vv = -4*(2*v*y^2 + 2*v*z^2 - (u^2 + v^2 + 1)*y - (u^2 + v^2 + 1)*z + 2*v)/(sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*((sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*y^2 + (sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2))*z^2 + sqrt(2)*u^2 + sqrt(2)*v^2 + sqrt(2)))
v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2))

Side note: your declaration of phi_inv is not correct: with your code, we have

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (u, v)

As you can see, the output is ill-formed, because (u, v) should be a function of (x, y, z).

The correct declaration should be

sage: phi_inv = P.diff_map(Q, {(CP, CQ) : list(CP[1:])})

which leads to

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (y, z)

Forturnately, phi_inv plays no role in the computation of the extrinsic curvature.

Thanks for reporting this issue. This is a bug, which is now fixed in the ticket #28462 (this is a 1-line fix, as you can see here !). Hopefully, this ticket will be merged in Sage 8.9 (to come out soon).

With #28462, your code leads to

sage: K = Q.extrinsic_curvature()
sage: K.display_comp()
K_uu = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u - v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_uv = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u + v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_vu = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u + v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_vv = 4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u - v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2))

Side note: your declaration of phi_inv is not correct: with your code, we have

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (u, v)

As you can see, the output is ill-formed, because (u, v) should be a function of (x, y, z).

The correct declaration should be

sage: phi_inv = P.diff_map(Q, {(CP, CQ) : list(CP[1:])})

which leads to

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (y, z)

Forturnately, phi_inv plays no role in the computation of the extrinsic curvature.

Thanks for reporting this issue. This is a bug, which is now fixed in the ticket #28462 (this is a 1-line fix, as you can see here !). . Hopefully, this ticket will be merged in Sage 8.9 (to come out soon).

With #28462, your code leads to

sage: K = Q.extrinsic_curvature()
sage: K.display_comp()
K_uu = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u - v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_uv = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u + v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_vu = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u + v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_vv = 4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u - v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2))

Side note: your declaration of phi_inv is not correct: with your code, we have

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (u, v)

As you can see, the output is ill-formed, because (u, v) should be a function of (x, y, z).

The correct declaration should be

sage: phi_inv = P.diff_map(Q, {(CP, CQ) : list(CP[1:])})

which leads to

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (y, z)

Forturnately, phi_inv plays no role in the computation of the extrinsic curvature.

Thanks for reporting this issue. This is a bug, which is now fixed in the ticket #28462. Hopefully, this ticket will be merged in Sage 8.9 (to come out soon).

With #28462, your code leads to

sage: K = Q.extrinsic_curvature()
sage: K.display_comp()
K_uu = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u - v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_uv = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u + v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_vu = -4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u + v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2)) 
K_vv = 4*sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(u - v)/(sqrt(2)*u^6 + sqrt(2)*v^6 + 3*(sqrt(2)*u^2 + sqrt(2))*v^4 + 3*sqrt(2)*u^4 + (3*sqrt(2)*u^4 + 6*sqrt(2)*u^2 + 5*sqrt(2))*v^2 + 5*sqrt(2)*u^2 + 3*sqrt(2))

Side note: your declaration of phi_inv is not correct: with your code, we have

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (u, v)

As you can see, the output is ill-formed, because (u, v) should be a function of (x, y, z).

The correct declaration should be

sage: phi_inv = P.diff_map(Q, {(CP, CQ) : list(CP[1:])})

which leads to

sage: phi_inv.display()
P --> Q
   (x, y, z) |--> (u, v) = (y, z)

Forturnately, phi_inv plays no role in the computation of the extrinsic curvature.