AdS3 metrics using SageManifolds

[ShayanJ]

I've written the following code using SageManifolds both for practice and for when I need different quantities related to the $AdS_3$ spacetime.

AdS3=Manifold(3,'AdS3')

AdS3_Polar=AdS3.open_subset('AdS3_Polar')
#AdS3_Poincare=AdS3.open_subset('AdS3_Poincare')

Global.<tau,rho,phi>=AdS3_Polar.chart(r'tau:(-oo,+oo) rho:(0,+oo) phi:(0,2*pi)')
#Poincare.<t,y,z>=AdS3_Poincare.chart(r't:(-oo,+oo) y:(-oo,+oo) z:(0,+oo)')

R22=Manifold(4,'R22')
X22.<X,Y,Z,W>=R22.chart()
h=R22.metric('h',signature=0)
h[0,0],h[1,1],h[2,2],h[3,3]=-1,-1,1,1

var('R',domain='real')
assume(R>0)
Phi=AdS3.diff_map( R22, [ R*cosh(rho)*cos(tau),R*cosh(rho)*sin(tau),R*sinh(rho)*cos(phi),R*sinh(rho)*sin(phi) ] , name='Phi' )
#Phi=AdS3.diff_map( R22, [ (R/(2*z))*(y^2-t^2+z^2+1),R*t/z,R*y/z,(R/(2*z))*(y^2-t^2+z^2-1) ], name='Phi' )

g=AdS3.lorentzian_metric('g')
g.set( Phi.pullback(h) )

#g=AdS3.lorentzian_metric('g')
#g.set( Phi.pullback(h) )

As you can see, the codes for computing the metric in the Poincare coordinates are commented. That's because I need the metric in both global and Poincare coordinates but I don't know how to associate both of them to the manifold. So one of them is commented at a given time. How can I associate more than one metric to a single manifold?

Thanks

 

[egourgoulhon]

You can find some SageManifolds code for AdS spacetime, including the use of both global and Poincaré coordinates, here.
This is for $AdS_4$, but I guess you can easily adapt it to $AdS_3$.