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Extrinsic curvature of Riemannian submanifold: no basis found for computing the components

asked 2020-10-26 18:31:16 +0200

I'm trying to compute the extrinsic curvature of a constant r slice of the following manifold:

M = Manifold(3, 'M', structure='Lorentzian')
X.<r,xplus,xminus> = M.chart('r:(0,+oo):r xplus:(-oo,+oo):x_+ xminus:(-oo,+oo):x_-')
g = M.riemannian_metric('g')

function('Lplus',latex_name='L_+',)(xplus)
function('Lminus',latex_name='L_-')(xminus)
var('l',domain='real')

g[0,0] = l^2/r^2
g[1,1] = l^2*Lplus(xplus)
g[2,2] = l^2*Lminus(xminus)
g[1,2] = 1/2*(-r^2-l^4*Lplus(xplus)*Lminus(xminus)/r^2)

I'm defining the submanifold using the following commands:

N = Manifold(2, 'N', ambient=M, structure='Lorentzian', start_index=1)
Y.<Xplus,Xminus> = N.chart('Xplus:(-oo,+oo):X_+ Xminus:(-oo,+oo):X_-')
var('r0',domain='real')
assume(r0>0)
phi = N.diff_map(M, {(Y,X): [r0,Xplus,Xminus]})
phi_inv = M.diff_map(N, {(X,Y): [xplus,xminus]})
phi_inv_t = M.scalar_field({X: r})
N.set_embedding(phi, inverse=phi_inv, var=r0, t_inverse = {r0: phi_inv_t})

But when computing the extrinsic curvature, I get an error:

N.extrinsic_curvature()

ValueError: no basis could be found for computing the components in the Coordinate frame (M, (d/dr,d/dxplus,d/dxminus))

Computing the normal vector leads to a similar error:

N.normal()
ValueError: no common basis for the contraction

What am I doing wrong? The examples at (https:// doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/pseudo_riemannian_submanifold.html) work fine for me.

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answered 2020-10-26 21:37:30 +0200

eric_g gravatar image

Replace g = M.riemannian_metric('g') by

g = M.metric()

and , after N.set_embedding(...), add

N.adapted_chart()

Then everything works fine.

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Comments

That solved it. Thanks!

Marius Gerbershagen gravatar imageMarius Gerbershagen ( 2020-10-27 18:32:46 +0200 )edit

@Marius Gerbershagen -- you can accept the answer to mark the question as solved. Click the tick mark at the top left of the answer, below the answer's score and the "upvote" and "downvote" buttons.

slelievre gravatar imageslelievre ( 2021-01-27 13:38:55 +0200 )edit

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Asked: 2020-10-26 18:30:50 +0200

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Last updated: Oct 26 '20