2022-12-26 23:31:20 +0200 received badge ● Popular Question (source) 2021-01-27 13:38:03 +0200 received badge ● Student (source) 2020-10-27 18:32:50 +0200 received badge ● Scholar (source) 2020-10-27 18:32:46 +0200 commented answer Extrinsic curvature of Riemannian submanifold: no basis found for computing the components That solved it. Thanks! 2020-10-26 18:48:34 +0200 asked a question Extrinsic curvature of Riemannian submanifold: no basis found for computing the components I'm trying to compute the extrinsic curvature of a constant r slice of the following manifold: M = Manifold(3, 'M', structure='Lorentzian') X. = 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. = 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.