Error computing curvature of graph submanifold

Paul Bryan
77 ●2 ●4 ●8

tmonteil
27483 ●32 ●205 ●519 http://wiki.sagemath.o...

The following appears to be a bug in sagemanifolds curvature. I see the error running it in cocalc.com. Other things like computing the induced_metric() and normal() and fine. Am I doing something wrong?

sage: P = Manifold(3, 'P', structure='Riemannian', metric_name='g', start_index=0)
sage: Q = Manifold(2, 'Q', ambient=P, structure='Riemannian', metric_name='k', start_index=1)

sage: CP.<x,y,z> = P.chart()
sage: CQ.<u,v> = Q.chart()

sage: g = P.metric()
sage: c = 2/(sum(y^2 for y in CP[1:]) + 1)
sage: g[0,0], g[1,1], g[2,2] = 1, c^2, c^2

sage: f = sum(CQ[:])
sage: f = u + v
sage: phi = Q.diff_map(P,{(CQ, CP):[f] + list(CQ[:])})
sage: phi_inv = P.diff_map(Q, {(CP, CQ) : list(CQ[:])})

sage: Q.set_embedding(phi, inverse=phi_inv)

sage: K = Q.extrinsic_curvature()

Error in lines 13-13

Traceback (most recent call last):

File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags), namespace, locals)

File "", line 1, in <module>

File "/ext/sage/sage-8.8_1804/local/lib/python2.7/site-packages/sage/manifolds/differentiable/pseudo_riemannian_submanifold.py", line 1067, in second_fundamental_form range(self._dim + 1))

File "sage/matrix/matrix0.pyx", line 1424, in sage.matrix.matrix0.Matrix.__setitem__ (build/cythonized/sage/matrix/matrix0.c:8796) self.set_unsafe(row, col, self._coerce_element(value))

File "sage/matrix/matrix0.pyx", line 1529, in sage.matrix.matrix0.Matrix._coerce_element (build/cythonized/sage/matrix/matrix0.c:10273) return self._base_ring(x)

File "sage/structure/parent.pyx", line 900, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9197) return mor._call_(x)

File "sage/structure/coerce_maps.pyx", line 288, in sage.structure.coerce_maps.NamedConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:5942) cdef Element e = method(C)

File "sage/symbolic/expression.pyx", line 1087, in sage.symbolic.expression.Expression._integer_ (build/cythonized/sage/symbolic/expression.cpp:8689) raise TypeError("unable to convert %r to an integer" % self)

TypeError: unable to convert 1/2sqrt(2)(u^4 + v^4 + 2(u^2 + 1)v^2 + 2u^2 + 1)y/(sqrt(u^4 + v^4 + 2(u^2 + 1)v^2 + 2u^2 + 3)(y^2 + z^2 + 1)) + 1/2sqrt(2)(u^4 + v^4 + 2(u^2 + 1)v^2 + 2u^2 + 1)z/(sqrt(u^4 + v^4 + 2(u^2 + 1)v^2 + 2u^2 + 3)(y^2 + z^2 + 1)) to an integer

answered 5 years ago

eric_g
8544 ●10 ●62 ●168 https://luth.obspm.fr/...

updated 5 years ago

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.

EDIT (14 Sep. 2019): the fix introduced in #28462 has been merged in Sage 8.9.rc0, so the next stable release of Sage will be free from this bug.

link

Comments

Awesome! Thanks for pointing out the error in my code too.

Paul Bryan ( 5 years ago )

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