Opened 3 years ago
Closed 3 years ago
#28462 closed defect (fixed)
Bug in computing the second fundamental form of a Riemannian submanifold
Reported by:  egourgoulhon  Owned by:  

Priority:  major  Milestone:  sage8.9 
Component:  geometry  Keywords:  submanifolds, pseudoRiemannian, second_fundamental_form 
Cc:  tscrim, ghFlorentinJ  Merged in:  
Authors:  Eric Gourgoulhon  Reviewers:  Florentin Jaffredo 
Report Upstream:  N/A  Work issues:  
Branch:  f815378 (Commits, GitHub, GitLab)  Commit:  f8153780899df2a1c936b1c368e8adb85e0b562d 
Dependencies:  Stopgaps: 
Description (last modified by )
As reported in this ask.sagemath question, we have currently (Sage 8.8 and 8.9.beta9):
sage: P = Manifold(3, 'P', structure='Riemannian') sage: Q = Manifold(2, 'Q', ambient=P, structure='Riemannian') sage: CP.<x,y,z> = P.chart() sage: CQ.<u,v> = Q.chart() sage: g = P.metric() sage: c = 2/(1 + y^2 + z^2) sage: g[0,0], g[1,1], g[2,2] = 1, c^2, c^2 sage: phi = Q.diff_map(P, (u+v, u, v)) sage: phi_inv = P.diff_map(Q, (y, z)) sage: Q.set_embedding(phi, inverse=phi_inv) sage: Q.second_fundamental_form() TypeError: unable to convert 1/2*sqrt(2)*(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1)*y/ (sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(y^2 + z^2 + 1)) + 1/2*sqrt(2)*(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1)*z/ (sqrt(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 3)*(y^2 + z^2 + 1)) to an integer
This results from the mention of the ring SR
missing in the declaration of a matrix involved in the computation.
The issue is fixed by this ticket.
Change History (14)
comment:1 Changed 3 years ago by
 Description modified (diff)
comment:2 Changed 3 years ago by
 Branch set to public/manifolds/bug_second_fund_form28462
 Commit set to 7f96e90c78b57358f7e72e7e887df5c6c42e40bd
comment:3 Changed 3 years ago by
 Cc tscrim ghFlorentinJ added
 Status changed from new to needs_review
comment:4 followup: ↓ 5 Changed 3 years ago by
I completely agree with the changes, but why was there no errors in the doctests ? The example in second_fundamental_form
features a symbolic computation, and the result was not an integer.
comment:5 in reply to: ↑ 4 Changed 3 years ago by
Replying to ghFlorentinJ:
I completely agree with the changes, but why was there no errors in the doctests ? The example in
second_fundamental_form
features a symbolic computation, and the result was not an integer.
Actually, the error does not show off when the coefficients of the ambient connection are zero, since then gamma_n
in line 1079 is filled with zeros, which is compatible with a matrix declared on the integers (the default, when the ring is not specified in matrix()
). This is the case of the doctests.
comment:6 Changed 3 years ago by
 Status changed from needs_review to positive_review
comment:7 followup: ↓ 8 Changed 3 years ago by
 Status changed from positive_review to needs_work
Argh, there is another issue: I wanted to add the following doctest to check that the original issue is solved:
sage: M = Manifold(2, 'M', structure='Riemannian') sage: N = Manifold(1, 'N', ambient=M, structure='Riemannian', ....: start_index=1) sage: CM.<x,y> = M.chart() sage: CN.<u> = N.chart() sage: g = M.metric() sage: g[0, 0], g[1, 1] = 1, 1/(1 + y^2)^2 sage: phi = N.diff_map(M, (u, u)) sage: N.set_embedding(phi) sage: N.second_fundamental_form() Field of symmetric bilinear forms K on the 1dimensional Riemannian submanifold N embedded in the 2dimensional Riemannian manifold M sage: N.second_fundamental_form().display() K = 2*sqrt(u^4 + 2*u^2 + 2)*(2*u*y^2  (u^2 + 1)*y + 2*u) /(u^6 + 3*u^4 + (u^6 + 3*u^4 + 4*u^2 + 2)*y^2 + 4*u^2 + 2) du*du
As you can see, the expression of the component K_{uu} involves y
, while it should be a function of u
only. I guess this is because the expression of the ambient connection coefficients returned by expr()
in line 1098 involves (x,y)
; x
and y
should be expressed in terms of u
via the embedding formulas before continuing the computation.
comment:8 in reply to: ↑ 7 ; followup: ↓ 10 Changed 3 years ago by
Replying to egourgoulhon:
x
andy
should be expressed in terms ofu
via the embedding formulas before continuing the >computation.
So I guess phi_inv
was really needed...
comment:9 Changed 3 years ago by
 Commit changed from 7f96e90c78b57358f7e72e7e887df5c6c42e40bd to f8153780899df2a1c936b1c368e8adb85e0b562d
Branch pushed to git repo; I updated commit sha1. New commits:
f815378  Fix another issue with second fundatemental form of pseudoRiemannian submanifolds

comment:10 in reply to: ↑ 8 Changed 3 years ago by
Replying to ghFlorentinJ:
Replying to egourgoulhon:
x
andy
should be expressed in terms ofu
via the embedding formulas before continuing the >computation.So I guess
phi_inv
was really needed...
No it was not. It was only a matter of using phi
to substitute (x,y)
by their expressions in terms of u
. This is done in the above commit. I've also added a simplification of the result by a call to chart.simplify()
. I think that everything is fixed now.
comment:11 Changed 3 years ago by
 Status changed from needs_work to needs_review
comment:12 Changed 3 years ago by
 Status changed from needs_review to positive_review
comment:14 Changed 3 years ago by
 Branch changed from public/manifolds/bug_second_fund_form28462 to f8153780899df2a1c936b1c368e8adb85e0b562d
 Resolution set to fixed
 Status changed from positive_review to closed
New commits:
Fix bug in computation of second fundamental form