# manifolds: antisymmetrize a tensor field creates a vector field instead of a tensor field [closed]

Tensor.antisymmetrize(...) creates a vector field instead of a tensor field as expected. See example below: CODE:

M = Manifold(3, 'M')
c_xyz.<x,y,z> = M.chart()
e_=c_xyz.frame()
K  =M.tensor_field(2,0,{e_:[[0,1,2],[4,5,6],[8,9,10],[12,13,14]]},name='K')
print('K :',K)
K.symmetries()
KS=K.symmetrize(0,1);print('KS:',KS)
KS.symmetries()
KA=K.antisymmetrize(0,1);print('KA:',KA)
KA.symmetries()


OUTPUT print :

K : Tensor field K of type (2,0) on the 3-dimensional differentiable manifold M
no symmetry; no antisymmetry
KS: Tensor field of type (2,0) on the 3-dimensional differentiable manifold M
symmetry: (0, 1); no antisymmetry
KA: 2-vector field on the 3-dimensional differentiable manifold M
no symmetry; antisymmetry: (0, 1)

edit retag reopen merge delete

### Closed for the following reason the question is answered, right answer was accepted by LPsFR close date 2020-12-04 17:59:26.162725

Got it. Thanks.

( 2020-12-04 16:36:30 +0200 )edit

Sort by » oldest newest most voted

Sage output is correct: it says that KA is a 2-vector field, not that it is a vector field. A 2-vector field is a tensor field of type (2,0), as you can check:

sage: KA.tensor_type()
(2, 0)


You can also take a look at KA (/\ stands for the wedge product):

sage: KA.display()
-3/2 d/dx/\d/dy - 3 d/dx/\d/dz - 3/2 d/dy/\d/dz


See the multivector field documentation for more details about 2-vector fields.

more