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Hairy ball theorem  Wikipedia is not as good or as wellreferenced as I'd hoped, and it mainly discusses vector fields on the 2sphere, the ordinary sort of sphere.
In particular, it does not mention the minimum number of zero points of a continuous vector field on a sphere. I would guess that it's 2, or more generally,
Euler characteristic
Sphere: 2, torus: 0, etc.
There's also the question of generalization to higherorder tensors. What's the minimum number of zero points for a 2tensor field? A 3tensor one? A 4tensor one? Etc.
A torus is topologically equivalent to a rectangle with periodic boundary conditions. It's easy to show that it's possible to construct an everywherenonzero tensor of any order  all one needs to do is construct a constant one.

To simplify this problem a bit, we ought to consider irreducible tensors. The 2tensor (a)*(metric) is reducible into the scalar a, for instance. These are connected with the irreducible representations of each surface point's neighborhood's symmetry group.
For a realvalued nsurface, this group is SO(n), and for a complexvalued one, SU(n). The surface need not have that global symmetry; only that local symmetry.
For SO(n), every irreducible tensor is traceless, because a nonzero trace would enable separating out a tensor with a 2subtracted order. For SU(n), tensors do not have that constraint.
Strictly speaking, for SO(n), we ought to also consider spinors and tensorspinors. The tensor parts of the latter are also traceless.

For real 2space and 3space, the irreducible tensors are all symmetric. Orderm tensors are composed of these numbers of basis tensors:
2D: m = 0 > 1, m > 0 > 2
3D; 2m+1
For 2D, the tensors can be related to Chebyshev polynomials, while for 3D, the tensors can be related to spherical harmonics.
Irreducible tensors need not be symmetric for at least 4 real dimensions or for complex dimensions.
In Liealgebra highestweight notation, the highestweight vector for a symmetric (traceless) mtensor is m * that for a vector.

Applications of these tensor constructs?
Electromagneticwave polarization can be expressed as Stokes parameters composed from outer products of the electricfield vector with itself or a phaseshifted version of itself. Since electric fields of electromagnetic waves are perpendicular to the direction of motion, they thus span a 2space. The Stokes parameters are thus
I  overall intensity  scalar
V  circular polarization  scalar
Q, U  linear polarization  symmetric traceless 2tensor
One can make the same construction for gravitational radiation, where one starts with the metricdistortion ST 2tensor. Its counterparts of the Stokes parameters are
I  overall intensity  scalar
V  circular polarization  scalar
Q, U  (bi)linear polarization  ST 4tensor
So the hairyball theorem extended to tensor fields may imply that the Cosmic Microwave Background may have spots with zero linear polarization, and likewise for its gravitationalradiation counterpart.
In particular, it does not mention the minimum number of zero points of a continuous vector field on a sphere. I would guess that it's 2, or more generally,
Euler characteristic
Sphere: 2, torus: 0, etc.
There's also the question of generalization to higherorder tensors. What's the minimum number of zero points for a 2tensor field? A 3tensor one? A 4tensor one? Etc.
A torus is topologically equivalent to a rectangle with periodic boundary conditions. It's easy to show that it's possible to construct an everywherenonzero tensor of any order  all one needs to do is construct a constant one.

To simplify this problem a bit, we ought to consider irreducible tensors. The 2tensor (a)*(metric) is reducible into the scalar a, for instance. These are connected with the irreducible representations of each surface point's neighborhood's symmetry group.
For a realvalued nsurface, this group is SO(n), and for a complexvalued one, SU(n). The surface need not have that global symmetry; only that local symmetry.
For SO(n), every irreducible tensor is traceless, because a nonzero trace would enable separating out a tensor with a 2subtracted order. For SU(n), tensors do not have that constraint.
Strictly speaking, for SO(n), we ought to also consider spinors and tensorspinors. The tensor parts of the latter are also traceless.

For real 2space and 3space, the irreducible tensors are all symmetric. Orderm tensors are composed of these numbers of basis tensors:
2D: m = 0 > 1, m > 0 > 2
3D; 2m+1
For 2D, the tensors can be related to Chebyshev polynomials, while for 3D, the tensors can be related to spherical harmonics.
Irreducible tensors need not be symmetric for at least 4 real dimensions or for complex dimensions.
In Liealgebra highestweight notation, the highestweight vector for a symmetric (traceless) mtensor is m * that for a vector.

Applications of these tensor constructs?
Electromagneticwave polarization can be expressed as Stokes parameters composed from outer products of the electricfield vector with itself or a phaseshifted version of itself. Since electric fields of electromagnetic waves are perpendicular to the direction of motion, they thus span a 2space. The Stokes parameters are thus
I  overall intensity  scalar
V  circular polarization  scalar
Q, U  linear polarization  symmetric traceless 2tensor
One can make the same construction for gravitational radiation, where one starts with the metricdistortion ST 2tensor. Its counterparts of the Stokes parameters are
I  overall intensity  scalar
V  circular polarization  scalar
Q, U  (bi)linear polarization  ST 4tensor
So the hairyball theorem extended to tensor fields may imply that the Cosmic Microwave Background may have spots with zero linear polarization, and likewise for its gravitationalradiation counterpart.