Hairy ball theorem - Wikipedia is not as good or as well-referenced as I'd hoped, and it mainly discusses vector fields on the 2-sphere, 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 higher-order tensors. What's the minimum number of zero points for a 2-tensor field? A 3-tensor one? A 4-tensor 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 everywhere-nonzero 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 2-tensor (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 real-valued n-surface, this group is SO(n), and for a complex-valued 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 2-subtracted order. For SU(n), tensors do not have that constraint. Strictly speaking, for SO(n), we ought to also consider spinors and tensor-spinors. The tensor parts of the latter are also traceless. - For real 2-space and 3-space, the irreducible tensors are all symmetric. Order-m 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 Lie-algebra highest-weight notation, the highest-weight vector for a symmetric (traceless) m-tensor is m * that for a vector. - Applications of these tensor constructs? Electromagnetic-wave polarization can be expressed as Stokes parameters composed from outer products of the electric-field vector with itself or a phase-shifted version of itself. Since electric fields of electromagnetic waves are perpendicular to the direction of motion, they thus span a 2-space. The Stokes parameters are thus I - overall intensity - scalar V - circular polarization - scalar Q, U - linear polarization - symmetric traceless 2-tensor One can make the same construction for gravitational radiation, where one starts with the metric-distortion ST 2-tensor. Its counterparts of the Stokes parameters are I - overall intensity - scalar V - circular polarization - scalar Q, U - (bi)linear polarization - ST 4-tensor So the hairy-ball theorem extended to tensor fields may imply that the Cosmic Microwave Background may have spots with zero linear polarization, and likewise for its gravitational-radiation counterpart.