Differentiable Manifolds and Tangent Spaces

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In Rn, there is a globally defined orthonormal frame

E1p=(1,0,⋯,0)p, E2p=(0,1,0,⋯,0)p,⋯,Enp=(0,⋯,0,1)p.

For any tangent vector Xp∈Tp(Rn), Xp=∑ni=1αiEip. Note that the coefficients αi are the ones that distinguish tangent vectors in Tp(Rn). For a differentiable function f, the directional derivative X∗pf of f with respect to Xp is given by

X∗pf=∑i=1nαi(∂f∂xi).

We identify each Xp with the differential operator

X∗p=∑i=1nαi∂∂xi:C∞(p)⟶R.

Then the frame fields E1p,E2p,⋯,Enp are identified with

(∂∂x1)p,(∂∂x2)p,⋯,(∂∂xn)p

respectively. Unlike Rn, we cannot always have a globally defined frame on a differentiable manifold. So it is necessary for us to use local coordinate neighborhoods that are homeomorphic toRn and the associated frames ∂∂x1,∂∂x2,⋯,∂∂xn.

Example. The points (x,y,z) are represented in terms of the spherical coordinates (ϕ,θ) as

x=sinϕcosθ,y=sinϕsinθ,z=cosϕ, 0≤ϕ≤π, 0≤θ≤2π.

By chain rule, one finds the standard basis ∂∂ϕ,∂∂θ for T∗S2:

∂∂ϕ∂∂θ=cosϕcosθ∂∂x+cosϕsinθ∂∂y−sinϕ∂∂z,=−sinϕsinθ∂∂x+sinϕcosθ∂∂y.

The frame field is not globally defined on S2 since ∂∂θ at ϕ=0,π. More generally, the following theorem holds.

Frame field on 2-sphere

Theorem. [Hairy Ball Theorem] If n is even, a non-vanishing C∞ vector field on Sn does not exist i.e. a C∞ vector field on Sn must take zero value at some point of Sn.

The Hairy Ball Theorem tells us why we have ball spots on our heads. It can be also stated as “you cannot comb a hairy ball flat.” There may also be a meteorological implication of this theorem. It may implicate that there must be at least one spot on earth where there is no wind at all. No-wind spot may be the eye of a hurricane. So, as long as there is wind (and there always is) on earth, there must be a hurricane somewhere at all times.

It has been known that all odd-dimensional spheres have at least one non-vanishing C∞ vector field and that only spheres S1,S3,S7 have a C∞ field of basis. For instance, there are three mutually perpendicular unit vector fields on S3⊂R4 i.e. a frame field: Let S3={(x1,x2,x3,x4)∈R4:∑4i=1(xi)2=1}. Then

XYZ=−x2∂∂x1+x2∂∂x2+x4∂∂x3−x3∂∂x4,=−x3∂∂x1−x4∂∂x2+x1∂∂x3+x2∂∂x4,=−x4∂∂x1+x3∂∂x2−x2∂∂x3+x1∂∂x4

form an orthonormal basis of C∞ vector fields on S3.