Analysis of Vector Fields on S2

We demonstrate analysis of vector fields on the sphere using the tensor field of contravariant derivatives. Shown are: a vector field in the top-left corner, the scaler field of vector lengths in top-right, the scalar field of the traces of the contravariant tensor field in botton-left, and the scalar field of the determinants of the contravariant tensor field in the bottom-right.

Let M be a n-manifold and X and Y are two smooth vector fields on M. With LXY we denote the Lie derivative of Y with respect to X. Define the contra-variant 2-tensor field

T W = (LX Y )(LX Y ) .
If in addition M is a Riemannian manifold with metric tensor G, then L = GW ∈ T11(M) is a linear operator field on M. For any similarity invariant h, h(GW) is a smooth function on M.

Let (x1,x2) be parametrizatiom on S2 and let X1 and X2 be the canonical vector fields -∂1 ∂x and -∂2 ∂x. Then the Lie derivatives with respect to X and Y are in fact the covariant derivatives

L 1Y = ∇ ∂ Y and L 2Y = ∇ ∂ Y. X ∂x1- X ∂x2
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