Connection (mathematics) - Motivation: The Unsuitability of Coordinates

Motivation: The Unsuitability of Coordinates

Consider the following problem. Suppose that a tangent vector to the sphere S is given at the north pole, and we are to define a manner of consistently moving this vector to other points of the sphere: a means for parallel transport. Naïvely, this could be done using a particular coordinate system. However, unless proper care is applied, the parallel transport defined in one system of coordinates will not agree with that of another coordinate system. A more appropriate parallel transportation system exploits the symmetry of the sphere under rotation. Given a vector at the north pole, one can transport this vector along a curve by rotating the sphere in such a way that the north pole moves along the curve without axial rolling. This latter means of parallel transport is the Levi-Civita connection on the sphere. If two different curves are given with the same initial and terminal point, and a vector v is rigidly moved along the first curve by a rotation, the resulting vector at the terminal point will be different from the vector resulting from rigidly moving v along the second curve. This phenomenon reflects the curvature of the sphere. A simple mechanical device that can be used to visualize parallel transport is the South Pointing Chariot.

For instance, suppose that S is given coordinates by the stereographic projection. Regard S as consisting of unit vectors in R3. Then S carries a pair of coordinate patches: one covering a neighborhood of the north pole, and the other of the south pole. The mappings

cover a neighborhood U₀ of the north pole and U₁ of the south pole, respectively. Let X, Y, Z be the ambient coordinates in R3. Then φ₀ and φ₁ have inverses

so that the coordinate transition function is inversion in the circle:

Let us now represent a vector field in terms of its components relative to the coordinate derivatives. If P is a point of U₀ ⊂ S, then a vector field may be represented by

where denotes the Jacobian matrix of φ₀, and v₀ = v₀(x, y) is a vector field on R2 uniquely determined by v. Furthermore, on the overlap between the coordinate charts U₀ ∩ U₁, it is possible to represent the same vector field with respect to the φ₁ coordinates:

To relate the components v₀ and v₁, apply the chain rule to the identity φ₁ = φ₀ o φ₀₁:

Applying both sides of this matrix equation to the component vector v₁(φ₁−1(P)) and invoking (1) and (2) yields

We come now to the main question of defining how to transport a vector field parallelly along a curve. Suppose that P(t) is a curve in S. Naïvely, one may consider a vector field parallel if the coordinate components of the vector field are constant along the curve. However, an immediate ambiguity arises: in which coordinate system should these components be constant?

For instance, suppose that v(P(t)) has constant components in the U₁ coordinate system. That is, the functions v₁(φ₁−1(P(t))) are constant. However, applying the product rule to (3) and using the fact that dv₁/dt = 0 gives

But is always a non-singular matrix (provided that the curve P(t) is not stationary), so v₁ and v₀ cannot ever be simultaneously constant along the curve.