Levi-Civita Connection - Example: The Unit Sphere in ℝ3

Example: The Unit Sphere in ℝ3

Let be the usual scalar product on ℝ3. Let S2 be the unit sphere in ℝ3. The tangent space to S2 at a point m is naturally identified with the vector sub-space of ℝ3 consisting of all vectors orthogonal to m. It follows that a vector field Y on S2 can be seen as a map Y: S2 → ℝ3, which satisfies

Denote by dY the differential of such a map. Then we have:

Lemma: The formula

defines an affine connection on S2 with vanishing torsion.
Proof: It is straightforward to prove that ∇ satisfies the Leibniz identity and is C∞(S2) linear in the first variable. It is also a straightforward computation to show that this connection is torsion free. So all that needs to be proved here is that the formula above does indeed define a vector field. That is, we need to prove that for all m in S2

Consider the map

begin{align} f: mathbf{S}^2 & longrightarrow R\ m & longmapsto langle Y(m), mrangle. end{align}

The map f is constant, hence its differential vanishes. In particular

The equation (1) above follows.

In fact, this connection is the Levi-Civita connection for the metric on S2 inherited from R3. Indeed, one can check that this connection preserves the metric.