**Example: The Unit Sphere in ℝ3**

Let be the usual scalar product on ℝ3. Let **S**2 be the unit sphere in ℝ3. The tangent space to **S**2 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 **S**2 can be seen as a map *Y*: **S**2 → ℝ3, which satisfies

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

**Lemma:** The formula

defines an affine connection on **S**2 with vanishing torsion.

**Proof:** It is straightforward to prove that ∇ satisfies the Leibniz identity and is *C*∞(**S**2) 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 **S**2

Consider the map

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 **S**2 inherited from **R**3. Indeed, one can check that this connection preserves the metric.

Read more about this topic: Levi-Civita Connection

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