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.

Read more about this topic:  Levi-Civita Connection

Famous quotes containing the words sphere and/or unit:

    “In the new science of the twenty-first century, not physical force but spiritual force will lead the way. Mental and spiritual gifts will be more in demand than gifts of a physical nature. Extrasensory perception will take precedence over sensory perception. And in this sphere woman will again predominate.”
    —Elizabeth Gould Davis (b. 1910)

    “During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner’s jury. In the case of elected bodies the only way of effecting this is by the Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
    —George Bernard Shaw (1856–1950)