Definitions

Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is the torsion-free Riemannian connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties.

In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called Christoffel symbols.

The Levi-Civita connection is named for Tullio Levi-Civita, although originally discovered by Elwin Bruno Christoffel. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's connection to define a means of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.

Formal definition

Let $\left(M,g\right)$ be a Riemannian manifold (or pseudo-Riemannian manifold). Then an affine connection $nabla$ is called a Levi-Civita connection if

1. it preserves the metric, i.e., for any vector fields $X$, $Y$, $Z$ we have $X\left(g\left(Y,Z\right)\right)=g\left(nabla_X Y,Z\right)+g\left(Y,nabla_X Z\right)$, where $X\left(g\left(Y,Z\right)\right)$ denotes the derivative of the function $g\left(Y,Z\right)$ along the vector field $X$.
2. it is torsion-free, i.e., for any vector fields $X$ and $Y$ we have $nabla_XY-nabla_YX=\left[X,Y\right]$, where $\left[X,Y\right]$ is the Lie bracket of the vector fields $X$ and $Y$.

The unique connection satisfying these conditions has the form:

$g\left(nabla_X Y, W\right) = frac\left\{1\right\}\left\{2\right\} \left\{ X \left(g\left(Y,W\right)\right) + Y \left(g\left(X,W\right)\right) - W \left(g\left(X,Y\right)\right) + g\left(\left[X,Y\right],W\right) - g\left(\left[X,W\right],Y\right) - g\left(\left[Y,W\right],X\right) \right\}$

Derivative along curve

The Levi-Civita connection (like any affine connection) defines also a derivative along curves, sometimes denoted by $D$.

Given a smooth curve $gamma$ on $\left(M,g\right)$ and a vector field $V$ along $gamma$ its derivative is defined by

$D_tV=nabla_\left\{dotgamma\left(t\right)\right\}V.$
(Formally D is the pullback connection on the pullback bundle γ*TM.)

In particular, $dot\left\{gamma\right\}\left(t\right)$ is a vector field along the curve $gamma$ itself. If $nabla_\left\{dotgamma\left(t\right)\right\}dotgamma\left(t\right)$ vanishes, the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.

Parallel transport

In general, parallel transport along a curve with respect to a connection defines isomorphisms between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are orthogonal – that is, they preserve the inner products on the various tangent spaces.

Example

The unit sphere in $mathbb\left\{R\right\}^3$

Let $langle cdot,cdot rangle$ be the usual scalar product on $mathbb\left\{R\right\}^3$. Let $S^2$ be the unit sphere in $mathbb\left\{R\right\}^3$. The tangent space to $S^2$ at a point $m$ is naturally identified with the vector sub-space of $mathbb\left\{R\right\}^3$ consisting of all vectors orthogonal to $m$. It follows that a vector field $Y$ on $S^2$ can be seen as a map

$Y:S^2longrightarrow mathbb\left\{R\right\}^3,$

which satisfies

$langle Y\left(m\right), mrangle = 0, forall min S^2.$

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

Lemma The formula

$left\left(nabla_X Yright\right)\left(m\right) = d_mY\left(X\right) + langle X\left(m\right),Y\left(m\right)rangle m$

defines an affine connection on $S^2$ with vanishing torsion.
Proof
It is straightforward to prove that $nabla$ satisfies the Leibniz identity and is $C^infty\left(S^2\right)$ 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 $S^2$
$langleleft\left(nabla_X Yright\right)\left(m\right),mrangle = 0qquad \left(1\right).$
Consider the map
begin\left\{align\right\} f: S^2 & longrightarrow mathbb\left\{R\right\}

`    m & longmapsto      langle Y(m), mrangle.`
end{align}
The map $f$ is constant, hence its differential vanishes. In particular
$d_mf\left(X\right) = langle d_m Y\left(X\right),mrangle + langle Y\left(m\right), X\left(m\right)rangle = 0.$ The equation (1) above follows. $Box$

In fact, this connection is the Levi-Civita connection for the metric on $S^2$ inherited from $mathbb\left\{R\right\}^3$. Indeed, one can check that this connection preserves the metric.

References

• Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish press.

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