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In Riemannian geometry, the Levi-Civita connection is the torsion-free metric 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 after 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.^[1]

[edit] Formal definition

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

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

Assuming we have a connection that satisfies the two conditions we can find a unique connection that has the form:

$g(\nabla_X Y, Z) = X (g(Y,Z)) - g(Y,\nabla_X Z)$

$g(\nabla_X Y, Z) = X (g(Y,Z)) - g(Y,\nabla_Z X) - g(Y,[X,Z])$

$g(\nabla_X Y, Z) = X (g(Y,Z)) - Z (g(Y,X)) + g(\nabla_Z Y,X) - g(Y,[X,Z])$

$g(\nabla_X Y, Z) = X (g(Y,Z)) - Z (g(Y,X)) + g(\nabla_Y Z,X) + g([Z,Y],X) - g(Y,[X,Z])$

$g(\nabla_X Y, Z) = X (g(Y,Z)) - Z (g(Y,X)) + Y (g(Z,X))- g(Z,\nabla_Y X) + g([Z,Y],X) - g(Y,[X,Z])$

$g(\nabla_X Y, Z) = X (g(Y,Z)) - Z (g(Y,X)) + Y (g(Z,X)) - g(Z,\nabla_X Y) - g(Z,[Y,X]) + g([Z,Y],X) - g(Y,[X,Z])$

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

Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. DoCarmo's text.

[edit] Riemannian Metric

Let $\nabla$ be the connection of the Riemannian metric. We then find that:

Γ j k l = Γ k j l

And we then find using the definition above that we can write our christoffel symbols in terms of the metric:

$\Gamma_{jk}^l = \frac{1}{2}\sum_r g^{lr} \{\partial _j g_{rk} + \partial _k g_{jr} - \partial _r g_{jk} \}$

[edit] 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 $γ$ on $(M, g)$ and a vector field $V$ along $γ$ its derivative is defined by

$D_tV=\nabla_{\dot\gamma(t)}V.$

(Formally D is the pullback connection on the pullback bundle γ*TM.)

In particular, $\dot{\gamma}(t)$ is a vector field along the curve $γ$ itself. If $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ 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.

[edit] 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.

[edit] Example

[edit] The unit sphere in $\mathbb{R}^3$

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

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

which satisfies

$\langle Y(m), m\rangle = 0, \forall m\in S^2.$

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

Lemma The formula

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

defines an affine connection on $S 2$ with vanishing torsion.
Proof
It is straightforward to prove that $\nabla$ satisfies the Leibniz identity and is $C^\infty(S^2)$ 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 $S^2 \,$

$\langle\left(\nabla_X Y\right)(m),m\rangle = 0\qquad (1).$

Consider the map

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

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

$d_mf(X) = \langle d_m Y(X),m\rangle + \langle Y(m), X(m)\rangle = 0.$

The equation (1) above follows.

$\Box$

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

[edit] Notes

^ See Spivak (1999) Volume II, page 238.

[edit] References

Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish press. ISBN 0-914098-71-3.

[edit] See also

Weitzenbock connection

[edit] External links

[0] See Spivak (1999) Volume II, page 238.

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