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

Formal definition

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

# "it preserves the metric", i.e., for any vector fields $X$, $Y$, $Z$ we have $X(g(Y,Z))=g(\; abla\_X\; Y,Z)+g(Y,\; abla\_X\; Z)$, 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 $abla\_XY-\; abla\_YX=\; [X,Y]$, where $[X,Y]$ is the Lie bracket of the vector fields $X$ and $Y$.

The unique connection satisfying these conditions has the form::$g(\; abla\_X\; Y,\; W)\; =\; frac\{1\}\{2\}\; \{\; X\; (g(Y,W))\; +\; Y\; (g(X,W))\; -\; W\; (g(X,Y))\; +\; g(\; [X,Y]\; ,W)\; -\; g(\; [X,W]\; ,Y)\; -\; g(\; [Y,W]\; ,X)\; \}$

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 $(M,g)$ and a vector field $V$ along $gamma$ its derivative is defined by:$D\_tV=\; abla\_\{dotgamma(t)\}V.$(Formally "D" is the pullback connection on the pullback bundle "γ"*T"M".)

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

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 $mathbb\{R\}^3$

Let $langle\; cdot,cdot\; angle$ 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

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

which satisfies

:$langle\; Y(m),\; m\; angle\; =\; 0,\; forall\; min\; S^2.$

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

LemmaThe formula

:$left(\; abla\_X\; Y\; ight)(m)\; =\; d\_mY(X)\; +\; langle\; X(m),Y(m)\; angle\; m$

defines an affine connection on $S^2$ with vanishing torsion.
"Proof"
It is straightforward to prove that $abla$ 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 $S^2$
$langleleft(\; abla\_X\; Y\; ight)(m),m\; angle\; =\; 0qquad\; (1).$
Consider the map

The map $f$ is constant, hence its differential vanishes. In particular
$d\_mf(X)\; =\; langle\; d\_m\; Y(X),m\; angle\; +\; langle\; Y(m),\; X(m)\; angle\; =\; 0.$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.

Notes

References

*

ee also

*Weitzenbock connection

External links

* [http://mathworld.wolfram.com/Levi-CivitaConnection.html MathWorld: Levi-Civita Connection]
* [http://planetmath.org/encyclopedia/LeviCivitaConnection.html PlanetMath: Levi-Civita Connection]