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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.## Formal definition

## Derivative along curve

## Parallel transport

## Example

### 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

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

$d\_mf(X)\; =\; langle\; d\_m\; Y(X),mrangle\; +\; langle\; Y(m),\; X(m)rangle\; =\; 0.$ 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.## Notes

## References

## See also

## External links

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.

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 $Y$.

The unique connection satisfying these conditions has the form:

- $g(nabla\_X\; Y,\; W)\; =\; frac\{1\}\{2\}\; \{\; X\; (g(Y,W))\; +\; Y\; (g(X,W))\; -\; W\; (g(X,Y))\; +\; g([X,Y],W)\; -\; g([X,W],Y)\; -\; g([Y,W],X)\; \}$

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

- $D\_tV=nabla\_\{dotgamma(t)\}V.$

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

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.

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

which satisfies

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

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

Lemma
The formula

- $left(nabla\_X\; Yright)(m)\; =\; d\_mY(X)\; +\; langle\; X(m),Y(m)rangle\; m$

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 $S^2$

$langleleft(nabla\_X\; Yright)(m),mrangle\; =\; 0qquad\; (1).$

Consider the map

$begin\{align\}\; f:\; S^2\; \&\; longrightarrow\; mathbb\{R\}$

m & longmapsto langle Y(m), mrangle.end{align}

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

$d\_mf(X)\; =\; langle\; d\_m\; Y(X),mrangle\; +\; langle\; Y(m),\; X(m)rangle\; =\; 0.$ 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.

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

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Last updated on Thursday July 10, 2008 at 13:55:26 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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