Levi-Civita connection


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
egin{align}f: S^2 & longrightarrow mathbb{R}\ m & longmapsto langle Y(m), m angle.end{align}
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]


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