- Levi-Civita connection
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.
fundamental theorem of Riemannian geometrystates that there is a unique connection which satisfies these properties.
In the theory of Riemannian and
pseudo-Riemannian manifolds the term covariant derivativeis 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 transportand explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy. [See Spivak (1999) Volume II, page 238.]
Let be a
Riemannian manifold(or pseudo-Riemannian manifold).Then an affine connectionis called a Levi-Civita connection if
# "it preserves the metric", i.e., for any vector fields , , we have , where denotes the
derivativeof the function along the vector field .
# "it is torsion-free", i.e., for any vector fields and we have , where is the Lie bracket of the
vector fields and .
The unique connection satisfying these conditions has the form::
Derivative along curve
The Levi-Civita connection (like any affine connection) defines also a derivative along
curves, sometimes denoted by .
Given a smooth curve on and a
vector fieldalong its derivative is defined by:(Formally "D" is the pullback connection on the pullback bundle"γ"*T"M".)
In particular, is a vector field along the curve itself. If 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
geodesicsof the metric that are parametrised proportionally to their arc length.
parallel transportalong 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.
The unit sphere in
Let be the usual
scalar producton .Let be the unit sphere in . The tangent space to at a point is naturally identified with the vector sub-space of consisting of all vectors orthogonal to . It follows that a vector field on can be seen as a map
Denote by the differential of such a map. Then we have:
defines an affine connection on with vanishing torsion.
It is straightforward to prove that satisfies the Leibniz identity and is 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 in
Consider the map
The map is constant, hence its differential vanishes. In particular
The equation (1) above follows.
In fact, this connection is the Levi-Civita connection for the metric on inherited from . Indeed, one can check that this connection preserves the metric.
* [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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