Levi-Civita connection
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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.
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.[1]
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[edit] Formal definition
Let (M,g) be a Riemannian manifold (or pseudo-Riemannian manifold). Then an affine connection is called a Levi-Civita connection if
- it preserves the metric, i.e., for any vector fields X, Y, Z we have , where 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 , where [X,Y] is the Lie bracket of the vector fields X and Y.
[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
(Formally D is the pullback connection on the pullback bundle γ*TM.)
In particular, is a vector field along the curve γ 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 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
Let be the usual scalar product on . Let S2 be the unit sphere in . The tangent space to S2 at a point m is naturally identified with the vector sub-space of consisting of all vectors orthogonal to m. It follows that a vector field Y on S2 can be seen as a map
which satisfies
Denote by dY the differential of such a map. Then we have:
Lemma The formula
defines an affine connection on S2 with vanishing torsion.
Proof
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 m in S2
Consider the map
The map f 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 S2 inherited from . 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.