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

Contents

Formal definition

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

  1. 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.
  2. 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) \}

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

D_tV=\nabla_{\dot\gamma(t)}V.

(Formally D is the pullback connection on the pullback bundle γ*TM.)

In particular, \dot{\gamma}(t) is a vector field along the curve γ itself. If \nabla_{\dot\gamma(t)}\dot\gamma(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 \rangle be the usual scalar product on \mathbb{R}^3. Let S2 be the unit sphere in \mathbb{R}^3. The tangent space to S2 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 S2 can be seen as a map

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

which satisfies

\langle Y(m), m\rangle = 0, \forall m\in S^2.

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

Lemma The formula

\left(\nabla_X Y\right)(m) = d_mY(X) + \langle X(m),Y(m)\rangle m

defines an affine connection on S2 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 S2
\langle\left(\nabla_X Y\right)(m),m\rangle = 0\qquad (1).
Consider the map
\begin{align} f: S^2 & \longrightarrow \mathbb{R}\\ m & \longmapsto \langle Y(m), m\rangle. \end{align}
The map f is constant, hence its differential vanishes. In particular
d_mf(X) = \langle d_m Y(X),m\rangle + \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 S2 inherited from \mathbb{R}^3. Indeed, one can check that this connection preserves the metric.

Notes

  1. ^ See Spivak (1999) Volume II, page 238.

References

  • Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish press. ISBN 0-914098-71-3. 

See also

External links

Wikipedia content modification information:

  • This page was last modified on 5 November 2008, at 18:05.

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