Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric 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 after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.

The Levi-Civita notions of intrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding $M^n \subset R^{n(n+1)/2}\,\!$, since the definition of the Christoffel symbols make sense in any Riemannian manifold. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. It was not until 1917 that Levi-Civita interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space.

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., $\nabla g = 0$.
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.

Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. DoCarmo's text.

Assuming a Levi Civita connection exists it is uniquely determined. Using conditions 1 and the symmetry of the metric tensor g we find:

$X (g(Y,Z)) + Y (g(Z,X))- Z (g(Y,X)) = g(\nabla_X Y + \nabla_Y X, Z) + g(\nabla_X Z - \nabla_Z X, Y) + g(\nabla_Y Z - \nabla_Z Y, X)$

By condition 2 the right hand side is equal to

$2g(\nabla_X Y, Z) - g([X,Y], Z) + g([X,Z],Y) + g([Y,Z],X)$

so we find

$g(\nabla_X Y, Z) = \frac{1}{2} \{ X (g(Y,Z)) + Y (g(Z,X)) - Z (g(X,Y)) + g([X,Y],Z) - g([Y,Z], X) - g([X,Z], Y) \}$

Since Z is arbitrary, this uniquely determines $\nabla_X Y$. Conversely, using the last line as a definition one shows that the expression so defined is a connection compatible with the metric, i.e. is a Levi Civita connection.

Christoffel symbols

Let $\nabla$ be the connection of the Riemannian metric. Choose local coordinates $x^1 \ldots x^n$ and let $\Gamma_{jk}^l$ be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry

$\Gamma_{jk}^l = \Gamma_{kj}^l$

The definition of the Levi Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as

$\Gamma_{jk}^l = \frac{1}{2}\sum_r g^{lr} \{\partial _j g_{rk} + \partial _k g_{jr} - \partial _r g_{jk} \}$

where as usual g^ij are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix (g_kl).

Derivative along curve

The Levi-Civita connection (like any affine connection) also defines 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 $S^2 \,$

$\langle\left(\nabla_X Y\right)(m),m\rangle = 0\qquad (1).$

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 $S^2 \,$ inherited from $\mathbb{R}^3$. Indeed, one can check that this connection preserves the metric.

See also

• Affine connection
• Weitzenböck connection

References

Primary historical references

• Christoffel, Elwin Bruno (1869), "Über die Transformation der homogenen Differentialausdrücke zweiten Grades", J. für die Reine und Angew. Math. 70: 46–70 
• Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana", Rend. Circ. Mat. Palermo 42: 73–205 

Secondary references

• Boothby, William M. (1986). An introduction to differential manifolds and Riemannian geometry. Academic Press. ISBN 0-12-116052-1. 
• Kobayashi, S., and Nomizu, K. (1963). Foundations of differential geometry. John Wiley & Sons. ISBN 0-470-49647-9.  See Volume I pag. 158
• Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish Press. ISBN 0-914098-71-3. 

External links

• MathWorld: Levi-Civita Connection
• PlanetMath: Levi-Civita Connection