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,^[1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols ^[2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.^[3]

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 $M^n \subset R^{n(n%2B1)/2}\,\!$ , 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.

1 Formal definition
2 Christoffel symbols
3 Derivative along curve
4 Parallel transport
5 Example
- 5.1 The unit sphere in
6 Notes
7 See also
8 References
- 8.1 Primary historical references
- 8.2 Secondary references
9 External links

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

it preserves the metric, i.e., $\nabla g = 0$ .
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:

$X (g(Y,Z)) %2B Y (g(Z,X))- Z (g(Y,X)) = g(\nabla_X Y %2B \nabla_Y X, Z) %2B g(\nabla_X Z - \nabla_Z X, Y) %2B g(\nabla_Y Z - \nabla_Z Y, X)$

By condition 2 the right hand side is equal to

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

so we find

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

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

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

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 $\gamma$ on $(M,g)$ and a vector field $V$ along $\gamma$ 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 $\gamma$ 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 $S^2$ 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

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

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

$\left(\nabla_X Y\right)(m) = d_mY(X) %2B \langle X(m),Y(m)\rangle m$

defines an affine connection on $S^2$ 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

$d_mf(X) = \langle d_m Y(X),m\rangle %2B \langle Y(m), X(m)\rangle = 0.$

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.

Notes

^ See Levi-Civita (1917)
^ See Christoffel (1869)
^ See Spivak (1999) Volume II, page 238

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.