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.

History

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

$M^n \subset \mathbf{R}^{\frac{n(n+1)}{2}},$

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.

Notation

• (M, g) denotes a Riemannian or pseudo-Riemannian manifold.
• TM is the tangent bundle of M.
• g is the Riemannian or pseudo-Riemannian metric of M.
• X, Y, Z are smooth vector fields on M, i. e. smooth sections of TM.
• [X, Y] is the Lie bracket of X and Y. It is again a smooth vector field.

The metric g can take up to two vectors or vector fields X, Y as arguments. In the former case the output is a number, the (pseudo-)inner product of X and Y. In the latter case, the inner product of X_p, Y_p is taken at all points p on the manifold so that g(X, Y) defines a smooth function on M. Vector fields act as differential operators on smooth functions. In a basis, the action reads

$Xf = X^i\frac{\partial}{\partial x^i}f = X^i\partial_i f,$

where Einstein's summation convention is used.

Formal definition

An affine connection ∇ is called a Levi-Civita connection if

1. it preserves the metric, i.e., ∇g = 0.
2. it is torsion-free, i.e., for any vector fields X and Y we have ∇_XY − ∇_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 ∇_XY. 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 ∇ be the connection of the Riemannian metric. Choose local coordinates $x^1 \ldots x^n$ and let $\Gamma^l{}_{jk}$ be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry

$\Gamma^l{}_{jk} = \Gamma^l{}_{kj}.$

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

$\Gamma^l{}_{jk} = \tfrac{1}{2} g^{lr} \left \{\partial _k g_{rj} + \partial _j g_{rk} - \partial _r g_{jk} \right \}$

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 $\gamma^*\nabla$ 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. Formally, the condition can be restated as the vanishing of the pullback connection applied to $\dot{\gamma}$ :

$(\gamma^*\nabla) \dot{\gamma}\equiv 0.$

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.

Parallel Transports Under Levi-Civita Connections

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This transport is given by the metric ds² = dx² + dy².

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This transport is given by the metric ds² = dr² + r²dθ².

Example: The unit sphere in R³

Let $\langle \cdot,\cdot \rangle$ be the usual scalar product on R³. Let S² be the unit sphere in R³. The tangent space to S² at a point m is naturally identified with the vector sub-space of R³ consisting of all vectors orthogonal to m. It follows that a vector field Y on S² can be seen as a map Y: S² → R³, which satisfies

$\langle Y(m), m\rangle = 0, \qquad \forall m\in \mathbf{S}^2.$

Denote by d_mY(X) the covariant derivative of the map Y in the direction of the vector X. 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 ∇ satisfies the Leibniz identity and is C^∞(S²) 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 f that sends every m in S² to <Y(m), m>, which is always 0. 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 R³. Indeed, one can check that this connection preserves the metric.

See also

• Affine connection
• Weitzenböck connection

References

Secondary references

• See Volume I pag. 158

External links

• MathWorld: Levi-Civita Connection
• PlanetMath: Levi-Civita Connection
• Levi-Civita connection at the Manifold Atlas