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

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 Xp, Yp 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

$D_{t}V=\nabla _{{\dot {\gamma }}(t)}V.$ 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.

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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: S2R3, which satisfies

$\langle Y(m),m\rangle =0,\qquad \forall m\in {\mathbf {S} }^{2}.$ Denote by dmY(X) the covariant derivative of the map Y in the direction of the vector X. Then we have:

Lemma: The formula

$\left(\nabla _{X}Y\right)(m)=d_{m}Y(X)+\langle X(m),Y(m)\rangle m$ 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

$d_{m}f(X)=\langle d_{m}Y(X),m\rangle +\langle Y(m),X(m)\rangle =0.$ 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.