Levi-Civita connection

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.

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 gamma on (M,g) and a vector field V along gamma its derivative is defined by

(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_{dotgamma(t)}dotgamma(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.


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^2longrightarrow mathbb{R}^3,

which satisfies

langle Y(m), mrangle = 0, forall min S^2.

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

Lemma The formula

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

defines an affine connection on S^2 with vanishing torsion.
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
langleleft(nabla_X Yright)(m),mrangle = 0qquad (1).
Consider the map
begin{align} f: S^2 & longrightarrow mathbb{R}

    m & longmapsto      langle Y(m), mrangle.
The map f is constant, hence its differential vanishes. In particular
d_mf(X) = langle d_m Y(X),mrangle + 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.



  • Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish press.

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