is torsion free iff i.e., if and only if is symmetric in its lower two indices. As one checks by taking for , coordinate vector fields (or computes directly), the Koszul expression of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as where as usual are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix . Derivative along curve The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by . Given a smooth curve on and a vector field along its derivative is defined by Formally, is the pullback connection on the pullback bundle . In particular, is a vector field along the curve itself. If 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 : 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 images below show parallel transport of the Levi-Civita connection associated to two different Riemannian metrics on the plane, expressed in polar coordinates. The metric of left image corresponds to the standard Euclidean metric , while the metric on the right has standard form in polar coordinates (when ), and thus preserves the vector tangent to the circle. This second metric has a singularity at the origin, as can be seen by expressing it in Cartesian coordinates: Example: the unit sphere in Let be the usual scalar product on . Let be the unit sphere in . The tangent space to at a point is naturally identified with the vector subspace of consisting of all vectors orthogonal to . It follows that a vector field on can be seen as a map , which satisfies Denote as the covariant derivative of the map in the direction of the vector . Then we have: In fact, this connection is the