Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 15 / 5 / a

An affine connection $\nabla$ is a metric connection for the Riemannian metric $g$ whenEquivalently the induced covariant derivative of the metric tensor satisfies $\nabla g=0$. This is metric compatibility; it imposes no condition on the torsion tensor. A Levi-Civita connection has the additional property of being a torsion-free connection.

A vector field along a map $\gamma$ assigns $V(t)\in T_{\gamma (t)}M$ smoothly to each parameter value. Equivalently, it is a section of a vector bundle ${\gamma }^{\ast }TM$. The pullback connection defines its covariant derivative along a curve, $D_{t}V$. To make the definition explicit, use connection coefficients with the conventionFor $V=V^k(t){\partial }_k{|}_{\gamma (t)}$ the derivative isThe change of frame of a vector-bundle connection ensures this expression is independent of the coordinate chart. It remains meaningful when $\dot{\gamma }=0$ and does not require extending $V$ to one ambient vector field.

The field is parallel exactly when $\boxed{D_{t}V=0}$. A geodesic for $\nabla$, with its specified affine parameter, is a smooth curve whose own tangent is parallel:This definition applies to any affine connection, whether or not it is a metric connection. A non-affine change of parameter generally gives a pregeodesic instead of this parametrized geodesic.