Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 17 / 4 / a

A connection on $TM$ is a metric connection for $g$ if for all smooth vector fields $X,Y,Z$,Equivalently, its induced covariant derivative of the metric satisfies $\nabla g=0$. This is also called a metric-compatible connection.

The torsion tensor of an affine connection isThe connection is symmetric, or a torsion-free connection, exactly whenIn a coordinate frame, write ${\nabla }_{{\partial }_i}{\partial }_j={\sum }_k{\Gamma }_{ij}^k{\partial }_k$. Since coordinate fields commute, the condition is ${\Gamma }_{ij}^k={\Gamma }_{ji}^k$; this is the reason for the word symmetric.

The Levi-Civita connection is the unique affine connection which is both a metric connection and a torsion-free connection. Uniqueness is encoded by the Koszul formulaIt follows by adding the metric-compatibility identities with derivative fields $X,Y$, subtracting the one with derivative field $Z$, and replacing differences of covariant derivatives by Lie brackets. Nondegeneracy of $g$ determines ${\nabla }_{X}Y$ uniquely from all its pairings with $Z$. The existence and uniqueness of the Levi-Civita connection supplies existence; no existence proof is needed here.

In Cartesian coordinates on Euclidean space, defineThis is an affine connection by the ordinary product rule. Its connection coefficients in this coordinate frame are all zero. The component formula for the Lie bracket of vector fields gives ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$, so it is torsion-free. Also, since the Euclidean metric has constant coefficients,This is exactly metric compatibility. By uniqueness it is the Euclidean Levi-Civita connection. In particular constant vector fields are parallel, while the derivative of a general field is given by the boxed formula.