Question: Suppose we are given a torsion-free (i.e. the torsion tensor vanishes) affine connection on a smooth connected manifold . Does there exist a Riemannian metric such that its Levi-Civita connection is ? If so, is it unique if we prescribe its value at a point?
Let be a local coordinates on and let (using Einstein’s notation). The torsion-free condition is then equivalent to . We want to solve the linear system of partial differential equations (Note that are as is a connection.)
We impose an initial condition at by
for some fixed symmetric .
It is clear from the initial condition and from (1) that is symmetric if the solution exists. According to the theory of first order partial differential equations system (cf. e.g. Stoker’s Differential Geometry Appendix B), the system (1) is uniquely solvable if and only if the compatibility conditions hold:
In view of (1), this is equivalent to
To shed some light on the following computation, let us introduce the curvature tensor associated with .
Definition 1 Given a connection and vector fields , , , we define the curvature tensor by
Here is the Lie bracket of vector fields. It is readily checked that is a tensor field regardless of whether is torsion-free or not. In local coordinates, we define by
It is not hard to see that if is torsion-free and the is compatible with in the sense of (1), then we have the local formula
We subtract RHS from LHS of (3) and get
Subtracting the two equations,
This implies that if is defined by . This is easily seen to be equivalent to . We conclude that (2) holds, and therefore we have proved the existence and uniqueness of the solution to the system
Gometrically, if we prescribe an inner product on for some , then there is a unique Riemannian metric which is compatible with the torsion-free connection . So we have proved
Theorem 2 Suppose is a torsion-free affine connection on a smooth connected manifold . Let be an inner product on , where . Then there exists a unique Riemannian metric such that is the Levi-Civita connection of and .