Levi-Civita connection

Remarks

Example: on $\R^n$ with $g = \sum_i dx_i^2$, let $\nabla_X Y$ be the vector field whose components are $X(Y^1), \ldots, X(Y^n)$. Then $\nabla_X Y - \nabla_Y X$ has components $X(Y^i) - Y(X^i)$, which is $[X, Y]$, and the product rule gives clause 2 for $g(Y, Z) = \sum_i Y^i Z^i$.

Non-example: on $\R^3$ with the Euclidean metric, let $\nabla_X Y = D_X Y + X \times Y$, where $D$ is the flat connection above and $\times$ is the cross product. This is an affine connection, and clause 2 holds because $\ip{X \times Y}{Z} + \ip{Y}{X \times Z} = 0$. Its torsion is $\nabla_X Y - \nabla_Y X - [X, Y] = 2 \, X \times Y$, which is not zero, so clause 1 fails.

The two clauses determine $\nabla$; this is the fundamental theorem of Riemannian geometry. In coordinates, $\nabla_{\partial_i} \partial_j = \sum_k \Gamma^k_{ij} \partial_k$ with the Christoffel symbols $$\Gamma^k_{ij} = \frac{1}{2} \sum_l g^{kl} \left( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \right).$$