Fundamental theorem of riemannian geometry

Take the connection supplied by [F1]. In $K(X,Y,Z)+K(X,Z,Y)$ the two $X$-derivative terms add to $2Xg(Y,Z)$, the $Y$ and $Z$ derivative terms cancel, and the bracket terms cancel in pairs by metric symmetry and bracket skew-symmetry. Dividing by two gives $g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z)=Xg(Y,Z)$, proving compatibility.

In $K(X,Y,Z)-K(Y,X,Z)$ all metric derivative terms cancel. The bracket terms pairing against $X$ and $Y$ cancel because $[Z,X]=-[X,Z]$ and $[Z,Y]=-[Y,Z]$; the two remaining terms give $2g(Z,[X,Y])$. Thus $g({\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y],Z)=0$ for every local $Z$. Nondegeneracy gives torsion zero, so the connection is Levi–Civita.

Any second Levi–Civita connection has exactly the same pairing with every $Z$ by [F2], so nondegeneracy identifies its derivative on every $X,Y$ with the constructed one. This proves uniqueness. Empty and zero-dimensional manifolds have the unique zero operator; dimension one still requires compatibility, supplied in step 1.1. The construction and nondegeneracy argument remain valid at boundary points and use no connection-existence choice theorem.