Levi civita connection

For a supplied Riemannian metric $g$, a Levi–Civita connection is an affine connection on $TM$ that is both metric compatible and torsion free, in the senses of Affine connection on a smooth manifold, Metric compatible connection on a riemannian vector bundle and Torsion tensor of an affine connection. Thus for all local fields $Xg(Y,Z)=g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z),{\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y]\mathrm{.}$ Existence and uniqueness are proved below from the supplied metric by the Koszul formula. This definition does not choose an arbitrary connection first and does not assume the general connection-existence theorem. In dimension zero the unique tangent-bundle connection satisfies both identities; in dimension one torsion freeness alone imposes no restriction, so metric compatibility remains essential.