Levi-Civita connection

The Levi-Civita connection is the unique torsion-free connection compatible with the metric; its coefficients are the Christoffel symbols formed from first metric derivatives.

For metric tensors related by $\tilde{g}=e^{2\omega }g$, differentiating the rescaled metric tensor in the Christoffel symbol formula gives the displayed difference, where ${\omega }_a={\partial }_a\omega$ and the index on ${\omega }^a$ is raised with $g$. Consequently ${\tilde{\nabla }}_{V}V={\nabla }_{V}V+2V(\omega )V-g(V,V){\mathrm{grad}}_g\omega$. The last term vanishes on a null vector, proving conformal preservation of null geodesic paths; for a non-null geodesic its orthogonal component generally changes the path.

For a family of Levi-Civita connections, the difference of affine connections is a tensor. Dividing by the family parameter and taking the limit shows that its variation is tensorial. Vary the metric compatibility equation and retain the symmetry of the two lower indices. Adding the two cyclic identities and subtracting the third gives the displayed result. In a normal coordinate frame it is the direct derivative of the Christoffel formula; its tensorial form then holds in every frame.

Two Riemannian metrics have the same Levi-Civita connection precisely when their relative positive self-adjoint endomorphism $A$ is parallel for that connection. Parallel transport preserves both metrics. Consequently agreement at one point implies agreement everywhere on a connected manifold. Without that agreement, constant scalar multiples always give examples; constant nonproportional Euclidean forms show that a common connection does not force scalar proportionality.

A Riemannian metric determines a unique torsion-free connection that is a metric connection. Metric compatibility and the torsion identity imply the Koszul formula, which proves uniqueness. Conversely the right-hand side of the Koszul formula is a smooth one-form in its last vector-field argument. The musical isomorphism defines $D_{X}Y$ from it; expanding the Lie bracket of vector fields verifies the connection rules, metric compatibility and zero torsion. Thus it constructs the Levi-Civita connection intrinsically.

The Lie derivative of a connection is the tensorAn isometry preserves the Levi-Civita connection, so ${\mathcal{L}}_K\nabla =0$ for a Killing vector field.