Definition
Let be a Riemannian manifold. There exists a unique affine connection that is
-
Compatible with the metric: i.e.
-
Torsion free (symmetric):
This is called the Levi-Civita connection of .
Search
Let (M,g=⟨,⟩) be a Riemannian manifold. There exists a unique affine connection ∇ that is
Compatible with the metric: i.e. X⟨Y,Z⟩=⟨∇XY,Z⟩+⟨Y,∇XZ⟩
Torsion free (symmetric): ∇XY−∇YX=[X,Y]
This is called the Levi-Civita connection of g.