Editing Riemannian manifold (section)

=== Levi-Civita connection ===
{{Main|Levi-Civita connection}}

Two Riemannian manifolds with different connections have different geometry. Thankfully, there is a natural connection associated to a Riemannian manifold called the [[Levi-Civita connection]].

A connection <math>\nabla</math> is said to ''preserve the metric'' if
: <math>X\bigl(g(Y,Z)\bigr) =  g(\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
A connection <math>\nabla</math> is ''torsion-free'' if
: <math>\nabla_X Y - \nabla_Y X = [X,Y], </math>
where <math>[\cdot,\cdot]</math> is the [[Lie bracket of vector fields|Lie bracket]].

A ''Levi-Civita connection'' is a torsion-free connection that preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.{{sfn|Lee|2018|pp=122–123}} Note that the definition of preserving the metric uses the regularity of <math>g</math>.