Derivative Along Curve
The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by D.
Given a smooth curve γ on (M,g) and a vector field V along γ its derivative is defined by
(Formally D is the pullback connection on the pullback bundle γ*TM.)
In particular, is a vector field along the curve γ itself. If vanishes, the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.
Read more about this topic: Levi-Civita Connection
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