Geodesic for a linear connection

From Diffgeom
Jump to: navigation, search

Definition

Given data

Definition part

Consider a smooth curve \gamma:[0,1] \to M. Let D/dt denote the connection along \gamma induced by \nabla, and consider the transport along \gamma for the connection D/dt. Then, we say that \gamma is a geodesic for \nabla if, under that transport, the tangent vector \gamma'(0) at \gamma(0) gets transported, at time t, to the tangent vector \gamma'(t) at \gamma(t).

Equivalently, we say that \gamma is a geodesic if:

\frac{D}{dt}(\gamma'(t)) = 0 \forall t

(with the derivative interpreted as a suitable one-sided derivative at the endpoints).