Geodesic for a linear connection

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Given data

Definition part

Consider a smooth curve \gamma:[0,1] \to M. Consider the connection along \gamma induced by \nabla, and consider the transport along \gamma for that connection. 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).