Geodesic for a linear connection

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Revision as of 21:17, 6 January 2012 by Vipul (talk | contribs) (Definition part)
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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:

Failed to parse (syntax error): \frac{D}{dt}(\gamma'(t)} = 0 \forall t

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