Difference between revisions of "Geodesic for a linear connection"

From Diffgeom
Jump to: navigation, search
(Definition part)
(Definition part)
 
Line 12: Line 12:
 
Equivalently, we say that <math>\gamma</math> is a geodesic if:
 
Equivalently, we say that <math>\gamma</math> is a geodesic if:
  
<math>\frac{D}{dt}(\gamma'(t)} = 0 \forall t</math>
+
<math>\frac{D}{dt}(\gamma'(t)) = 0 \forall t</math>
  
 
(with the derivative interpreted as a suitable one-sided derivative at the endpoints).
 
(with the derivative interpreted as a suitable one-sided derivative at the endpoints).

Latest revision as of 21:17, 6 January 2012

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).