Geodesic for a linear connection

From Diffgeom
Revision as of 21:13, 6 January 2012 by Vipul (talk | contribs) (Created page with "==Definition== ===Given data=== * A connected differential manifold <math>M</math> with tangent bundle denoted by <math>TM</math>. * A [[defining ingredient::linear ...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Definition

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).