Editing Exponential map (Riemannian geometry) (section)

== Definition ==
Let {{math|''M''}} be a [[differentiable manifold]] and {{math|''p''}} a point of {{math|''M''}}.  An [[affine connection]] on {{math|''M''}} allows one to define the notion of a [[straight line]] through the point {{math|''p''}}.<ref>A source for this section is {{harvtxt|Kobayashi|Nomizu|1996|loc=§III.6}}, which uses the term "linear connection" where we use "affine connection" instead.</ref>

Let {{math|''v'' ∈ T<sub>''p''</sub>''M''}} be a [[tangent vector]] to the manifold at {{math|''p''}}.  Then there is a unique [[geodesic]] {{math|''γ''<sub>''v''</sub>}}:[0,1] → {{math|''M''}} satisfying {{math|''γ''<sub>''v''</sub>(0) {{=}} ''p''}} with initial tangent vector {{math|''γ''′<sub>''v''</sub>(0) {{=}} ''v''}}.  The corresponding '''exponential map''' is defined by {{math|exp<sub>''p''</sub>(''v'') {{=}} ''γ''<sub>''v''</sub>(1)}}. In general, the exponential map is only ''locally defined'', that is, it only takes a small neighborhood of the origin at {{math|T<sub>''p''</sub>''M''}}, to a neighborhood of {{math|''p''}} in the manifold. This is because it relies on the theorem of [[Picard–Lindelöf theorem|existence and uniqueness]] for [[ordinary differential equation]]s which is local in nature.  An affine connection is called complete if the exponential map is well-defined at every point of the [[tangent bundle]].