Definition
Given data
Let 
be a metric space with metric 
. Then, a curve ![{\displaystyle \gamma :[0,1]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaf1c0ebc6e628b53498811c91b9df46e394d07)
(or 
or 
) is termed a geodesic if for any ![{\displaystyle a\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/254834e1cbe5c10e41397c0985566bb1cef07712)
, there exists an open subset 
of 
in ![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
such that the restriction of 
to ![{\displaystyle [b,c]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2aeaa2066f9a5b244a6ce860206302d11090076)
is the shortest path between them in the metric space, for any 
.
If it is true that the geodesic is the shortest path between its endpoints, it is termed a minimizing geodesic.