• subFinsler geometry
• needle variations
• end-point map
• geodesics
• subRiemannian geometry
• Carnot groups

After a brief presentation of Carnot groups, we define the extended end-point map and show the relation between the openness of this map and the property of curves to be (locally) length-minimizing. Moreover, we consider a suitable polynomial map from the space of controls to the real line that has the property to being open at 0 if and only if the extended end-point map is open at a given control.
We compute the first and second variation of the end-point map and we apply first and second-order open mapping theorems to get necessary conditions for curves to be geodesics. Finally, perturbing geodesics with appropriate needle variations, we prove that under certain hypothesis the projection of a geodesic to a Carnot group of lower step is a quasi-geodesic. As a consequence, we prove that the projection of blow-ups of geodesics are length-minimizing in Carnot groups of lower step and we give an alternative proof for differentiability of normal extremals.