Existence of tangent lines to Carnot-Carath\ue9odory geodesics

We show that length minimizing curves in Carnot\u2013Carath\ue9odory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line. This is the first regularity result for length minimizers that holds with no assumption on either the space (e.g., its rank, step, or analyticity) or the curve, and it is novel even in the setting of Carnot groups