Infinite Geodesics and Isometric Embeddings in Carnot Groups of Step 2

Abstract

In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are lines exactly when the induced norm on the horizontal space is strictly convex. As a further consequence, we show that all isometric embeddings between such homogeneous groups are affine. The core of the proof is an asymptotic study of the extremals given by the Pontryagin Maximum Principle.

Carnot rectifiability of sub-Riemannian manifolds with constant tangent

“…The consequence of Corollary 4.6 can be informally rephrased by saying that the projection of each geodesic is not close to any vector subspace of co-dimension 2. This property is closely related to the notion of minimal height of a configuration of points, which we next recall from [8].…”

“…Proof. Let C 1 and K 1 be the constants given in (8) and Lemma 5.5, respectively. We prove the result for K 2 -2C 1 K 1 pn ´1q 2 .…”

Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems

“…In a recent paper [17], E. Hakavuori proved that for step 2 sub-Finsler Carnot groups with strictly convex norms, only lines are infinite geodesics. Thus in the case k = 3, in view of Th.…”

Geodesic fields for Pontryagin type C0-Finsler manifolds

“…The first case where all the minimizing paths were calculated explicitly was on the two dimensional non-abelian simply connected Lie group (see [23]). Several cases were studied since then (see, for instance, [7], [8], [16], [24], [27], [35], [36]) and they emphasize that several phenomena that do not occur in the Finsler case can happen in this setting, as the sudden change of derivatives along minimizing paths. Here, as it frequently happens in the study of Lie groups endowed with invariant geometrical structures, the problem is usually faithfully represented on its Lie algebra and on its dual space.…”