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Section 1 Introduction

Carnot groups have rich algebraic and metric structures, and share many properties with normed spaces. Recently several articles have generalized classical regularity results of isometric embeddings in normed spaces into the setting of Carnot groups. In real normed spaces, there are two simple criteria for an isometric embedding to be affine: surjectivity or strict convexity of the norm on the target. Both regularity criteria have analogues for isometric embeddings of Carnot groups.

Surjective isometric embeddings behave in the Carnot group case similarly as they do in the normed-space case. Namely, isometries between arbitrary (open subsets of) Carnot groups are affine [16], i.e., compositions of left translations and group homomorphisms. For globally defined isometries, there is an even more general result that isometries between connected nilpotent metric Lie groups are affine [14].

For non-surjective isometric embeddings, it was proved in [13] that if \(G\) is a sub-Riemannian Carnot group of step 2, then all isometric embeddings \(\RR\hookrightarrow G\text{,}\) i.e., all infinite geodesics, are affine. This property was coined the geodesic linearity property in [6], and was used as an alternative to the strict convexity criterion as the two conditions are equivalent in normed spaces. More precisely, it was shown in [6] that if \(\HH^n\) is a Heisenberg group with a homogeneous distance satisfying the geodesic linearity property, then all isometric embeddings \(\RR^m\hookrightarrow \HH^n\) and \(\HH^m\hookrightarrow\HH^n\) are affine.

It was conjectured in [6] and subsequently proved in [5] that for Heisenberg groups the geodesic linearity property is equivalent to strict convexity of the projection norm, see Definition 2.4. While the point of view presented in [6] is purely metric, the essential tools of the proof in [5] arise from considering an isometric embedding \(\RR\hookrightarrow \HH^n\) as an optimal control problem, and reformulating the first order necessary criterion of the Pontryagin Maximum Principle in the language of convex analysis.

The goal of this paper is to extend the main results of [6] and [5] to arbitrary Carnot groups of step 2. The central object of study is again the Pontryagin Maximum Principle, from which relevant invariants will be extracted by an asymptotic study of the optimal controls. The main result of the paper is the following:

The necessity of the strict convexity assumption is a direct consequence of the necessity of strict convexity for linearity of geodesics in the normed-space case, see Proposition 5.2. Examples may also be found from the singular geodesics for the non-strictly convex \(\ell^\infty\) sub-Finsler norm exhibited in [7] and [3].

The restriction to step 2 is motivated by the known counterexample in the simplest Carnot group of step 3, the sub-Riemannian Engel group. The complete study of geodesics in the sub-Riemannian Engel group in [4] gives the first (and to date essentially only) known example of a non-affine infinite geodesic in a sub-Riemannian Carnot group. Note that in general, very little is known about geodesics even in the sub-Riemannian case, see [11][18][12][8][9] for some recent results.

The proof for Heisenberg groups in [6] that the geodesic linearity property of the target implies that all isometric embeddings are affine works also more generally for stratified groups, see Proposition 6.2. Consequently, Corollary 1.2 leads to the analogous rigidity result for arbitrary isometric embeddings:

It is worth remarking that although there are no explicit restrictions on the domain \((H,d_H)\) in Theorem 1.3, the mere existence of an isometric embedding \((H,d_H)\hookrightarrow (G,d_G)\) implies some restrictions. In particular, Pansu's Rademacher theorem [20] implies that there must exist an injective homogeneous homomorphism \(H\to G\text{.}\) It follows that \(H\) has step at most 2 and rank at most the rank of \(G\text{.}\)

Subsection 1.1 Structure of the paper

Section 2 presents the relevant definitions that will be used throughout the rest of the paper and some basic lemmas. The main points of interest are properties of blowdowns of geodesics, i.e., geodesics “viewed from afar”, and the observations about subdifferentials of convex functions.

Sections 3–5 are devoted to the proofs of Theorem 1.1 and Corollary 1.2 about infinite geodesics. Section 3 rephrases the classical first order optimality condition of the Pontryagin Maximum Principle in the setting of a step 2 sub-Finsler Carnot group. In the sub-Riemannian case the PMP reduces to a linear ODE for the controls. This is no longer true in the sub-Finsler case, making explicit solution of the system unfeasible. Nonetheless, the PMP has a form (Proposition 3.1) that is well suited to the study of the asymptotic behavior of optimal controls. The key object is the bilinear form \(B\colon V_1\times V_1\to\RR\text{.}\)

Section 4 covers the aforementioned asymptotic study. The goal of the section is to study blowdowns of infinite geodesics through the behavior of their controls. Using integral averages of controls, it is shown that any blowdown control must in fact be contained in the kernel of the bilinear form \(B\text{.}\)

Section 5 wraps up the proof of Theorem 1.1 using the conclusions of the previous sections. This section is where the strict convexity of the norm is critical. The importance of the assumption is that any linear map has a unique maximum on the ball. By observing that any element of \(\ker B\) defines an invariant along the corresponding optimal control, the uniqueness is exploited to prove that infinite geodesics must be invariant under blowdowns. Corollary 1.2 follows from the sub-Finsler case by the observation that the length metric associated with a homogeneous norm is always a sub-Finsler metric.

Section 6 covers the proof of Theorem 1.3 about isometric embeddings as a consequence of Corollary 1.2. The link between geodesics and general isometric embeddings arises from considering a foliation by horizontal lines in the domain and studying the induced foliation by infinite geodesics in the image. The affinity of isometric embeddings follows from the observation that two lines are at a sublinear distance from each other if and only if they are parallel.