Affinity of infinite geodesics

Section 5 Affinity of infinite geodesics

Subsection 5.1 Sub-Finsler Carnot groups

The proof of Theorem 1.1 will now be concluded. The key ingredients are the sub-Finsler PMP 3.1, the knowledge of asymptotic behavior of blowdown controls from Lemma 4.2, and the convex analysis arguments from Subsection 2.5.

Proof of Theorem 1.1.

Let \(\gamma\colon[0,\infty)\to G\) be an infinite geodesic and let \(u\colon[0,\infty)\to V_1\) be its control. Let \(a\colon[0,\infty)\to V_1^*\) be the curve of subdifferentials of the squared norm \(\frac{1}{2}\norm{\cdot}^2\) and let \(B\colon V_1\times V_1\to\RR\) be the skew-symmetric bilinear form given by the PMP 3.1.

By Lemma 2.16, there exists a sequence \(\dilationfactor_k\to\infty\) such that the blowdown \(\tilde{\gamma}=\lim\limits_{k\to\infty}\delta_{1/\dilationfactor_k}\circ\gamma\circ\delta_{\dilationfactor_k}\colon [0,\infty)\to G\) is affine, i.e., a left translation of a one parameter subgroup. By Lemma 2.15, taking a subsequence if necessary, the dilated controls \(u_{\dilationfactor_k}(t) = u(\dilationfactor_kt)\) converge in \(\Lloc([0,\infty);V_1)\) to the control \(\tilde{u}\) of the curve \(\tilde{\gamma}\text{.}\) Since the curve \(\tilde{\gamma}\) is affine, the control \(\tilde{u}\) is constant. That is, there exists a constant vector \(Y\in V_1\text{,}\) which for almost every \(t\in[0,\infty)\) is the limit

\begin{equation} Y = \tilde{u}(t) = \lim\limits_{k\to\infty}u(\dilationfactor_k t)\text{.}\label{eq-Y-as-limit}\tag{5.1} \end{equation}

By Lemma 4.2, \(Y\in\ker B\text{,}\) so the derivative condition PMP 3.1 i implies that the curve \(t\mapsto a(t)Y\) is constant \(a(t)Y\equiv: C\text{.}\)

Fix any \(t\in[0,\infty)\) such that the limit (5.1) holds. By Lemma 2.18, up to taking a further subsequence, the subdifferentials \(a(\dilationfactor_k t)\) of the squared norm \(\frac{1}{2}\norm{\cdot}^2\) at the points \(u(\dilationfactor_k t)\) converge to a subdifferential \(\tilde{a}\colon V_1\to\RR\) of the squared norm \(\frac{1}{2}\norm{\cdot}^2\) at the point \(Y\text{.}\) Moreover, since \(a(t)Y\equiv C\) is constant, also the limit evaluates to \(\tilde{a}Y = C\text{.}\) Applying Lemma 2.19 for the subdifferential \(\tilde{a}\) shows that \(C = \tilde{a}Y = \norm{Y}^2\text{.}\) Similarly applying Lemma 2.19 for the subdifferential \(a(t)\) shows that \(a(t)u(t) = \norm{u(t)}^2\text{.}\) Since the curves \(\gamma\) and \(\tilde{\gamma}\) are both geodesics, \(\norm{u(t)} = 1 = \norm{Y}\text{,}\) so combining all of the above gives the equality

\begin{equation*} a(t)Y = 1 = a(t)u(t)\text{.} \end{equation*}

Consequently for any convex combination \(X\in V_1\) of \(u(t)\) and \(Y\text{,}\) Lemma 2.19 implies that

\begin{equation*} 1 = a(t)X \leq \norm{X}\norm{u(t)} = \norm{X}\text{.} \end{equation*}

By strict convexity of the norm this is only possible when \(u(t)=Y\text{.}\)

Repeating the same argument at all the times \(t\) satisfying the limit (5.1), it follows that \(u(t)=Y\) for almost every \(t\in[0,\infty)\text{,}\) so the geodesic \(\gamma\) is affine.

Subsection 5.2 Arbitrary homogeneous distances

The proof of Corollary 1.2 about infinite geodesics for arbitrary homogeneous distances follows from the sub-Finsler case by passing to the induced length metric. The relevant properties are captured in the next lemma.

Lemma 5.1.

Let \((G,d)\) be a stratified group equipped with a homogeneous distance \(d\) and let \(d_\ell\) be the length metric of \(d\text{.}\) Then

\((G,d_\ell)\) is a sub-Finsler Carnot group.
All geodesics of \((G,d)\) are also geodesics of \((G,d_\ell)\text{.}\)
The projection norm of \(d\) is the sub-Finsler norm of \(d_\ell\text{.}\)

Proof.

(i).

In Theorem 1.1 of [15] sub-Finsler Carnot groups are characterized as the only geodesic metric spaces that are locally compact, isometrically homogeneous, and admit a dilation. Therefore it suffices to verify that the length metric associated with a homogeneous distance satisfies these properties.

The claims of isometric homogeneity and admitting a dilation follow directly from the corresponding properties of the metric \(d\text{.}\) Namely, since left-translations are isometries of the metric \(d\text{,}\) they preserve the length of curves, and hence are also isometries of the length metric \(d_\ell\text{.}\) Similarly since dilations scale the length of curves linearly, they are dilations for the length metric \(d_\ell\text{.}\)

Finiteness of the length metric \(d_\ell\) follows from the stratification assumption: each element \(g\in G\) can be written as a product of elements in \(\exp(V_1)\) and the horizontal lines \(t\mapsto \exp(tX)\) are all geodesics. Therefore concatenation of suitable horizontal line segments defines a finite length curve from the identity \(e\) to any desired point \(g\text{.}\) It follows that the length metric \(d_\ell\) determines a well defined homogeneous distance on \(G\text{,}\) so by Proposition 2.26 of [17] it induces the manifold topology of \(G\text{.}\) In particular \((G,d_\ell)\) is a boundedly compact length space, so it is a geodesic metric space (see Corollary 2.5.20 of [10]). Applying Theorem 1.1 of [15] shows that \((G,d_\ell)\) is a sub-Finsler Carnot group.

(ii).

The lengths of all rectifiable curves in the original metric \(d\) and its associated length metric \(d_\ell\) always agree (see Proposition 2.3.12 of [10]). In particular, the claim that the geodesics of \((G,d)\) are geodesics of \((G,d_\ell)\) follows.

(iii).

The horizontal projection \(\pi\colon (G,d)\to V_1\) is a submetry both for the sub-Finsler norm \(\norm{\cdot}_{SF}\) (by definition) and for the projection norm \(\norm{\cdot}_d\) (by Lemma 2.5). Hence the norms \(\norm{\cdot}_{SF}\) and \(\norm{\cdot}_d\) have exactly the same balls, so \(\norm{\cdot}_{SF} = \norm{\cdot}_d\text{.}\)

Proof of Corollary 1.2.

Let \((G,d)\) be a stratified group of step 2 equipped with a homogeneous distance \(d\) whose projection norm is strictly convex, and let \(\gamma\colon[0,\infty)\to (G,d)\) be an infinite geodesic.

Let \(d_\ell\) be the length-metric associated with \(d\text{.}\) By Lemma 5.1 i and ii, the curve \(\gamma\) is also a geodesic of \((G,\norm{\cdot})\text{,}\) where \(\norm{\cdot}\colon V_1\to\RR\) is the sub-Finsler norm of the sub-Finsler metric \(d_\ell\text{.}\) Moreover by Lemma 5.1 iii the norm \(\norm{\cdot}=\norm{\cdot}_d\) is by assumption strictly convex.

Consequently by Theorem 1.1, the geodesic \(\gamma\) is affine.

The necessity of the strict convexity assumption is an immediate consequence of the classical case of normed spaces by the following simple lifting argument.

Proposition 5.2.

Let \(G\) be a stratified group equipped with an arbitrary homogeneous distance \(d\text{.}\) If the projection norm of \(d\) is not strictly convex, then there exist an infinite geodesic \(\gamma\colon\RR\to G\) which is not affine.

Proof.

If the projection norm \(\norm{\cdot}_d\colon V_1\to\RR\) is not strictly convex, then there exists a non-linear geodesic \(\beta\colon\RR \to V_1\text{.}\) For example, if the norm \(\norm{X+cY}_d\) is constant for \(-\epsilon\leq c\leq \epsilon\text{,}\) then the curve \(\beta(t) = tX+\epsilon\sin(t)Y\) is an infinite geodesic.

By Lemma 2.5, the projection \(\pi\colon (G,d)\to (V_1,\norm{\cdot})\) is a submetry, so the geodesic \(\beta\colon \RR\to V_1\) lifts to an infinite geodesic \(\gamma\colon \RR\to G\text{.}\) Since the projection is a homomorphism and the geodesic \(\beta\) is not affine, neither is the geodesic \(\gamma\text{.}\)