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.

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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.

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Subsection 5.2 Arbitrary homogeneous distances

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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.

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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 $d_\ell\text{.}$

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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 $\norm{\cdot}_{SF} = \norm{\cdot}_d\text{.}$

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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.

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

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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.

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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 $\gamma\text{.}$