soft open fences

# Distortion coefficients of the $\alpha$-Grushin plane

Samuël Borza  Department of Mathematical Sciences, Durham University

(September 3, 2024)

###### Abstract

We compute the distortion coefficients of the $\alpha$-Grushin plane. They are
expressed in terms of generalised trigonometric functions. Estimates for the
distortion coefficients are then obtained and a conjecture of a curvature-
dimension condition for the generalised Grushin planes is suggested.

Keywords— Grushin plane, Sub-Riemannian geometry, Distortion coefficients,
Synthetic curvature
MSC (2010)— 51F99, 53B99, 28A75, 28C10

##  1 Introduction

Grushin structures first appeared in the work of Grushin on hypoelliptic
operators in the seventies, see [Gru70] for example. The $\alpha$-Grushin
plane, denoted by $\mathbb{G}_{\alpha}$, consists of equipping the
2-dimensional Euclidean space with the metric generated by the vector fields
$X=\partial_{x},Y_{\alpha}=|x|^{\alpha}\partial_{y}$.

These structures form a class of rank-varying sub-Riemannian manifolds, in a
precise way that will be explained in Section 2.2. In this work, we will focus
on the case $\alpha\geqslant 1$. The $\alpha$-Grushin plane has Hausdorff
dimension $\alpha+1$ and is not bracket-generating unless $\alpha$ is an
integer. Furhtermore, they constitute a natural generalisation of the
traditional Grushin plane, corresponding to the case $\alpha=1$. Along with
the Heisenberg groups $\mathbb{H}_{n}$, they are considered as fundamental
examples of sub-Riemannian geometry, exhibiting key characteristics of the
theory.

Since the work initiated by Juillet in [Jui09] and [Jui20], it is known that,
unlike Riemannian manifolds, no (ideal) sub-Riemannian manifold satisfies the
curvature-dimension condition introduced by Sturm, Lott and Villani. However,
it has been shown by Barilari and Rizzi in [BR19] that they do support
interpolation inequalities and even a Brunn-Minkowski inequality. For the
Heisenberg group, this was first established by Balogh, Kristaly and Sipos in
[BKS18]. Distortion coefficients, which capture some curvature information,
play a key role in these results. In this work, distortion coefficients for
the $\alpha$-Grushin plane are established.

A fine study of the geodesics of $\mathbb{G}_{\alpha}$ will be important in
this work. Because of the lack of a natural connection in sub-Riemannian
geometry, geodesics are obtained with the Pontryagin Maximum Principle. This
is the Hamiltonian point of view: a minimizing path between two points can be
lifted to one on the cotangent bundle that satisfies Hamilton’s equations.

The geodesics of the $\alpha$-Grushin plane were first studied by Li and Chang
in [CL12]. They are expressed with a generalisation of trigonometric
functions, defined as inverses of special functions. Section 2.3 and 3.1 are
devoted to these topics while in Section 3.2, we use an extended Hadamard
technique to find the cut locus of the geodesics. In what follows, the
notation $\mathrm{Cut}(q_{0})$ stands for the set of cut locci of $q_{0}$,
i.e. the set of points in $\mathbb{G}_{\alpha}$ where the geodesics starting
at $q_{0}$ stop being minimizing.

######  Theorem 1 (Distortion coefficients of the $\alpha$-Grushin plane).

Let $q_{0}$ and $q$ be two points of $\mathbb{G}_{\alpha}$ such that
$q\notin\mathrm{Cut}(q_{0})$. We denote by
$(x(t),y(t)):\interval{0}{1}\to\mathbb{G}_{\alpha}$ the unique constant speed
minimizing geodesic joining $q_{0}$ to $q$ and by
$(u(t),v(t))\in\mathrm{T}_{(x(t),y(t))}^{*}(\mathbb{G}_{\alpha})$ its lift.
For all $t\in\interval{0}{1}$, we have

$\beta_{t}(q,q_{0})=\dfrac{\mathrm{J}(t,x_{0},u_{0},v_{0})}{\mathrm{J}(1,x_{0},u_{0},v_{0})}$

where

$\displaystyle\mathrm{J}(t,x_{0},u_{0},v_{0})=\ $ $\displaystyle
x(t)\left[t\alpha
u_{0}\left(u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha}\right)\left((3\alpha-1)u_{0}^{2}+\alpha(\alpha+1)v_{0}^{2}x_{0}^{2\alpha}\right)\right.$
(1) $\displaystyle\quad\left.+(\alpha-1)^{2}u_{0}^{4}x_{0}\right]$
$\displaystyle-u(t)\left[t^{2}\alpha
u_{0}\left(u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha}\right)\left((3\alpha-1)u_{0}^{2}+\left(2\alpha^{2}-\alpha+1\right)v_{0}^{2}x_{0}^{2\alpha}\right)\right.$
$\displaystyle\quad\left.+tx_{0}\left(\left(4\alpha^{2}-3\alpha+1\right)u_{0}^{4}+2\alpha^{2}(\alpha+1)u_{0}^{2}v_{0}^{2}x_{0}^{2\alpha}+\alpha^{2}(\alpha+1)v_{0}^{4}x_{0}^{4\alpha}\right)\right.$
$\displaystyle\quad\left.+(\alpha-1)^{2}u_{0}^{3}x_{0}^{2}\right]$
$\displaystyle-x(t)u(t)^{2}(\alpha-1)^{2}\left[tu_{0}\left(u_{0}^{2}+\alpha
v_{0}^{2}x_{0}^{2\alpha}\right)+u_{0}^{2}x_{0}\right]$
$\displaystyle-u(t)x(t)^{2}(\alpha-1)^{2}u_{0}^{3}.$

Because of the analyticity of the geodesic flow, the case $v_{0}=0$ can be
seen as taking the limit $v_{0}\to 0$. Geometrically, this means that the
points $q_{0}$ and $q$ are joined by a straight horizontal line. This limit is
performed in Proposition 16.

Although the $\mathrm{CD}$ condition is not suited for this type of spaces,
the weaker measure contraction property introduced independently by Ohta and
Sturm (in [Oht07] and [Stu06b]) seems more adapted to sub-Riemannian geometry.
Indeed, there are numerous examples of sub-Riemannian manifolds that do
satisfy a $\mathrm{MCP}$-condition, including the Heisenberg group
$\mathbb{H}_{n}$ (cf. [Jui09]) and the Grushin plane $\mathbb{G}_{1}$ (see
[BR19]). We therefore investigate the $\mathrm{MCP}$-condition for
$\mathrm{G}_{\alpha}$ and we obtain a relevant estimate on the distortion
coefficients for singular points, that is to say, those on the $y$-axis, and
for those lying on the same horizontal line. In light of this, we suggest the
following conjecture.

######  Conjecture 2 (Curvature dimension of the $\alpha$-Grushin plane).

For $\alpha\geqslant 1$, the $\alpha$-Grushin plane satisfies the synthetic
curvature-dimension condition $\mathrm{MCP}(K,N)$ if and only if $K\leqslant
0$ and

$N\geqslant 2\left[\dfrac{(\alpha+1)m+1}{m+1}\right]$

with $m\in\interval{-3}{-2}$ the unique non zero solution of

$(m+1)^{2\alpha}(m+1)-\left((2\alpha+1)m+1\right)=0.$

Note that in the rest of this work, we will write $x^{2\alpha}$ for
$(x^{2})^{\alpha}$. We will provide evidence in favour of this conjecture in
Section 4.2. As we will see, the $\mathrm{MCP}(0,N)$ condition is related to a
lower bound for the distortion coefficients of the form
$\beta_{t}(q_{0},q)\geqslant t^{N}$. It will be proven that the lower bound
holds for singular points. Furthermore, it is sharp for the points lying on
the same horizontal lines.

### Acknowledgements

I would like to thank Prof Wilhelm Klingenberg for his supervision and
precious encouragements. I am also grateful to Prof Nicolas Juillet, Prof
Alpár Mészáros and Prof Alexandru Kristály for carefully reading the
manuscript and for providing invaluable feedback. I benefited greatly from
instructive discussions with Prof Shingo Takeuchi on the topic of generalised
trigonometry. I am thankful to Prof Li Yutian for introducing me to the study
of the $\alpha$-Grushin plane and for his constant support. This work was
supported by the UK Engineering and Physical Sciences Research Council (EPSRC)
grant EP/N509462/1 (1888382) for Durham University.

##  2 Preliminaries

###  2.1 Synthetic curvature-dimension conditions

In this section, we give an overview of metric geometry, synthetic notions of
curvature and distortion coefficients. A metric space $(X,\mathrm{d})$ is a
length space if the distance is induced from a length structure; that is to
say, $\mathrm{d}(x,y):=\inf\\{\mathrm{L}(\gamma)|\gamma:[a,b]\to X\text{ is
admissible, }\gamma(a)=x\text{ and }\gamma(b)=y\\}$ where
$\mathrm{L}:\mathcal{A}\to\mathbb{R}\cap\\{+\infty\\}$ is a length functional
on a complete set of admissible paths $\mathcal{A}\subseteq\mathcal{C}(X)$. A
minimal geodesic is an admissible path $\gamma:[a,b]\to X$ in $\mathcal{A}$
such that $\mathrm{d}(x,y)=\mathrm{L}(\gamma)$. We refer to [BBI01] for more
on metric geometry.

If the space has the property that every two points can be joined by a minimal
geodesic, we will say that $(X,\mathrm{d})$ is a geodesic space. We equip the
metric space $(X,\mathrm{d})$ with a Radon measure $\mathfrak{m}$. The
structure $(X,\mathrm{d},\mathfrak{m})$ is called a geodesic metric measure
space. The notion of distortion coefficients fits into this context.

######  Definition 3.

Let $x,y\in X$. The distortion coefficient from $x$ to $y$ at time
$t\in\interval{0}{1}$ is

$\beta_{t}(x,y)=\limsup_{r\to
0^{+}}\frac{\mathfrak{m}(\mathrm{Z}_{t}(x,B_{r}(y)))}{\mathfrak{m}(B_{r}(y))}$

where $\mathrm{Z}_{t}(x,B_{r}(y))$ stands for the set of $t$-intermediate
points from $x$ to the ball centred in $y$ of radius $r$;

$\mathrm{Z}_{t}(A,B):=\\{\gamma(t)|\gamma\in\mathrm{Geo}(X),\gamma(0)\in
A\text{ and }\gamma(1)\in B\\}$

whenever $A$ and $B$ are $\mathfrak{m}$-measurable subsets of $X$.

There is an intuitive physical interpretation of the distortion coefficients:
$\beta_{t}(x,y)$ "compares the volume occupied by the light rays emanating
from the light source [$x$], when they arrive close to $\gamma(t)$, to the
volume that they would occupy in a flat space." (see [Vil09, Chapter 14.]). In
particular, we can heuristically expect that these distortion coefficients
would be related to the curvature of the space.

The theory of synthetic curvature was developed by Lott, Sturm, and Villani
(see [LV09], [Stu06a], and [Stu06b]). We summarise here what is necessary in
this work. We denote by $\mathcal{P}(X)$ the set of Borel probability measures
and by $\mathcal{P}^{2}(X)$ the subset of those with finite second moment. We
write $\mathrm{Geo}(X)$ for the set of all minimal geodesics of $X$
parametrised on $[0,1]$. For all $t\in\interval{0}{1}$, the evaluation map is
defined as

$\mathrm{e}_{t}:\mathrm{Geo}(X)\to X:\gamma\mapsto\gamma(t).$

A dynamical transference plan $\Pi$ is a Borel probability measure on
$\mathrm{Geo}(X)$ while a displacement interpolation associated to $\Pi$ is a
path $(\mu_{t})_{t\in\interval{0}{1}}\subseteq\mathcal{P}^{2}(X)$ such that
$\mu_{t}=(\mathrm{e}_{t})_{\\#}\Pi$ for all $t\in\interval{0}{1}$. We equip
$\mathcal{P}^{2}(X)$ with the $L_{2}$-Wasserstein distance $\mathcal{W}_{2}$:
for any $\mu,\nu\in\mathcal{P}^{2}(X)$,

$\mathcal{W}_{2}(\mu_{0},\mu_{1}):=\inf_{\pi\in\Pi(\mu,\nu)}\int_{X}\mathrm{d}(x,y)^{2}\pi(\mathrm{d}x\mathrm{d}y)$

with
$\Pi(\mu,\nu):=\\{\omega\in\mathcal{P}(X^{2})|(\mathrm{proj_{1}})_{\\#}\omega=\mu\text{
and }(\mathrm{proj_{2}})_{\\#}\omega=\nu\\}$. For
$\mu_{0},\mu_{1}\in\mathcal{P}^{2}(X)$, the set
$\mathrm{OptGeo}(\mu_{0},\mu_{1})$ is the space of all measures
$\nu\in\mathcal{P}(\mathrm{Geo}(X))$ such that $(e_{0},e_{1})_{\\#}\nu$
realises the minimum for the $L_{2}$-Wasserstein distance. A measure
$\nu\in\mathrm{OptGeo}(\mu_{0},\mu_{1})$ is called a dynamical optimal plan.
We now need to define the distortion coefficients of the $(K,N)$-model space.
For
$K\in\mathbb{R},N\in\interval{1}{+\infty},\theta\in\interval[open]{0}{+\infty}$
and $t\in\interval{0}{1}$, we set

$\tau_{K,N}^{(t)}(\theta)=t^{1/N}\sigma_{K,N-1}^{(t)}(\theta)^{1-1/N}$

with

$\sigma_{K,N}^{(t)}(\theta)=\begin{cases}+\infty&K\theta^{2}\geqslant
N\pi^{2}\\\ \dfrac{\sin(t\theta\sqrt{K/N})}{\sin(\theta\sqrt{K/N})}&\text{if
}0<K\theta^{2}<N\pi^{2}\\\ t&\text{if }K\theta^{2}<0\text{ and }N=0\text{ or
if }K\theta^{2}=0\\\
\dfrac{\sinh(t\theta\sqrt{-K/N})}{\sinh(\theta\sqrt{-K/N})}&\text{if
}K\theta^{2}\leqslant 0\text{ and }N>0\end{cases}$

The definition of the coefficients $\tau_{K,N}$ is not arbitrary. In fact, the
coefficients $\tau_{K,N}^{(t)}$ are nothing but the distortion coefficients of
the model space $X^{(K,N)}$; that is to say, $X^{(K,N)}$ is the $N$-sphere of
constant cuvature $K$ if $K>0$, $X^{(K,N)}$ is the $N$-Euclidean space if
$K=0$ and $X^{(K,N)}$ is the $N$-hyperbolic plane of constant curvature $K$ if
$K<0$ (see again [Vil09, Chapter 14.]).

We are ready to introduce a first notion of synthetic curvature: the
celebrated Curvature-Dimension condition.

######  Definition 4.

Let $K\in\mathbb{R}$ and $N\in\left[1,+\infty\right)$. A geodesic metric
measure space $(X,\mathrm{d},\mathfrak{m})$ verifies $\mathrm{CD}(K,N)$ if,
for any $\mu_{0},\mu_{1}\in\mathcal{P}^{2}(X,\mathfrak{m})$ with bounded
support, there exists $\nu\in\mathrm{OptGeo}(\mu_{0},\mu_{1})$ and
$\pi\in\mathcal{P}(X^{2})$ a $W_{2}$-optimal plan such that
$\mu_{t}:=(\mathrm{e}_{t})_{\\#}\nu\ll\mathfrak{m}$ and for any
$N^{\prime}\geqslant N$,

$\mathcal{E}_{N^{\prime}}(\mu_{t})\geqslant\int_{X^{2}}\tau_{K,N}^{(1-t)}(\mathrm{d}(x,y))\rho_{0}^{1/N^{\prime}}+\tau_{K,N}^{(t)}(\mathrm{d}(x,y))\rho_{1}^{-1/N^{\prime}}\pi(\mathrm{d}x\mathrm{d}y)$

where $\mathcal{E}_{N}$ stands for the Rényi functional

$\mathcal{E}_{N}:\mathcal{P}(X)\to\interval{0}{+\infty}:\rho\mathfrak{m}+\mu_{s}\mapsto\int_{X}\rho^{1-1/N}\mathfrak{m}(\mathrm{d}x).$

For an extensive treatment of the $\mathrm{CD}$-condition and more generally
of Optimal Transport theory, we refer the reader to [Vil09]. Alongside this
notion of curvature, a weaker condition was developed independently by Sturm
and Ohta: the Measure Contraction Property (see [Stu06b], [Oht07]).

######  Definition 5.

Let $K\in\mathbb{R}$ and $N\in\left[1,+\infty\right)$. A geodesic metric
measure space $(X,\mathrm{d},\mathfrak{m})$ verifies $\mathrm{MCP}(K,N)$ if,
for every $x\in X$ and measurable set $A\subseteq X$ with
$\mathfrak{m}(A)\in\left(0,+\infty\right)$, there exists
$\nu\in\mathrm{OptGeo}\left(\mu_{A},\delta_{x}\right)$ such that for all
$t\in\interval{0}{1}$

$\mu_{A}\geqslant(\mathrm{e}_{t})_{\\#}\left(\tau_{K,N}^{(1-t)}(\mathrm{d}(\gamma(0),\gamma(1)))\nu(\mathrm{d}\gamma)\right)$

where $\mu_{A}:=\frac{1}{\mathfrak{m}(A)}\mathfrak{m}\in\mathcal{P}(X)$ is the
normalization of $\mu|_{A}$.

These two definitions generalise the notion of Ricci curvature bounded from
below by $K\in\mathbb{R}$ and dimension bounded from above by $N\geqslant 1$.
Indeed, if $(M,g)$ is a Riemannian manifold and $\psi$ a positive
$\mathcal{C}^{2}$ function on $M$, the metric measure space
$(M,\mathrm{d}_{g},\psi\cdot\mathrm{vol}_{g})$ satisfies the
$\mathrm{CD}(K,N)$ condition if and only if it satisfies the
$\mathrm{MCP}(K,N)$ condition and if and only if
$\mathrm{Ric}_{g,\psi,N}\geqslant Kg$ where

$\mathrm{Ric}_{g,\psi,N}:=\mathrm{Ric_{g}}-(N-n)\frac{\nabla^{2}_{g}h^{\frac{1}{N-n}}}{h^{\frac{1}{N-n}}}.$

Note that in the case where $N=n$, it only makes sense to consider constant
functions $\psi$ in the definition of the generalised Ricci tensor. These
results can be found in [Stu06b], [LV09] for the equivalence with
$\mathrm{CD}$ and the one with $\mathrm{MCP}$ is proved in [Oht07].

For general metric measure spaces, the two notions of synthetic curvature are
not equivalent. The $\mathrm{CD}$ condition does imply the $\mathrm{MCP}$
condition however (see [Oht07] and [CS12]). As we will see later, this
discordant behaviour already appears for sub-Riemannian manifolds.

###  2.2 Sub-Riemannian geometry

In what follows, we set up the basics of sub-Riemannian geometry. We rely on
[ABB20] for the general theory. We combine the exposition presented in [LD17]
(see also [LLP19]) with the one of [BL05] for geometric controls that are not
necessary of class $C^{\infty}$.

A manifold is a set equipped with an equivalence class of differentiable
atlases such that its manifold topology is connected, Hausdorff and second-
countable. We emphasise here on the theory of sub-Riemannian manifolds of
class $\mathcal{C}^{r}$ instead of class $\mathcal{C}^{\infty}$. As we will
see later, the $\alpha$-Grushin plane is a sub-Riemannian manifold that is
generated by global vector fields that are not smooth.

######  Definition 6.

Let $M$ be a smooth manifold of class $\mathcal{C}^{r}$ for
$r\in\mathbb{N}^{\geqslant 1}\cup\\{\infty\\}\cup\\{\omega\\}$. A triple
$(E,g,\varphi)$ is said to be a sub-Riemannian structure of class
$\mathcal{C}^{r}$ on $M$ if

  1. 1.

$E$ is a $\mathcal{C}^{r}$-vector bundle on $M$,

  2. 2.

$g$ is a $\mathcal{C}^{r}$-metric on $E$,

  3. 3.

$\varphi:E\to\mathrm{T}(M)$ is a $\mathcal{C}^{r}$-morphism of vector bundles.

The family $\mathcal{D}$ of $\mathcal{C}^{r}$-horizontal vector fields is
defined as

$\mathcal{D}:=\\{\varphi\circ u|u\text{ is a section of }E\text{ of class
}\mathcal{C}^{r}\\}.$

We also define the distribution at point a $p\in M$ with

$\mathcal{D}_{p}:=\\{v(x)|v\in\mathcal{D}\\}.$

The rank of the sub-Riemannian structure at $p\in M$ is
$\mathrm{rank}(p):=\mathrm{dim}(\mathcal{D}_{p})$. Observe that in our
definition, a sub-Riemannian manifold can be rank-varying; i.e. the map
$\mathrm{rank}$ might not be constant.

######  Definition 7.

We say that a Lipschitz curve $\gamma:[0,T]\to M$ is horizontal if there
exists a measurable and essentially bounded function $u:[0,T]\to E$ such that
for all $t\in[0,T]$, we have $u(t)\in E_{\gamma(t)}$ and
$\partial\gamma(t)=\varphi(u(t))$. The sub-Riemannian length of $\gamma$ is
defined by

$\mathrm{L}_{\mathrm{CC}}(\gamma)=\int_{0}^{T}\lVert\partial\gamma\rVert_{\gamma(t)}\mathrm{d}t$

where $\lVert v\rVert_{p}:=\min\\{\sigma(u)|u\in E_{p}\text{ and
}\varphi(u)=(p,v)\\}$ for $v\in\mathcal{D}_{p}$ and $p\in M$.

######  Remark.

It can be proven that $\lVert\cdot\rVert_{p}$ is well-defined and is a norm on
$\mathcal{D}_{p}$, induced by an inner product
$\langle\cdot,\cdot\rangle_{p}$.

In the case where every two points can be joined by a horizontal curve, we
have a well defined distance function on $M$.

######  Definition 8.

Let $M$ be a sub-Riemannian manifold. The sub-Riemannian distance
$\mathrm{d_{CC}}$ of $M$, also called the Carnot-Caratheodory distance, is
defined by

$\mathrm{d_{CC}}(x,y):=\inf\\{\mathrm{L_{CC}}(\gamma)|\gamma:[0,T]\to M\text{
is horizontal and }\gamma(0)=x\text{ and }\gamma(T)=y\\}.$

Traditionally, the definition of a sub-Riemannian structure demands that
$\mathcal{D}$ is a $\mathcal{C}^{\infty}$-distribution and that it satisfies
the Hörmander condition; that is to say,
$\mathrm{Lie}_{p}(\mathcal{D})=\mathrm{T}_{p}(M)$ for all $p\in M$. This is
motivated by the following well-known result.

######  Theorem 9 (Chow–Rashevskii theorem).

Let $M$ be a sub-Riemannian manifold such that its distribution $\mathcal{D}$
is $\mathcal{C}^{\infty}$ and satisfies the Hörmander condition. Then,
$(M,\mathrm{d_{CC}})$ is a metric space and the manifold and metric topology
of $M$ coincides.

We refrain from this convention here, as the Grushin planes that we will study
do not always satisfy this property. However, we will assume from now on that
every two points of the sub-Riemannian manifold $M$ can be joined by a
horizontal curve, making $\mathrm{d_{CC}}$ a distance of $M$, and that the
metric and manifold topology do coincide.

Finally, the horizontal distribution of a sub-Riemannian manifold $M$ is
defined by

$\mathrm{H}(M):=\bigsqcup_{p\in M}\mathcal{D}_{p}.$

Now that we have turned our sub-Riemannian manifold into a metric space, we
would like to study the geodesics associated with $\mathrm{d_{CC}}$. These
would be horizontal curves that are locally a minimiser for the length
functional $\mathrm{L_{CC}}$. Because of the lack of a torsion-free metric
connection, we cannot study geodesics through a covariant derivative. Rather,
we will characterise these geodesics via Hamilton’s equations.

######  Definition 10.

Let $M$ a sub-Riemannian manifold and $(X_{1},\dotsc,X_{m})$ a generating
family of vector fields. The Hamiltonian of the sub-Riemannian structure is
defined by

$H:\mathrm{T^{*}}(M)\to\mathbb{R}:\lambda\mapsto
H(\lambda):=\frac{1}{2}\sum_{i=0}^{m}\langle\lambda,X_{i}\rangle^{2}.$

We therefore approach the problem via the cotangent bundle
$\mathrm{T}^{*}(M)$, on which there is a natural symplectic form $\sigma$. We
can now characterise length minimisers of a sub-Riemannian manifold.

######  Theorem 11.

Let $\gamma:[0,T]\to M$ be a horizontal curve which is a length minimiser and
parametrised by constant speed. Then, there exists a Lipschitz curve
$\lambda(t)\in\mathrm{T}^{*}_{\gamma(t)}(M)$ such that one and only one of the
following is satisfied:

  1. (N)

$\partial\lambda=\vec{H}(\lambda)$, where $\vec{H}$ is the unique vector field
in $\Gamma(\mathrm{T}^{*}(M))$ such that
$\sigma(\cdot,\vec{H}(\lambda))=\mathrm{d}_{\lambda}H$ for all
$\lambda\in\mathrm{T}^{*}(M)$;

  2. (A)

$\sigma(\partial\lambda(t),\mathrm{T}_{\lambda(t)}(\mathcal{D})^{\perp})=0$
for all $t\in[0,T]$.

If $\lambda$ satisfies $(N)$ (resp. $(A)$), we will say that $\lambda$ is a
normal extremal (resp. abnormal extremal) and $\gamma$ is a normal geodesic
(resp. abnormal geodesic). Note that a geodesic may be both normal and
abnormal.

If $\gamma$ is a normal geodesic associated with a normal extremal $\lambda$,
then $(N)$ is nothing but Hamilton’s equation for $H$ in the natural
coordinates of the cotangent bundle:

$\left\\{\begin{array}[]{rcl}\partial x_{i}&=&\dfrac{\partial H}{\partial
p_{i}}\\\ \partial p_{i}&=&-\dfrac{\partial H}{\partial x_{i}}.\\\
\end{array}\right.$ (2)

The study of abnormal geodesics is an area of intensive research. It does
happen that a sub-Riemannian structure do not have any non trivial abnormal
geodesic. In this case, a complete sub-Riemannian manifold is said to be
ideal.

As explained in the introduction, no ideal sub-Riemannian manifold satisfies
the $\mathrm{CD}(K,N)$ condition (see [Jui20]). However, it is known that they
often satisfy an $\mathrm{MCP}$ condition: the Heisenberg groups (see
[Jui09]), generalised H-type groups, Sasakian manifolds (see [BR19, Section
7.]), etc. We conclude this section with the following theorem that relates an
$\mathrm{MCP}$ condition to a lower bound on the distortion coefficients of a
ideal sub-Riemannian manifold.

######  Theorem 12 ([BR19, Theorem 9.]).

Let $M$ be an ideal sub-Riemannian manifold equipped with a measure $\mu$. Let
$N\geqslant 1$. Then, the following conditions are equivalent:

  1. 1.

$\beta_{t}(q_{0},q)\geqslant t^{N}$ for all $q_{0},q\notin\mathrm{Cut}(M)$ and
$t\in[0,1]$;

  2. 2.

the measure contraction property $\mathrm{MCP}(0,N)$ is satisfied, i.e. for
all non-empty Borel sets $B\subseteq M$ and $q\in M$ we have
$\mu(\mathrm{Z}_{t}(q,B))\geqslant t^{N}\mu(B)$.

###  2.3 Generalised trigonometric functions

In this section, we give an account of $(p,q)$-trigonometry. The generalised
sine and cosine functions will be essential in the study of the geometry of
the $\alpha$-Grushin plane, as shown by Li in [CL12]. Generalised trigonometry
has a long history. The theory as presented here was pioneered by Edmunds in
[EGL12]. For recent developments, we point out the work of Takeuchi (see
[Tak17] and the references therein).

Consider

$F_{p,q}:\interval{0}{1}\to\mathbb{R}:t\mapsto\int_{0}^{x}\dfrac{1}{\sqrt[p]{1-t^{q}}}\mathrm{d}t.$

The map $F_{p,q}$ being strictly increasing, we may define its inverse

$\sin_{p,q}:\interval[scaled]{0}{\dfrac{\pi_{p,q}}{2}}\to\mathbb{R}:x\mapsto
F_{p,q}^{-1}(x)$

where the $(p,q)$-pi constant is defined as

$\pi_{p,q}:=2\int_{0}^{1}\dfrac{1}{\sqrt[p]{1-t^{q}}}\mathrm{d}t=\mathrm{B}\left(\dfrac{1}{p},1-\dfrac{1}{q}\right).$

Here the function $\mathrm{B}(\cdot,\cdot)$ stands for the complete beta
function.

We will extend the $(p,q)$-sine function to the whole real line. We first note
that $\sin_{p,q}(0)=0$ and $\sin_{p,q}(\pi_{p,q}/2)$ = 1. For
$x\in\interval[scaled]{\pi_{p,q}/2}{\pi_{p,q}}$, we set
$\sin_{p,q}(x)=\sin_{p,q}(\pi_{p,q}-x)$. Then, the $(p,q)$-sine is extended to
the whole $\mathbb{R}$ by $2\pi_{p,q}$-periodicity. We then define the
$(p,q)$-cosine by setting $\cos_{p,q}=(\sin_{p,q})^{\prime}$. These two
functions are of class $\mathcal{C}^{1}$. In fact, they are also of class
$\mathcal{C}^{\infty}$ except at the points $x=k\pi_{p,q}$ for
$k\in\mathbb{Z}$.

We have the following identities:

$|\sin_{p,q}|^{q}+|\cos_{p,q}|^{p}=1,\ \
(\sin_{p,q})^{\prime\prime}=(\cos_{p,q})^{\prime}=\dfrac{-q}{p}|\cos_{p,q}|^{2-p}|\sin_{p,q}|^{q-2}\sin_{p,q}.$

Therefore, the $(p,q)$-sine function can be alternatively defined as the
solution to the following ordinary differential equation

$\partial^{2}f=\dfrac{-q(p-1)}{p}|f|^{q-2}f,\ \ f(0)=0,\ \ \partial f(0)=1.$
(3)

Unlike in the case of classical trigonometric functions, additions formulas
are not known for $\sin_{p,q}(x+y)$ and $\cos_{p,q}(x+y)$. This problem
basically boils down to finding a function $F_{p,q}(x,y)$ that solves the
integral equation

$\int_{0}^{F_{p,q}(x,y)}\dfrac{1}{\sqrt[p]{1-t^{q}}}\mathrm{d}t=\int_{0}^{x}\dfrac{1}{\sqrt[p]{1-t^{q}}}\mathrm{d}t+\int_{0}^{y}\dfrac{1}{\sqrt[p]{1-t^{q}}}\mathrm{d}t.$

We would then have $\sin_{p,q}(x+y)=F(\sin_{p,q}(x),\sin_{p,q}(y))$. This is a
very hard problem, even for integers value for $p$ and $q$. For $(p,q)=(2,2)$,
the classical addition formula for $\sin$ emerges. When $(p,q)=(2,4)$, the
corresponding addition formula is the one for the lemniscate function that
Euler investigated in [Eul61]: let $\mathrm{sl}(x):=\sin_{2,4}(x)$ (resp.
$\mathrm{cl}(x):=\cos_{2,4}(x)$) stand for the sinlem function (resp. the
coslem function), then we have

$\mathrm{sl}(x+y)=\dfrac{\mathrm{sl}(x)\mathrm{cl}(y)+\mathrm{sl}(y)\mathrm{cl}(x)}{1+\mathrm{sl}^{2}(x)\mathrm{sl}^{2}(y)}$

with an analogous formula for $\mathrm{cl}(x+y)$.

##  3 Geometry of the $\alpha$-Grushin plane

###  3.1 Geodesics of the $\alpha$-Grushin plane

For $\alpha\in[1,+\infty)$, the $\alpha$-Grushin plane $\mathbb{G}_{\alpha}$
is defined as the sub-Riemannian structure on $\mathbb{R}^{2}$ generated by
the global vector fields $X=\partial_{x}$ and $Y=|x|^{\alpha}\partial_{y}$.
This genetating family of vector fields are
$\mathcal{C}^{\lfloor\alpha\rfloor}$ if $\alpha$ is not an integer and
$\mathcal{C}^{\infty}$ otherwise. The horizontal space at
$p\in\mathbb{G}_{\alpha}$ is
$H_{p}\left(\mathbb{G}_{\alpha}\right)=\mathrm{span}\\{X(p),Y_{\alpha}(p)\\}$
and the horizontal bundle is the disjoint union of these
$\mathrm{H}_{\alpha}=\sqcup_{p\in\mathbb{G}_{\alpha}}H_{p}(\mathbb{G}_{\alpha})$.
The rank of $\mathcal{D}=\mathrm{span}\left\\{X,Y_{\alpha}\right\\}$ is not
constant: it is a singular distribution if $x=0$ and Riemannian otherwise. We
then consider the bundle metric $g^{\alpha}$ to be the one that makes
$(X,Y_{\alpha})$ an orthonormal family of vector fields:
$g_{(x,y)}^{\alpha}(v,w)=v^{2}+\frac{1}{x^{2\alpha}}w^{2}$. This turns the
structure $(\mathbb{G}_{\alpha},\mathcal{H}_{\alpha},g^{\alpha})$ into a sub-
Riemannian manifold. It is easy to see that it does not satisfy the Hörmander
unless $\alpha\in\mathbb{N}\setminus\\{0\\}$.

Let $I$ be a non-empty interval of $\mathbb{R}$. A path
$\gamma:I\to\mathbb{G}_{\alpha}$ is said to be horizontal if
$\partial\gamma(t)\in H_{\gamma(t)}(\mathbb{G}_{\alpha})$ for almost every
$t\in I$. We can compute the length of a horizontal curve
$L(\gamma)=\int_{I}\sqrt{g_{\alpha}(\partial\gamma(t),\partial\gamma(t))}\mathrm{d}t$.
We denote by $\mathrm{d}_{\alpha}$ the Carnot-Caratheodory distance associated
with $L_{\alpha}$. This means that a path $\gamma$ is horizontal if and only
if $\partial\gamma(t)=u(t)X(\gamma(t))+v(t)Y_{\alpha}(\gamma(t))$ for some
measurable maps $u,v:I\to\mathbb{R}$ and for almost every $t\in I$. Equipping
the $\alpha$-Grushin plane with the Lebesgue measure $\mathcal{L}^{2}$, we
obtain a metric measure space
$(\mathbb{G}_{\alpha},\mathrm{d}_{\alpha},\mathcal{L}^{2})$.

The theory of sub-Riemannian geometry informs us that the geodesics of the
space are found by solving Hamilton’s equations. Here, the Hamiltonian is
$H:T^{*}(\mathbb{G}_{\alpha})\to\mathbb{R}$ with
$H(x,y,u,v)=\frac{1}{2}(u^{2}+v^{2}x^{2\alpha})$. A simple calculation shows
that there are no non-trivial abnormal geodesics in the $\alpha$-Grushin
plane. Consequently, the sub-Riemannian manifold $\mathbb{G}_{\alpha}$ is
ideal. In this context, Hamilton’s equations (2) become

$\left\\{\begin{array}[]{rcl}\partial x&=&u\\\ \partial y&=&vx^{2\alpha}\\\
\partial u&=&-\alpha v^{2}x^{2(\alpha-1)}x\\\ \partial v&=&0.\\\
\end{array}\right.$ (4)

We observe that $\partial^{2}x=-\alpha v^{2}x^{2(\alpha-1)}$, which is the
equation (3) for $(p,q)=(2,2\alpha)$. In what follows, we will therefore
denote $\sin_{\alpha}$ for $\sin_{2,2\alpha}$ (and respectively
$\cos_{\alpha}$).

######  Theorem 13.

Let $\gamma:I\to\mathbb{G}_{\alpha}$ be horizontal path with initial value
$\gamma(0)=(x_{0},y_{0})$ and initial covector $\lambda(0)=(u_{0},v_{0})$. In
the case where $v_{0}\neq 0$ and $(x_{0},u_{0})\neq 0$, the curve $\gamma$ is
a geodesic if and only if

$\left\\{\begin{array}[]{rcl}x(t)&=&A\sin_{\alpha}(\omega t+\phi)\\\
y(t)&=&y_{0}+v_{0}\dfrac{A^{2\alpha}}{(\alpha+1)\omega^{2}}\left(\omega^{2}t+\omega\cos_{\alpha}(\phi)\sin_{\alpha}(\phi)\right.\\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\left.-\omega\cos_{\alpha}(\omega
t+\phi)\sin_{\alpha}(\omega t+\phi)\right)\\\
u(t)&=&A\omega\cos_{\alpha}(\omega t+\phi)\\\ v(t)&=&v_{0}\\\
\end{array}\right.$ (5)

for uniquely determined parameters $A>0,\ \omega\in\mathbb{R}\setminus\\{0\\}$
and $\phi\in\left[0,2\pi_{\alpha}\right)$ satisfying

$A^{2}\omega^{2}=u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha},\ \
\omega^{2}=v_{0}^{2}A^{2(\alpha-1)},\ \ x_{0}=A\sin_{\alpha}(\phi)\text{ and
}u_{0}=A\omega\cos_{\alpha}(\phi).$ (6)

If $v_{0}=0$ or $(x_{0},u_{0})=0$, the geodesic is
$(x(t),y(t))=(u_{0}t+x_{0},y_{0})$ with its lift being constant:
$(u(t),v(t))=(u_{0},v_{0})$.

######  Remark.

Since the right-hand side of the equation is continuous with respect to the
initial condition $v_{0}$, we an obtain the $\gamma(t|x_{0},y_{0},u_{0},0)$
from $\lim_{v_{0}\to 0}\gamma(t|x_{0},y_{0},u_{0},v_{0})$.

###### Proof.

The case when $v_{0}=0$ or $(x_{0},u_{0})=0$ is straightforward. We assume
that $v_{0}\neq 0$ and $(x_{0},u_{0})\neq 0$. For $A>0,\
\omega\in\mathbb{R}\setminus\\{0\\}$ and
$\phi\in\left[0,2\pi_{\alpha}\right)$, we have

$\displaystyle\partial^{2}(A\sin_{\alpha}(\omega t+\phi))$
$\displaystyle=\partial(A\omega\cos_{\alpha}(\omega t+\phi))$
$\displaystyle=-\alpha A\omega^{2}\sin_{\alpha}(\omega
t+\phi)^{2(\alpha-1)}\sin_{\alpha}(\omega t+\phi)$
$\displaystyle=-\alpha\frac{\omega^{2}}{A^{2(\alpha-1)}}(A\sin_{\alpha}(\omega
t+\phi))^{2(\alpha-1)}(A\sin_{\alpha}(\omega t+\phi)).$

So by uniqueness of solutions to the differential equation (4), we get

$x(t)=A\sin_{\alpha}(\omega t+\phi)\quad\text{and}\quad
u(t)=A\omega\cos_{\alpha}(\omega t+\phi)$

if we set $\omega^{2}=v_{0}^{2}A^{2(\alpha-1)}$, $x_{0}=A\sin_{\alpha}(\phi)$
and $u_{0}=A\omega\cos_{\alpha}(\phi)$. Considering the integral of motion
$u^{2}+v^{2}x^{2\alpha}$ at $t=0$ yields

$\displaystyle u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha}$
$\displaystyle=(A\omega\cos_{\alpha}(\phi))^{2}+\frac{\omega^{2}}{A^{2(\alpha-1)}}(A\sin_{\alpha}(\phi))^{2\alpha}$
$\displaystyle=A^{2}\omega^{2}.$

Since $\partial^{2}x=-\alpha v_{0}^{2}x^{2(\alpha-1)}x$, we deduce that
$x^{2\alpha}=-x\partial^{2}x/\alpha v_{0}^{2}$ and thus, integrating by part,
we have

$\int_{0}^{t}x^{2\alpha}=\int_{0}^{t}\frac{-x\partial^{2}x}{\alpha
v_{0}^{2}}=\frac{-1}{\alpha v_{0}^{2}}\left([x\partial
x]_{0}^{t}-\int_{0}^{t}(\partial x)^{2}\right)=\frac{-1}{\alpha
v_{0}^{2}}\left([xu]_{0}^{t}-\int_{0}^{t}u^{2}\right).$

Now we remember that $A^{2}\omega^{2}=u^{2}+v^{2}x^{2\alpha}$. So,
$u^{2}=A^{2}\omega^{2}-v^{2}x^{2\alpha}$ and

$\displaystyle\int_{0}^{t}x^{2\alpha}$ $\displaystyle=\dfrac{-1}{\alpha
v_{0}^{2}}\left(x(t)u(t)-x(0)u(0)-\int_{0}^{t}A^{2}\omega^{2}+\int_{0}^{t}v^{2}x^{2\alpha}\right)$
$\displaystyle=\dfrac{A^{2}}{\alpha
v_{0}^{2}}\left(\omega^{2}t+\omega\cos_{\alpha}(\phi)\sin_{\alpha}(\phi)\right.$
$\displaystyle\qquad\left.-\omega\cos_{\alpha}(\omega
t+\phi)\sin_{\alpha}(\omega
t+\phi)-\dfrac{v_{0}^{2}}{A^{2}}\int_{0}^{t}x^{2\alpha}\right).$

Finally, we isolate $\int_{0}^{t}x^{2\alpha}$ and integrate $\partial
y=v_{0}x^{2\alpha}$ to get

$y(t)=y_{0}+v_{0}\dfrac{A^{2\alpha}}{(\alpha+1)\omega^{2}}\left(\omega^{2}t+\omega\cos_{\alpha}(\phi)\sin_{\alpha}(\phi)-\omega\cos_{\alpha}(\omega
t+\phi)\sin_{\alpha}(\omega t+\phi)\right).$

It remains to prove that there is a one-to-one correspondence between

$\\{(x_{0},v_{0},v_{0})\in(\mathbb{R}^{2})^{*}\times(\mathbb{R}^{*})\\}\quad\text{and}\quad\\{(A,\omega,\phi)\in\mathbb{R}^{>0}\times\mathbb{R}^{*}\times[0,2\pi_{\alpha})\\}$

via (6). Going from $(A,\omega,\phi)$ to $(x_{0},u_{0},v_{0})$ is clear. The
other direction is given by

$A=\left(\frac{u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha}}{v_{0}^{2}}\right)^{1/2\alpha},\quad\omega=v_{0}\left(\frac{u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha}}{v_{0}^{2}}\right)^{(\alpha-1)/2\alpha},$
(7)
$\sin_{\alpha}(\phi)=x_{0}\left(\dfrac{v_{0}^{2}}{u_{0}^{2}+v^{2}_{0}x_{0}^{2\alpha}}\right)^{1/2\alpha},\quad\cos_{\alpha}(\phi)=\mathrm{sgn}(v_{0})\dfrac{u_{0}}{(u_{0}^{2}+v^{2}_{0}x_{0}^{2\alpha})^{1/2}}.$
(8)

∎

By simply differentiating the relations (6) with respect to $x_{0},u_{0}$ and
$v_{0}$, we find the following identities:

$A_{x_{0}}=\dfrac{1-\cos_{\alpha}^{2}(\phi)}{\sin_{\alpha}(\phi)}\text{,}\quad
A_{u_{0}}=\dfrac{\cos_{\alpha}(\phi)}{\alpha\omega}\text{,}\quad
A_{v_{0}}=\dfrac{-\cos^{2}_{\alpha}(\phi)A}{\alpha v_{0}};$ (9)
$\omega_{x_{0}}=(\alpha-1)\left(\dfrac{\omega}{A}\right)\left(\dfrac{1-\cos_{\alpha}^{2}(\phi)}{\sin_{\alpha}(\phi)}\right)\text{,}\quad\omega_{u_{0}}=\left(\dfrac{\alpha-1}{\alpha}\right)\dfrac{\cos_{\alpha}(\phi)}{\alpha
A}\text{,}$
$\omega_{v_{0}}=\dfrac{\omega}{v_{0}}\left(1-\left(\dfrac{\alpha-1}{\alpha}\right)\cos^{2}_{\alpha}(\phi)\right);$
(10)
$\phi_{x_{0}}=\dfrac{\cos_{\alpha}(\phi)}{A}\text{,}\quad\phi_{u_{0}}=\dfrac{-\sin_{\alpha}(\phi)}{\alpha\omega
A}\quad\text{
and}\quad\phi_{v_{0}}=\dfrac{\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)}{\alpha
v_{0}}.$ (11)

We mention here the work of Li and Chang (cf. [CL12]). They obtained the
geodesics joining every two points in the $\alpha$-Grushin plane by solving
the boundary value problem corresponding to the differential equation in
Theorem 13. We note that their results are stated for
$\alpha\in\mathbb{N}\setminus\\{0\\}$. However, if we carefully define sub-
Riemannian manifolds of class $\mathcal{C}^{k}$ as it was done in Section 2.2,
we can see that their conclusions remain valid in the case $\alpha\geqslant
1$. In particular, their detailed study of the geodesics was used to derive an
expression for the Carnot-Caratheodory distance of $\mathbb{G}_{\alpha}$
between every two points. As a consequence, the $\alpha$-Grushin plane is a
uniquely geodesic space, that is to say, there is a unique minimizing geodesic
joining almost every two pairs of points in $\mathbb{G}_{\alpha}$ .

###  3.2 Cut locus of the $\alpha$-Grushin plane

When we look at the the geodesics of $\mathbb{G}_{\alpha}$, we observe three
types of behaviours: the straight lines corresponding to an initial covector
$v_{0}=0$; the geodesics for which $x_{0}=0$ (called singular or Grushin
points); and those for which $x_{0}\neq 0$ (called Riemannian points). In this
section, we investigate the cut locii and the cut times of the
$\alpha$-Grushin plane. The techniques used here were developed in [ABS08,
Section 3.2], [Riz18, Appendix A] and [ABB20, Section 13.5].

The case when $v_{0}=0$ is trivial: the corresponding geodesic is a straight
line and is length-minimizing for all times. Its cut locus is empty and its
cut time is infinite.

We then look at a geodesic $\gamma$ starting from a singular point $x_{0}=0$
with initial covector $v_{0}=\pm s$ with $s>0$. From (5), we see that the
geodesic $\gamma(\cdot|0,y_{0},u_{0},s)$ is a reflection of
$\gamma(\cdot|0,y_{0},u_{0},-s)$ with respect to the $y$-axis. Furthermore,
these two intersect at the $y$-axis a first time when
$t=\pi_{\alpha}/|\omega|$. Therefore, $\gamma$ must lose its optimality after
$t\geqslant\pi_{\alpha}/|\omega|$. From [CL12, Theorem 12], we know that there
is one and only one minimizing geodesic joining the singular point $(0,y_{0})$
to a point $(x_{1},y_{1})$ with $x_{1}\neq 0$. So, the geodesic $\gamma$ is
minimizing before $t=\pi_{\alpha}/|\omega|$.

It remains to study the case of a geodesic $\gamma$ starting at a Riemannian
point $(x_{0},y_{0})$, i.e. with $x_{0}\neq 0$. We will use an extended
Hadamard technique, as described in [ABB20, Section 13.4].

We firstly observe that

$\gamma\left(\dfrac{\pi_{\alpha}}{|\omega|}\middle|A,\omega,\phi\right)=\gamma\left(\dfrac{\pi_{\alpha}}{|\omega|}\middle|A,\omega,\pi_{\alpha}-\phi\right).$

This means that the points

$\left(-x_{0},y_{0}+\mathrm{sgn}(\omega)\left(\dfrac{x_{0}}{\sin_{\alpha}(\phi)}\right)^{\alpha+1}\dfrac{\pi_{\alpha}}{(\alpha+1)}\right)$

are joined by two distinct geodesics if $\phi\neq\pi_{\alpha}/2$,
$3\pi_{\alpha}/2$.

This leads us to conjecture that the cut time should still be
$t^{*}_{\mathrm{cut}}(u_{0},v_{0})=\pi_{\alpha}/|\omega|$ and that the cut
locus should be

$\mathrm{Cut}^{*}(q_{0})=\left\\{(-x_{0},y)\in\mathbb{G}_{\alpha}\middle||y-y_{0}|\geqslant|x_{0}|^{\alpha+1}\dfrac{\pi_{\alpha}}{(\alpha+1)}\right\\}.$

We know that $A^{2}\omega^{2}=u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha}=\kappa^{2}$
where the positive parameter $\kappa$ is the constant speed of the geodesic
$\gamma$. We can then parametrise $u_{0},v_{0}$ and the corresponding
parameters $A$ and $\omega$ with respect to
$\phi\in\ointerval{0}{2\pi_{\alpha}}\setminus\\{\pi_{\alpha}\\}$:

$u_{0}=\kappa\cos_{\alpha}(\phi),\ \
v_{0}=\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}\left|\dfrac{\sin_{\alpha}(\phi)}{x_{0}}\right|^{\alpha-1},A=\left|\dfrac{x_{0}}{\sin_{\alpha}(\phi)}\right|\text{
and }\omega=\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}.$

For $t\in\interval{0}{t^{*}_{\mathrm{cut}}(u_{0},v_{0})}$ and
$\phi\in\ointerval{0}{2\pi_{\alpha}}\setminus\\{\pi_{\alpha}\\}$, the
expression of the geodesics from Theorem 13 can thus be written as

$\left\\{\begin{array}[]{rcl}x(t,\phi)&=&\left|\dfrac{x_{0}}{\sin_{\alpha}(\phi)}\right|\sin_{\alpha}\left(\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}t+\phi\right)\\\
y(t,\phi)&=&y_{0}+\dfrac{1}{(\alpha+1)}\left|\dfrac{x_{0}}{\sin_{\alpha}(\phi)}\right|^{\alpha+1}\left[\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}t+\cos_{\alpha}(\phi)\sin_{\alpha}(\phi)\right.\\\
&&\left.-\cos_{\alpha}\left(\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}t+\phi\right)\sin_{\alpha}\left(\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}t+\phi\right)\right]\\\
\end{array}\right.$

This gives us the exponential map and we compute the determinant of its
differential:

$\begin{array}[]{rcl}D(t,\phi)&=&\dfrac{\kappa}{\sin_{\alpha}^{2}(\phi)}\left|\dfrac{x_{0}}{\sin_{\alpha}(\phi)}\right|^{\alpha}\left[x_{0}\sin_{\alpha}\left(\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}t+\phi\right)\cos_{\alpha}(\phi)\right.\\\
&&\left.-\sin_{\alpha}(\phi)(x_{0}+\kappa
t\cos_{\alpha}(\phi))\cos_{\alpha}\left(\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}t+\phi\right)\right]\\\
\end{array}$

One can check that $\lim\limits_{\phi\to
0}D(t,\phi)=\lim\limits_{\phi\to\pi_{\alpha}}D(t,\phi)=0$ unless $\alpha=1$,
in which case we have

$\lim\limits_{\phi\to
0}D(t,\phi)=\dfrac{|x_{0}|\kappa^{2}t}{3}\left(\dfrac{\kappa^{2}t^{2}}{x_{0}^{2}}+\dfrac{3\kappa
t}{x_{0}}+3\right)$

and

$\lim\limits_{\phi\to\pi_{\alpha}}D(t,\phi)=\dfrac{|x_{0}|\kappa^{2}t}{3}\left(\dfrac{\kappa^{2}t^{2}}{x_{0}^{2}}-\dfrac{3\kappa
t}{x_{0}}+3\right).$

We now claim that there are no conjugate points before
$t=\pi_{\alpha}/|\omega|$. Indeed, we firstly observe that $D(0,\phi)$
vanishes for every $\phi$. Secondly, with the help of the derivative of $D$
with respect to $t$;

$\begin{array}[]{rcl}\partial_{t}D(t,\phi)&=&\dfrac{\alpha\kappa^{2}}{x_{0}}(x_{0}+\kappa
t\cos_{\alpha}(\phi))\left|\dfrac{x_{0}}{\sin_{\alpha}(\phi)}\right|^{\alpha}\\\
&&\qquad\sin^{2(\alpha-1)}_{\alpha}\left(\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}t+\phi\right)\sin_{\alpha}\left(\kappa\dfrac{\sin_{\alpha}(\phi)}{x_{0}}t+\phi\right),\end{array}$

we see that $\partial_{t}D(t,\phi)=0$ if and only if

$t=-\dfrac{x_{0}}{\kappa\cos_{\alpha}(\phi)}\text{ or
}t=\dfrac{x_{0}}{\kappa\sin_{\alpha}(\phi)}(l\pi_{\alpha}-\phi),\
l\in\mathbb{Z}.$

The former is a local minimum that is positive while the later is a local
maximum that is also positive. Thirdly, we observe that

$D(t^{*}_{\mathrm{cut}},\phi)=\kappa\dfrac{\pi_{\alpha}}{\sin_{\alpha}(\phi)}\left|\dfrac{x_{0}}{\sin_{\alpha}(\phi)}\right|^{\alpha+1}\cos^{2}_{\alpha}(\phi)$

which is zero if and only if $\phi=\pi_{\alpha}/2$. So, the function $D$ is
never zero on $\ointerval{0}{t^{*}_{\mathrm{cut}}}$. Therefore, there are no
conjugate points in $\rinterval{0}{t^{*}_{\mathrm{cut}}}$.

Finally, we need to make some topological considerations in order to conclude.
Consider the sets

$\begin{array}[]{rcl}N&:=&\left\\{t\phi\ \middle|\
t\in\rinterval{0}{t^{*}_{\mathrm{cut}}},\
\phi\in\interval{0}{2\pi_{\alpha}}\right\\}=\left\\{(u_{0},v_{0})\in\mathrm{T}^{*}(\mathbb{G}_{\alpha})\
\middle|\ |v_{0}|<\pi_{\alpha}\right\\};\\\
M&:=&\exp(N)=\left\\{(x,y)\in\mathbb{G}_{\alpha}\ \middle|\
(x,y)\notin\mathrm{Cut}^{*}(q_{0})\right\\}.\\\ \end{array}$

The map $\exp:N\to\exp(M)$ is clearly proper. Since $M$ is simply-connected,
we can conclude that $\exp$ is a diffeomorphism by [ABB20, Corollary 13.24]
and the conjectured cut locii and time are thus the true ones by the extended
Hadamard technique.

Summarizing the findings of this section, we have proven the following result.

######  Theorem 14.

Let $\alpha\geqslant 1$ and $\gamma(t|A,\omega,\phi)=(x(t),y(t))$ be a
geodesic of $\mathbb{G}_{\alpha}$ with initial value $\gamma(0)=(x_{0},y_{0})$
and initial covector $\lambda(0)=(u_{0},v_{0})$, such as described in Theorem
13. If $v_{0}=0$, then there are no conjugate points along $\gamma$,
$t_{\mathrm{cut}}[\gamma]=+\infty$ and $\mathrm{Cut}(x_{0},y_{0})=\emptyset$.
If $v_{0}\neq 0$, the cut time is

$t_{\mathrm{cut}}[\gamma]=\dfrac{\pi_{\alpha}}{|\omega|}$

while the cut locus is $\mathrm{Cut}(0,y_{0})=\left\\{(0,y)\ \middle|\
y\in\mathbb{R}\right\\}$ and, if $x_{0}\neq 0$,

$\mathrm{Cut}(x_{0},y_{0})=\left\\{(-x_{0},y)\in\mathbb{G}_{\alpha}\middle||y-y_{0}|\geqslant|x_{0}|^{\alpha+1}\dfrac{\pi_{\alpha}}{(\alpha+1)}\right\\}.$

The cut locii and geodesics of $\mathbb{G}_{\alpha}$ are illustrated at the
Figure 1. With that in mind, we now turn to the analysis of distortion
coefficients on the $\alpha$-Grushin plane.

Figure 1: Geometry of $\mathbb{G}_{\alpha}$

Illustration of the geodesics of the $\alpha$-Grushin plane from a singular
point (on the left) and from a Riemannian point (on the right). The shaded
area represents a ball around the starting point and the thick line is the cut
locus.

##  4 Distortion of the $\alpha$-Grushin plane

###  4.1 Computation of the distortion coefficients

We present here our main result: an explicit computation of the distortion
coefficient of $\mathbb{G}_{\alpha}$. To this aim, we use the techniques
initiated by Balogh, Kristaly and Sipos in [BKS18] and generalised by Barilari
and Rizzi in [BR19]. In the latter, the authors prove interpolation
inequalities of optimal transport for ideal sub-Riemannian manifolds. They are
expressed in terms of the distortion coefficients for which the expression is
obtained through a fine analysis of sub-Riemannian Jacobi fields.

######  Theorem 15.

Let $q,q_{0}\in\mathbb{G}_{\alpha}$ such that $q\notin\mathrm{Cut}(q_{0})$.
Assume that $q$ and $q_{0}$ do not lie on the same horizontal line. Then, for
all $t\in\interval{0}{1}$, we have

$\beta_{t}(q,q_{0})=\dfrac{\mathrm{J}(t,A,\omega,\phi)}{\mathrm{J}(1,A,\omega,\phi)}$

where

$\displaystyle\mathrm{J}(t,A,\omega,\phi)=$ $\displaystyle\sin_{\alpha}(\omega
t+\phi)\cos_{\alpha}(\phi)\left(\omega
t\alpha(\alpha(\alpha+1)-(\alpha-1)^{2}\cos^{2}_{\alpha}(\phi))\right.$ (12)
$\displaystyle\qquad\left.+(\alpha-1)^{2}\sin_{\alpha}(\phi)\cos^{3}_{\alpha}(\phi)\right)$
$\displaystyle+\cos_{\alpha}(\omega t+\phi)\left((\omega
t)^{2}\alpha\cos_{\alpha}(\phi)(-1+\alpha-2\alpha^{2}+2(\alpha-1)^{2}\cos^{2}_{\alpha}(\phi))\right.$
$\displaystyle\qquad\left.-\omega
t\sin_{\alpha}(\phi)(\alpha^{2}(\alpha+1)-(\alpha-1)^{3}\cos^{4}_{\alpha}(\phi))\right.$
$\displaystyle\qquad\left.-(\alpha-1)^{2}\sin^{2}_{\alpha}(\phi)\cos^{3}_{\alpha}(\phi)\right)$
$\displaystyle-\sin_{\alpha}(\omega t+\phi)\cos^{2}_{\alpha}(\omega
t+\phi)(\alpha-1)^{2}\cos_{\alpha}(\phi)\left(\omega
t((\alpha-1)\cos^{2}_{\alpha}(\phi)-\alpha)\right.$
$\displaystyle\qquad\left.-\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)\right)$
$\displaystyle-\sin^{2}_{\alpha}(\omega t+\phi)\cos_{\alpha}(\omega
t+\phi)(\alpha-1)^{2}\cos^{3}_{\alpha}(\phi).$

######  Remark.

We consider geodesics parametrised by constant speed on $\interval{0}{1}$.
Consequently, since Theorem 14 states that
$t_{\mathrm{cut}}=\pi_{\alpha}/|\omega|$, we always have
$|\omega|\leqslant\pi_{\alpha}$.

###### Proof.

We let
$\lambda_{0}=(u_{0},v_{0})\in\mathrm{T}^{*}_{q_{0}}(\mathbb{G}_{\alpha})$ be
the covector corresponding to the unique minimizing geodesic joining
$q_{0}=(x_{0},y_{0})$ to $q=(x,y)$ in $\mathbb{G}_{\alpha}$. The assumption
that $q$ and $q_{0}$ do not lie on the same horizontal line means that
$v_{0}\neq 0$.

By choosing the global Darboux frame induced by the sections of
$\mathrm{T}(\mathrm{T}^{*}(\mathbb{G}_{\alpha}))$;
$E_{1}=\partial_{u},E_{2}=\partial_{v},F_{1}=\partial_{x},F_{2}=\partial_{y}$,
Lemma 44 in [BR19] yields that $\beta_{t}(q,q_{0})=J(t)/J(1)$ where the
function $J$ is the determinant of the exponential map
$(u,v)\to\mathrm{exp}_{(x_{0},y_{0})}(u,v)$ in these coordinates, computed at
$(u_{0},v_{0})$.

Taking the derivatives of (5), we find

$\displaystyle\partial_{u_{0}}x(t)$
$\displaystyle=A_{u_{0}}\sin_{\alpha}(\omega
t+\phi)+A\left(\omega_{u_{0}}t+\phi_{u_{0}}\right)\cos_{\alpha}(\omega
t+\phi)$ $\displaystyle\partial_{v_{0}}x(t)$
$\displaystyle=A_{v_{0}}\sin_{\alpha}(\omega
t+\phi)+A\left(\omega_{v_{0}}t+\phi_{v_{0}}\right)\cos_{\alpha}(\omega
t+\phi)$

and

$\displaystyle\partial_{u_{0}}y(t)$ $\displaystyle=\left[\left(2\alpha
A_{u_{0}}\omega-A\omega_{u_{0}}\right)\left(\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)-\sin_{\alpha}(\omega
t+\phi)\cos_{\alpha}(\omega t+\phi)\right)\right.$
$\displaystyle\qquad+A\omega(\alpha+1)(\phi_{u_{0}}\cos_{\alpha}^{2}(\phi)-\left(\omega_{u_{0}}t+\phi_{u_{0}}\right)\cos_{\alpha}^{2}(\omega
t+\phi))$ $\displaystyle\qquad\left.+\alpha\omega
t\left(2A_{u_{0}}\omega+A\omega_{u_{0}}\right)\right]\dfrac{v_{0}A^{2(\alpha-1)}A}{(\alpha+1)\omega^{2}};$
$\displaystyle\partial_{v_{0}}y(t)$ $\displaystyle=\left[\left(2\alpha\omega
v_{0}A_{v_{0}}+A(\omega-
v_{0}\omega_{v_{0}})\right)\left(\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)-\sin_{\alpha}(\omega
t+\phi)\cos_{\alpha}(\omega t+\phi)\right)\right.$
$\displaystyle\qquad+A\omega
v_{0}(\alpha+1)(\phi_{u_{0}}\cos_{\alpha}^{2}(\phi)-\left(\omega_{v_{0}}t+\phi_{v_{0}}\right)\cos_{\alpha}^{2}(\omega
t+\phi))$ $\displaystyle\qquad\left.+\omega t(2\alpha v_{0}\omega
A_{v_{0}}+A(\omega+\alpha
v_{0}\omega_{v_{0}}))\right]\dfrac{v_{0}A^{2(\alpha-1)}A}{(\alpha+1)\omega^{2}}.$

To make things clearer, we have used the subscript notation to indicate a
partial derivative and we also set
$[f,g]:=f_{u_{0}}g_{v_{0}}-f_{v_{0}}g_{u_{0}}$.

With this in mind, we calculate

$\displaystyle[x,y](t)=$
$\displaystyle\dfrac{A^{2\alpha}}{(\alpha+1)\omega^{2}}\left[\sin_{\alpha}^{2}(\omega
t+\phi)\cos_{\alpha}(\phi)\left(v[A,\omega]-\omega A_{u_{0}}\right)\right.$
$\displaystyle+\alpha v_{0}\omega\sin_{\alpha}^{2\alpha}(\omega
t+\phi)\sin_{\alpha}(\omega t+\phi)\left([A,\omega]t+[A,\phi]\right)$
$\displaystyle+\sin_{\alpha}(\omega
t+\phi)\left(\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)(\omega
A_{u_{0}}-v_{0}[A,\omega])-\alpha
v_{0}\omega\sin_{\alpha}^{2\alpha}(\phi)[A,\phi]\right.$
$\displaystyle\quad\left.+\omega(\omega
A_{u_{0}}t+v_{0}\cos_{\alpha}^{2}(\phi)[A,\phi])\right)$
$\displaystyle+\sin_{\alpha}(\omega t+\phi)\cos^{2}_{\alpha}(\omega
t+\phi)\left(t((2\alpha-1)v_{0}\omega[A,\omega]-A\omega\omega_{u_{0}})\right.$
$\displaystyle\quad\left.+(2\alpha-1)v_{0}\omega[A,\phi]+Av_{0}[\phi,\omega]-A\omega\phi_{u_{0}}\right)$
$\displaystyle+\cos_{\alpha}(\omega
t+\phi)\left(\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)(A\omega\phi_{u_{0}}+2\alpha
v_{0}\omega[\phi,A]+Av_{0}[\omega,\phi])\right.$
$\displaystyle\quad+\omega^{2}t^{2}(2\alpha
v_{0}[\omega,A]+A\omega_{u_{0}})-\omega
t(\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)(2\alpha
v_{0}[A,\omega])-A\omega_{u_{0}})$
$\displaystyle\left.\quad\left.+\omega(2\alpha
v_{0}[A,\phi])-A\phi_{u_{0}}+\alpha
v_{0}A\sin_{\alpha}^{2\alpha}(\phi)[\omega,\phi]+\alpha
v_{0}\cos_{\alpha}^{2}(\phi)[\phi,\omega]\right)\right].$

Using the identities $\sin_{\alpha}^{2\alpha}(x)+\cos_{\alpha}^{2}(x)=1$, (9),
(10) and (11), we find that

$\displaystyle[x,y](t)=$
$\displaystyle\dfrac{A^{2\alpha}}{(\alpha+1)\omega^{2}}\Biggl{[}\sin_{\alpha}(\omega
t+\phi)\cos_{\alpha}(\phi)\left(\omega
t\alpha(\alpha(\alpha+1)-(\alpha-1)^{2}\cos^{2}_{\alpha}(\phi))\right.$ (13)
$\displaystyle\quad\left.+(\alpha-1)^{2}\sin_{\alpha}(\phi)\cos^{3}_{\alpha}(\phi)\right)$
$\displaystyle+\cos_{\alpha}(\omega t+\phi)\left((\omega
t)^{2}\alpha\cos_{\alpha}(\phi)(-1+\alpha-2\alpha^{2}+2(\alpha-1)^{2}\cos^{2}_{\alpha}(\phi))\right.$
$\displaystyle\quad\left.-\omega
t\sin_{\alpha}(\phi)(\alpha^{2}(\alpha+1)-(\alpha-1)^{3}\cos^{4}_{\alpha}(\phi))-(\alpha-1)^{2}\sin^{2}_{\alpha}(\phi)\cos^{3}_{\alpha}(\phi)\right)$
$\displaystyle-\sin_{\alpha}(\omega t+\phi)\cos^{2}_{\alpha}(\omega
t+\phi)(\alpha-1)^{2}\cos_{\alpha}(\phi)\Bigl{(}\omega
t((\alpha-1)\cos^{2}_{\alpha}(\phi)-\alpha)$
$\displaystyle\quad-\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)\Bigr{)}$
$\displaystyle-\sin^{2}_{\alpha}(\omega t+\phi)\cos_{\alpha}(\omega
t+\phi)(\alpha-1)^{2}\cos^{3}_{\alpha}(\phi)\Biggl{]}.$

By simplifying, rearranging and performing
$\beta_{t}(q,q_{0})=[x,y](t)\big{/}[x,y](1)$, we finally obtain the desired
expression. ∎

We can transform (12) from the set of coordinates $A,\omega$ and $\phi$ to
$x_{0},u_{0}$ and $v_{0}$ via the identities (7) and (8). Performing
$\beta_{t}(q,q_{0})=[x,y](t)\big{/}[x,y](1)$ concludes the proof of the
Theorem 1.

It is interesting to look at the limit $\alpha\to 1$. In this case, the
$\alpha$-Grushin plane is the traditional Grushin plane while $\sin_{\alpha}$
and $\cos_{\alpha}$ are the usual sine and cosine functions. The formula (12)
simplifies to

$\displaystyle[x,y](t)$
$\displaystyle=t\dfrac{A^{2}}{\omega}\Biggl{[}\sin(\omega
t+\phi)\cos(\phi)-\omega\cos(\omega
t+\phi)\Big{(}\sin(\phi)+t\cos(\phi)\Big{)}\Biggr{]}$
$\displaystyle=t\dfrac{(u_{0}^{2}+tu_{0}v_{0}^{2}x_{0}+v_{0}^{2}x_{0}^{2})\sin(tv_{0})-tu_{0}^{2}v_{0}\cos(tv_{0})}{v_{0}^{3}}$

and thus, we find what was already established in [BR19, Proposition 61]: the
distortion coefficients of the usual Grushin plane are

$\beta_{t}(q,q_{0})=t\dfrac{(u_{0}^{2}+tu_{0}v_{0}^{2}x_{0}+v_{0}^{2}x_{0}^{2})\sin(tv_{0})-tu_{0}^{2}v_{0}\cos(tv_{0})}{(u_{0}^{2}+u_{0}v_{0}^{2}x_{0}+v_{0}^{2}x_{0}^{2})\sin(v_{0})-u_{0}^{2}v_{0}\cos(v_{0})}\text{,
for all }t\in\interval{0}{1}.$

We now want to investigate the behaviour of $\beta_{t}(q_{0},q)$ when $q_{0}$
and $q$ do lie on the same horizontal line, that is to say, when $v_{0}\to 0$.

######  Proposition 16.

When $v_{0}\to 0$, we have

$\beta_{t}(q_{0},q)=t\dfrac{(u_{0}t+x_{0})^{2\alpha}(u_{0}t+x_{0})-x_{0}^{2\alpha}x_{0}}{(u_{0}+x_{0})^{2\alpha}(u_{0}+x_{0})-x_{0}^{2\alpha}x_{0}}.$

###### Proof.

We aim to perform $\lim_{v_{0}\to 0}\mathrm{J}(t)\big{/}\mathrm{J}(1)$ where
$\mathrm{J}$ is defined by (1). We already know from Theorem 13 have that
$\lim_{v_{0}\to 0}x(t)=u_{0}t+x_{0}$ and $\lim_{v_{0}\to 0}u(t)=u_{0}$. Let us
make the following preliminary calculations:

$\begin{split}x_{v_{0}}(t)&=A_{v_{0}}\sin_{\alpha}(\omega
t+\phi)+A\left(\omega_{v_{0}}t+\phi_{v_{0}}\right)\cos_{\alpha}(\omega
t+\phi)\\\ &=\dfrac{A}{\alpha v_{0}}\left(\cos_{\alpha}(\omega
t+\phi)\left[\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)+\omega
t(\alpha-(\alpha-1)\cos^{2}_{\alpha}(\phi))\right]\right.\\\
&\qquad\left.-\sin_{\alpha}(\omega t+\phi)\cos^{2}_{\alpha}(\phi)\right)\\\
&=\dfrac{\left[t(u_{0}^{2}+\alpha
v_{0}^{2}x_{0}^{2\alpha})+u_{0}x_{0}\right]\cdot u(t)+u_{0}^{2}\cdot
x(t)}{\alpha v_{0}(u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha})}\end{split}$

and

$\begin{split}u_{v_{0}}(t)&=-\alpha
A\omega\left(\omega_{v_{0}}t+\phi_{v_{0}}\right)\sin^{2(\alpha-1)}_{\alpha}(\omega
t+\phi)\sin_{\alpha}(\omega t+\phi)\\\ &\quad+\cos_{\alpha}(\omega
t+\phi)(\omega A_{v_{0}}+A\omega_{v_{0}})\\\
&=\dfrac{A\omega}{\alpha}\left(\sin^{2(\alpha-1)}_{\alpha}(\omega
t+\phi)\sin_{\alpha}(\omega t+\phi)\right.\\\ &\quad\left.\left[\omega
t((\alpha-1)\cos^{2}_{\alpha}(\phi)-\alpha)-\sin_{\alpha}(\phi)\cos_{\alpha}(\phi)\right]+\cos_{\alpha}(\omega
t+\phi)(1-\cos^{2}_{\alpha}(\phi))\right)\\\
&=v_{0}\dfrac{x_{0}^{2\alpha}\cdot u(t)-\left[t(u_{0}^{2}+\alpha
v_{0}^{2}x_{0}^{2\alpha})+u_{0}x_{0}\right]\cdot
x(t)^{2(\alpha-1)}x(t)}{(u_{0}^{2}+v_{0}^{2}x_{0}^{2\alpha})}.\end{split}$

Since simply replacing $v_{0}$ with 0 in $\beta_{t}(q_{0},q)$ leads to $0/0$,
we will use l’Hospital theorem, as many times as needed, we find:

$\begin{split}\beta_{t}(q_{0},q)&=\lim\limits_{v_{0}\to
0}\dfrac{\mathrm{J}(t,x_{0},u_{0},v_{0})}{\mathrm{J}(1,x_{0},u_{0},v_{0})}=\lim\limits_{v_{0}\to
0}\dfrac{\partial_{v_{0}}\mathrm{J}(t,x_{0},u_{0},v_{0})}{\partial_{v_{0}}\mathrm{J}(1,x_{0},u_{0},v_{0})}\\\
&=\lim\limits_{v_{0}\to
0}\dfrac{\partial^{2}_{v_{0}}\mathrm{J}(t,x_{0},u_{0},v_{0})}{\partial^{2}_{v_{0}}\mathrm{J}(1,x_{0},u_{0},v_{0})}=t\dfrac{(u_{0}t+x_{0})^{2\alpha}(u_{0}t+x_{0})-x_{0}^{2\alpha}x_{0}}{(u_{0}+x_{0})^{2\alpha}(u_{0}+x_{0})-x_{0}^{2\alpha}x_{0}}.\\\
\end{split}$

∎

###  4.2 Relevant curvature-dimension estimates

Now that we have the expressions for the distortion coefficients, we would
like to find appropriate bounds on them. In [Jui20], Juillet proved that an
ideal sub-Riemannian manifold never satisfies the $\mathrm{CD}(K,N)$
condition. This means that we do not have
$\beta_{t}(q_{0},q)\geqslant\tau_{K,N}^{(t)}$ for whatever choice of
$K\in\mathbb{R}$ and $N\geqslant 1$. However, there is a chance that the
weaker curvature-dimension condition $\mathrm{MCP}(K,N)$ could hold for the
$\alpha$-Grushin plane.

In particular, the traditional Grushin plane, equivalent to
$\mathbb{G}_{\alpha}$ when $\alpha=1$, is $\mathrm{MCP}(K,N)$ if and only if
$N\geqslant 5$ and $K\leqslant 0$. We expect the $\alpha$-Grushin plane to
satisfy the $\mathrm{MCP}$ property for a minimal value of $N$ that would
depend on $\alpha$. By Theorem 12, the related bound on the distortion
coefficients should be of the form $\beta_{t}(q_{0},q)\geqslant t^{N}$. In
this section, we provide a bound in the case where $q_{0}$ and $q$ lie on the
same horizontal line and when $q_{0}$ is a Grushin point.

For $\alpha>1$, it is easy to see that the equation

$(m+1)^{2\alpha}(m+1)-\left((2\alpha+1)m+1\right)=0.$ (14)

has a unique non-zero solution $m\in\ointerval{-3}{-2}$. If $\alpha=1$, the
value of the root is $m=-3$.

######  Proposition 17.

Let $q_{0},q\in\mathbb{G}_{\alpha}$ lying on the same horizontal line. We have
that

$\beta_{t}(q_{0},q)\geqslant t^{N}$

if and only if

$N\geqslant 2\left[\dfrac{(\alpha+1)m+1}{m+1}\right]$

with $m\in\ointerval{-3}{-2}$ being the unique solution non zero solution of
(14).

###### Proof.

We are looking for the best $N\in\interval{1}{+\infty}$ such that

$t\dfrac{(u_{0}t+x_{0})^{2\alpha}(u_{0}t+x_{0})-x_{0}^{2\alpha}x_{0}}{(u_{0}+x_{0})^{2\alpha}(u_{0}+x_{0})-x_{0}^{2\alpha}x_{0}}\geqslant
t^{N}$ (15)

for all $t\in[0,1]$ and $x_{0},u_{0}\in\mathbb{R}$; that is to say,

$\dfrac{(u_{0}t+x_{0})^{2\alpha}(u_{0}t+x_{0})-x_{0}^{2\alpha}x_{0}}{(u_{0}+x_{0})^{2\alpha}(u_{0}+x_{0})-x_{0}^{2\alpha}x_{0}}\geqslant
t^{N-1}.$

If we take the logarithm of the above, we find that the inequality is
equivalent to

$\int_{u_{0}t}^{u_{0}}\partial_{z}\log\left[(z+x_{0})^{2\alpha}(z+x_{0})-x_{0}^{2\alpha}x_{0}\right]\mathrm{d}z\leqslant(N-1)\int_{u_{0}t}^{u_{0}}\partial_{z}\log|z|\mathrm{d}z.$

The latter is equivalent to the same inequality for the integrands:

$\partial_{z}\log\left[(z+x_{0})^{2\alpha}(z+x_{0})-x_{0}^{2\alpha}x_{0}\right]\leqslant(N-1)\partial_{z}\log|z|.$

This leads to

$\dfrac{(2\alpha+1)(z+x_{0})^{2\alpha}}{(z+x_{0})^{2\alpha}(z+x_{0})-x_{0}^{2\alpha}x_{0}}\leqslant(N-1)\dfrac{1}{z}$

and thus

$N\geqslant\dfrac{(2(\alpha+1)z+x_{0})(z+x_{0})^{2\alpha}-x_{0}x_{0}^{2\alpha}}{(z+x_{0})^{2\alpha}(z+x_{0})-x_{0}^{2\alpha}x_{0}}.$

We are therefore looking for the global maximum of the map

$f:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto\dfrac{(2(\alpha+1)x+y)(x+y)^{2\alpha}-yy^{2\alpha}}{(x+y)(x+y)^{2\alpha}-yy^{2\alpha}}.$

Firstly, let us compute the critical points of $f$. We find

$\partial_{x}f(x,y)=\dfrac{(2\alpha+1)y(x+y)^{2\alpha}\left[(x+y)(x+y)^{2\alpha}-((2\alpha+1)x+y)y^{2\alpha}\right]}{(x+y)\left[(x+y)(x+y)^{2\alpha}-yy^{2\alpha}\right]^{2}}$

and

$\partial_{y}f(x,y)=-\dfrac{(2\alpha+1)x(x+y)^{2\alpha}\left[(x+y)(x+y)^{2\alpha}-((2\alpha+1)x+y)y^{2\alpha}\right]}{(x+y)\left[(x+y)(x+y)^{2\alpha}-yy^{2\alpha}\right]^{2}}$

if $x+y\neq 0$ and $\partial_{x}f(x,y)=\partial_{y}f(x,y)=0$ otherwise. The
set of critical points is therefore

$\\{(x,y)\in\mathbb{G}_{\alpha}|x+y=0\\}\cup\\{(x,y)\in\mathbb{G}_{\alpha}|(x+y)(x+y)^{2\alpha}-((2\alpha+1)x+y)y^{2\alpha}=0\\}.$

The case $x+y=0$ corresponds to $f(x,y)=1$ and thus the trivial condition
$N\geqslant 1$. In the other case, we either have that $y=0$, or else we can
set $m=x/y$ and it follows that

$(m+1)^{2\alpha}(m+1)-\left((2\alpha+1)m+1\right)=0.$

The former leads to $f(x,y)=2(\alpha+1)$ at the limit and $N\geqslant
2(\alpha+1)$ while the later means that

$\displaystyle f(x,y)$
$\displaystyle=\dfrac{(m+1)^{2\alpha}(2(\alpha+1)m+1)-1}{(m+1)^{2\alpha}(m+1)-1}$
$\displaystyle=2\left[\dfrac{(\alpha+1)m+1}{m+1}\right];$

the function is constant at these points. Since $m\geqslant-3$ and
$\alpha\geqslant 1$, we get

$2\left[\dfrac{(\alpha+1)m+1}{m+1}\right]\geqslant 2(\alpha+1)\geqslant 1.$

We have therefore established that

$\max_{(x,y)\in\mathbb{R}^{2}}\dfrac{(2(\alpha+1)x+y)(x+y)^{2\alpha}-yy^{2\alpha}}{(x+y)(x+y)^{2\alpha}-yy^{2\alpha}}=2\left[\dfrac{(\alpha+1)m+1}{m+1}\right].$

As pointed out previously, this maximum provides the desired optimal $N$ in
the inequality (15). ∎

It seems that Grushin structures behave in such a way that the distortion
coefficients for points $q_{0}$ and $q$ lying on the same horizontal line
provides the sharpest $N$ such that $\beta_{t}(q_{0},q)\geqslant t^{N}$. This
is also what happens when $\alpha=1$ (see [BR19, Proposition 62.] and [Riz18,
Theorem 8.] for Grushin half-planes).

We thus expect that the optimal $N$ obtained in Proposition 17 is sharp. We
are able to verify this suggestion for singular points, i.e. when
$q_{0}=(0,y_{0})$.

######  Proposition 18.

Let $q_{0}=(x_{0},y_{0})\in\mathbb{G}_{\alpha}$ with $x_{0}=0$ and
$q\notin\mathrm{Cut}(q_{0})$. For all $t\in\interval{0}{1}$,

$\beta_{t}(q,q_{0})\geqslant t^{N}$

for all $N\geqslant 2\left[\dfrac{(\alpha+1)m+1}{m+1}\right]$ where $m<0$ is
such that (14) holds.

###### Proof.

In the case of $x_{0}=0$, the Jacobian determinant (13) becomes

$\begin{split}[x,y](t)=\
&\dfrac{A^{2\alpha}}{(\alpha+1)\omega^{2}}\left(\alpha(3\alpha-1)\omega
t-(\alpha-1)^{2}\sin_{\alpha}(\omega t)\cos_{\alpha}(\omega t)\right)\\\
&\qquad\left(\sin_{\alpha}(\omega t)-\omega t\cos_{\alpha}(\omega t)\right)\\\
=\
&\dfrac{1}{\alpha^{3}(\alpha+1)|u_{0}|v_{0}^{2}}\left(\alpha(3\alpha-1)u_{0}^{2}t-(\alpha-1)^{2}x(t)u(t)\right)\left(x(t)-tu(t)\right)\end{split}$
(16)

If we let $x_{0}=0$, then we get $\phi=0$ or $\phi=\pi_{\alpha}$. In both
cases, it follows from (16) that

$\beta_{t}(q,q_{0})=\dfrac{g(\omega t)h(\omega t)}{g(\omega)h(\omega)}.$ (17)

where we have set
$g(z)=\alpha(3\alpha-1)z-(\alpha-1)^{2}\sin_{\alpha}(z)\cos_{\alpha}(z)$ and
$h(z)=\sin_{\alpha}(z)-z\cos_{\alpha}(z)$. We first note that $g(0)=h(z)=0$.
Then, we check that

$\begin{split}g^{\prime}(z)&=\alpha
z\sin_{\alpha}^{2(\alpha-1)}(z)\sin_{\alpha}(z)>0\end{split}$

and

$\begin{split}h^{\prime}(z)&=\alpha(3\alpha-1)-(\alpha-1)^{2}\left[\cos_{\alpha}^{2}(z)-\alpha\sin_{\alpha}^{2\alpha}(z)\right]\\\
&=(\alpha+1)\left[\alpha^{2}-(\alpha-1)^{2}\cos_{\alpha}^{2}(z)\right]>0\\\
\end{split}$

since $z\in\interval{0}{\pi_{\alpha}},|\cos_{\alpha}|\leqslant 1$ and
$\alpha>1$. Therefore, the functions $g$ and $h$ are strictly increasing and
positive.

We want to prove that (17) is greater than $t^{N}$. We will find
$N_{1},N_{2}\geqslant 1$ such that

$\dfrac{h(\omega t)}{h(\omega)}\geqslant t^{N_{1}}\text{ and }\dfrac{g(\omega
t)}{g(\omega)}\geqslant t^{N_{2}}.$

Let us look at $N_{1}$ first. Similarly as we did in the proof of Proposition
17, we know that the desired inequality happens if and only if we have

$H(z):=N_{1}h(z)-zh^{\prime}(z)\geqslant 0\text{ for all
}z\in\interval{0}{\pi_{\alpha}}.$

We can see that $H(0)=0$ and

$\begin{split}H^{\prime}(z)&=\alpha
z\sin_{\alpha}^{2(\alpha-1)}(z)\left[(N-2)\sin_{\alpha}(z)-(2\alpha-1)z\cos_{\alpha}(z)\right]\\\
&\geqslant\alpha(2\alpha-1)z\sin_{\alpha}^{2(\alpha-1)}(z)\left[\sin_{\alpha}(z)-z\cos_{\alpha}(z)\right]\geqslant
0\end{split}$

if $N_{1}\geqslant 2\alpha+1$, in which case, $H(z)\geqslant 0$. Now, let us
find a condition on $N_{2}$. We have

$\begin{split}N_{2}f(z)-zf^{\prime}(z)&=\alpha(N_{2}(3\alpha-1)-\alpha(\alpha+1))z-(\alpha-1)^{2}N\sin_{\alpha}(z)\cos_{\alpha}(z)\\\
&\quad+(\alpha-1)^{2}(\alpha+1)\cos_{\alpha}^{2}(z)\\\
&\geqslant\alpha(N_{2}(3\alpha-1)-\alpha(\alpha+1))z-(\alpha-1)^{2}N\sin_{\alpha}(z)\cos_{\alpha}(z)\\\
&\geqslant\alpha(N_{2}(3\alpha-1)-\alpha(\alpha+1))z-(\alpha-1)^{2}Nz\\\
&=(\alpha+1)(N_{2}(2\alpha-1)-\alpha^{2})z\geqslant 0\text{ if
}N_{2}\geqslant\dfrac{\alpha^{2}}{2\alpha-1}.\end{split}$

Therefore,

$\beta_{t}(q,q_{0})\geqslant
t^{2\alpha+1}t^{\frac{\alpha^{2}}{2\alpha-1}}\geqslant t^{N}\geqslant t^{N}$

since $N\geqslant 2\left[\dfrac{(\alpha+1)m+1}{m+1}\right]\geqslant
2\alpha+1+\dfrac{\alpha^{2}}{2\alpha-1}$. Indeed, this last condition is
equivalent to $(\alpha-1)^{2}m+\alpha(5\alpha-4)+1\geqslant 0$ and this is
true because $m\geqslant-3$. ∎

The lower bound holds for very specific Riemannian points as well. More
specifically, the condition is satisfied along the geodesics that uniquely
join the cut locus.

######  Proposition 19.

Let $q_{0}=(x_{0},y_{0})\in\mathbb{G}_{\alpha}$ with $x_{0}\neq 0$ and
$q\notin\mathrm{Cut}(q_{0})$. Assume that the unique minimal geodesic joining
$q_{0}$ to $q$ has an initial covector $(u_{0},v_{0})$ with $u_{0}=0$ while
$v_{0}\neq 0$. Then, for all $t\in\interval{0}{1}$,

$\beta_{t}(q,q_{0})\geqslant t^{N}$

for all $N\geqslant 2\left[\dfrac{(\alpha+1)m+1}{m+1}\right]$ where $m<0$ is
such that (14) holds.

###### Proof.

The condition $u_{0}=0$ means that $\phi=\pi_{\alpha}/2$ or $3\pi_{\alpha}/2$.
In both cases, this yields

$\beta_{t}(q_{0},q)=t\dfrac{\cos_{\alpha}(\omega
t+\phi)}{\cos_{\alpha}(\omega+\phi)}=t\dfrac{\sin_{\alpha}(\omega
t)}{\sin_{\alpha}(\omega)}\geqslant t^{2}\geqslant t^{N}.$

Indeed, $N^{\prime}\sin_{\alpha}(z)-z\cos_{\alpha}(z)\geqslant 0$ for all
$N^{\prime}\geqslant 1$ and $N\geqslant
2\left[\dfrac{(\alpha+1)m+1}{m+1}\right]\geqslant 2$. ∎

By analysing in more details and looking at the graph of the distortion
coefficients (12), it does seem to us that the relevant condition is also
satisfied when $x_{0}\neq 0$, and with $\phi\neq,\pi_{\alpha}/2$,
$3\pi_{\alpha}/2$. We therefore suggest the Conjecture 2. A proof of this
could require further work, potentially requiring a more comprehensive study
of the $(2,2\alpha)$-trigonometric functions.

## References

  * [ABB20] Andrei Agrachev, Davide Barilari, and Ugo Boscain.  A comprehensive introduction to sub-Riemannian geometry, volume 181 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2020.  From the Hamiltonian viewpoint, With an appendix by Igor Zelenko.
  * [ABS08] Andrei Agrachev, Ugo Boscain, and Mario Sigalotti.  A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds.  Discrete Contin. Dyn. Syst., 20(4):801–822, 2008.
  * [BBI01] Dmitri Burago, Yuri Burago, and Sergei Ivanov.  A course in metric geometry, volume 33 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2001.
  * [BKS18] Zoltán M. Balogh, Alexandru Kristály, and Kinga Sipos.  Geometric inequalities on Heisenberg groups.  Calc. Var. Partial Differential Equations, 57(2):Paper No. 61, 41, 2018.
  * [BL05] Francesco Bullo and Andrew D. Lewis.  Geometric control of mechanical systems, volume 49 of Texts in Applied Mathematics.  Springer-Verlag, New York, 2005.  Modeling, analysis, and design for simple mechanical control systems.
  * [BR19] Davide Barilari and Luca Rizzi.  Sub-Riemannian interpolation inequalities.  Invent. Math., 215(3):977–1038, 2019.
  * [CL12] Der-Chen Chang and Yutian Li.  SubRiemannian geodesics in the Grushin plane.  J. Geom. Anal., 22(3):800–826, 2012.
  * [CS12] Fabio Cavalletti and Karl-Theodor Sturm.  Local curvature-dimension condition implies measure-contraction property.  J. Funct. Anal., 262(12):5110–5127, 2012.
  * [EGL12] David E. Edmunds, Petr Gurka, and Jan Lang.  Properties of generalized trigonometric functions.  J. Approx. Theory, 164(1):47–56, 2012.
  * [Eul61] Leonhard Euler.  Observationes de comparatione arcuum curvarum irrectificibilium.  Novi commentarii academiae scientiarum Petropolitanae, pages 58–84, 1761.
  * [Gru70] V. V. Grušin.  A certain class of hypoelliptic operators.  Mat. Sb. (N.S.), 83 (125):456–473, 1970.
  * [Jui09] Nicolas Juillet.  Geometric inequalities and generalized Ricci bounds in the Heisenberg group.  Int. Math. Res. Not. IMRN, 2009(13):2347–2373, 2009.
  * [Jui20] Nicolas Juillet.  SubRiemanniann structures do not satisify Riemannian Brunn–Minkowski inequalities.  arXiv e-prints, page arXiv:2002.01170, February 2020.
  * [LD17] Enrico Le Donne.  Lecture notes on sub-riemannian geometry.  preprint, 2017.  https://sites.google.com/site/enricoledonne/lecture_notes.
  * [LLP19] Enrico Le Donne, Danka Lučić, and Enrico Pasqualetto.  Universal infinitesimal Hilbertianity of sub-Riemannian manifolds.  arXiv e-prints, page arXiv:1910.05962, October 2019.
  * [LV09] John Lott and Cédric Villani.  Ricci curvature for metric-measure spaces via optimal transport.  Ann. of Math. (2), 169(3):903–991, 2009.
  * [Oht07] Shin-Ichi Ohta.  On the measure contraction property of metric measure spaces.  Comment. Math. Helv., 82(4):805–828, 2007.
  * [Riz18] Luca Rizzi.  A counterexample to gluing theorems for MCP metric measure spaces.  Bull. Lond. Math. Soc., 50(5):781–790, 2018.
  * [Stu06a] Karl-Theodor Sturm.  On the geometry of metric measure spaces. I.  Acta Math., 196(1):65–131, 2006.
  * [Stu06b] Karl-Theodor Sturm.  On the geometry of metric measure spaces. II.  Acta Math., 196(1):133–177, 2006.
  * [Tak17] Shingo Takeuchi.  Arithmetric properties of the generalized trigonometric functions (qualitative theory of ordinary differential equations and related areas).  RIMS Kokyuroku, 2032:76–100, 2017.
  * [Vil09] Cédric Villani.  Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 2009.  Old and new.