Differentiable sphere theorems whose comparison spaces are standard spheres or exotic ones

TL;DR

This paper establishes a differentiable sphere theorem for manifolds with a single cut point, showing that small radial curvature deviations imply diffeomorphism to the original manifold, extending classical results like the Cartan--Ambrose--Hicks theorem.

Contribution

It introduces a weak differentiable sphere theorem for manifolds with a single cut point, connecting curvature closeness to diffeomorphism, and generalizes previous results including Cheeger's theorem.

Findings

Manifolds with a single cut point are diffeomorphic under small radial curvature deviations.

The result provides a weak version of the Cartan--Ambrose--Hicks theorem.

It extends the Blaschke conjecture for spheres and includes exotic spheres in the scope.

Abstract

We show that for an arbitrarily given closed Riemannian manifold $M$ admitting a point $p \in M$ with a single cut point, every closed Riemannian manifold $N$ admitting a point $q \in N$ with a single cut point is diffeomorphic to $M$ if the radial curvatures of $N$ at $q$ are sufficiently close in the sense of $L^1$-norm to those of $M$ at $p$. Our result hence not only produces a weak version of the Cartan--Ambrose--Hicks theorem in the case where underlying manifolds admit a point with a single cut point, but also is a kind of a weak version of the Blaschke conjecture for spheres proved by Berger. In particular that result generalizes one of theorems in Cheeger's Ph.D. Thesis in that case. Remark that every exotic sphere of dimension $> 4$ admits a metric such that there is a point whose cut locus consists of a single point.

Equations

$${\tau }_{u_k}(t)={\exp }_{p_k}t{u}_k$$

$${\Psi }_t^{(u_1)}:=P_t^{(u_2)}\circ I_{p_1}.$$

$$\lambda (t):=\underset{\begin{array}{c}{u}_1\in {\mathbb{S}}_{p_1}^{n-1}\\ x_1\in {\mathbb{S}}_{u_1}^{n-2}\end{array}}{\max }\left|K^{(1)}(P_t^{(u_1)}(x_1)\wedge P_t^{(u_1)}(u_1))-K^{(2)}({\Psi }_t^{(u_1)}(x_1)\wedge {\Psi }_t^{(u_1)}(u_1))\right|$$

$$K^{(k)}(x\wedge y)=\frac{⟨R^{(k)}\left(x,y\right)x,y⟩}{\Vert x{\Vert }^2\Vert y{\Vert }^2-{⟨x,y⟩}^2}.$$

$$R^{(k)}(X,Y)Z:={\nabla }_Y{\nabla }_{X}Z-{\nabla }_X{\nabla }_{Y}Z+{\nabla }_{[X,Y]}Z$$

$${\int }_0^{\pi }\lambda (t)dt<{\epsilon }_n(M_1),$$

$$\eta_{n}(M_{1})\cdot x\exp(2(n-1)x)=\frac{1}{2}\left[\sqrt{1+\Big{\{}\frac{8}{\pi}(n-1)\Big{\}}^{-\frac{1}{2}}}-1\right]$$

$$\Big{\|}\frac{d^{2}c_{\,\gamma}}{dt^{2}}\Big{\|}^{2}-2\mathop{\mathrm{Lip^{b}}}\nolimits(\sigma)^{-2}\leq 2\Big{\{}\frac{\sqrt{2}-1}{2(e^{\pi}-1)}\Big{\}}^{2}-1$$

$$\mathrm{Li}{\mathrm{p}}^{\mathrm{b}}(\sigma ):=\inf \{a|a^{-1}\Vert u-v\Vert \le \Vert \sigma (u)-\sigma (v)\Vert \le a\Vert u-v\Vert \mathrm{for}\mathrm{all}u,v\in {\mathbb{S}}_{q_1}^{n-1}\},$$

$$E_i^{(k)}(t):=P_t^{(u_k)}(e_i^{(u_k)}),i=1,2,\dots ,n.$$

$$E_i^{(1)}(t)=P_t^{(u_1)}(e_i^{(u_1)}),E_i^{(2)}(t)=P_t^{(u_2)}(I_{p_1}(e_i^{(u_1)}))={\Psi }_t^{(u_1)}(e_i^{(u_1)})$$

$$\dot{\tau}_{u_{1}}(t)=E_{n}^{(1)}(t)=P_{t}^{(u_{1})}(u_{1}),\quad\dot{\tau}_{u_{2}}(t)=E_{n}^{(2)}(t)=\Psi_{t}^{(u_{1})}\big{(}u_{1}\big{)}.$$

$$a_{ij}^{(k)}(t):=\big{\langle}R^{(k)}(E_{i}^{(k)}(t),\dot{\tau}_{u_{k}}(t))\dot{\tau}_{u_{k}}(t),E_{j}^{(k)}(t)\big{\rangle}$$

$$A(t\,;u_{k}):=\left(\begin{array}[]{cc}0&I_{n-1}\\ \big{(}a_{ij}^{(k)}(t)\big{)}&0\end{array}\right),$$

$$\Vert A(t;u_1)-A(t;u_2)\Vert \le 2(n-1)\lambda (t)$$

$$\Vert A(t;u_2)\Vert \le c_1(n,p_1)+2(n-1)\lambda (t)$$

$$a_{ij}^{(k)}(t)=-\big{\langle}R^{(k)}(E_{i}^{(k)}(t),\dot{\tau}_{u_{k}}(t))E_{j}^{(k)}(t),\dot{\tau}_{u_{k}}(t)\big{\rangle}\quad(k=1,2),$$

$$K^{(k)}(E^{(k)}_{i}(t)\wedge\dot{\tau}_{u_{k}}(t))=\big{\langle}R^{(k)}(E^{(k)}_{i}(t),\dot{\tau}_{u_{k}}(t))E^{(k)}_{i}(t),\dot{\tau}_{u_{k}}(t)\big{\rangle}=-a_{ii}^{(k)}(t).$$

$$K^{(k)}((E_{i}^{(k)}(t)+E_{j}^{(k)}(t))\wedge\dot{\tau}_{u_{k}}(t))=\frac{1}{2}\big{(}-a_{ii}^{(k)}(t)-2a_{ij}^{(k)}(t)-a_{jj}^{(k)}(t)\big{)}.$$

$$a_{ij}^{(k)}(t)=$$

$$-K^{(k)}((E_i^{(k)}(t)+E_j^{(k)}(t))\wedge {\dot{\tau }}_{u_k}(t)).$$

$$\big{|}a_{ij}^{(1)}(t)-a_{ij}^{(2)}(t)\big{|}\leq 2\lambda(t).$$

$$\Vert A(t;u_1)-A(t;u_2)\Vert$$

$$\displaystyle\leq(n-1)\max_{i,\,j\,=\,1,\,2,\,\ldots,\,n-1}\big{|}a_{ij}^{(1)}(t)-a_{ij}^{(2)}(t)\big{|}$$

$$\le 2(n-1)\lambda (t),$$

$$\Vert A(t;u_2)\Vert =\Vert A(t;u_2)-A(t;u_1)+A(t;u_1)\Vert \le \Vert A(t;u_2)-A(t;u_1)\Vert +\Vert A(t;u_1)\Vert ,$$

$${\sigma }_{q_k}^{p_k}(u_k):=-{\dot{\tau }}_{u_k}(\pi ),$$

$$\sigma :={\sigma }_{q_2}^{p_2}\circ I_{p_1}\circ {\sigma }_{p_1}^{q_1}$$

$$I_{q_1}^{(u_1)}:={\Psi }_{\pi }^{(u_1)}\circ (P_{\pi }^{(u_1)}{)}^{-1}.$$

$$\delta_{n}:=\sqrt{1+\Big{\{}\frac{8}{\pi}(n-1)\Big{\}}^{-\frac{1}{2}}}-1.$$

$${\int }_0^{\pi }\lambda (t)dt<{\epsilon }_n(M_1),$$

$${\dot{f}}_i^{(k)}(t):=\frac{d{f}_i^{(k)}}{dt}(t),{\ddot{f}}_i^{(k)}(t):=\frac{d^2{f}_i^{(k)}}{d{t}^2}(t)$$

Full text

1 Introduction

In the global Riemannian geometry the relationship between curvatures and structures, especially topology, of Riemannian manifolds has been studied from various kinds of viewpoint, and a great number of results concerning with such a relation has been gotten. It is the topological $1/4$–pinching sphere theorem that is counted among the masterpieces of such results from the geodesic theory’s standpoint, which states the following:

Theorem 1.1

(Rauch–Berger–Klingenberg)* If a compact simply connected Riemannian manifold $X$ admits a metric whose sectional curvature $K_{X}$ lies in $(1/4,1]$, then the manifold is homeomorphic to a sphere.*

This masterpiece was very first proved by Rauch [23] in the case where $3/4<K_{X}\leq 1$, and worked out by Berger [3] and Klingenberg [18] in the case where $1/4<K_{X}\leq 1$. Note that the complex projective space admits a metric satisfying $1/4\leq K_{X}\leq 1$ and is not homeomorphic to a sphere.

Theorem 1.1 produced the $1/4$–pinching race as the problem if “homeomorphic” in the statement could be replaced by “diffeomorphic”. There were a large number of entrant for the race, e.g., Gromoll [9], Calabi, Shikata [26], Sugimoto–Shiohama–Karcher [29], Grove–Karcher–Ruh [11, 12], Im Hof–Ruh [15], and Suyama [30], et al. Using the Ricci flow introduced by Hamilton [13], Brendle and Schoen [5] finally proved that the masterpiece can be reinforced into the differentiable $1/4$–pinching sphere theorem, which implies that every exotic sphere does not admit a $1/4$–pinched metric.

By remembering that the $1/4$–pinching race (problem) had originated in Hopf’s curvature pinching conjecture, the solution to the problem by Brendle–Schoen asks the following natural question of us.

**Question. **

Replacing the unit standard sphere in the Hopf conjecture by an arbitrary compact simply connected Riemannian manifold $X$, should a compact simply connected Riemannian manifold whose radial curvature is close to that of $X$ be diffeomorphic to $X$? That is, can we weaken the assumption of the Cartan–Ambrose–Hicks theorem [1, 6] to closeness of radial curvatures of the manifolds?**

In the question above the radial curvature is, by definition, the restriction of the sectional curvature of a pointed Riemannian manifold to all $2$-dimensional planes which contain the unit tangent velocity vector, as one of its basis, of any minimal geodesic emanating from the base point.

The purpose of this article is to solve the question above by hypothesizing that underlying closed manifolds admit metrics such that there is a point whose cut locus consists of a single point. It is worthy of note that every homotopy $n$-sphere of dimension $n\geq 5$ admits such a metric, and so are all exotic $n$-spheres. This note follows from Smale’s $h$-cobordism theorem [27, 28] and Weinstein’s deformation technique [31] for metrics on twisted spheres (also see [4, Proposition 7.19]).

We are now going to state our main theorem precisely. For each $k=1,2$ let $M_{k}$ be a closed manifold of dimension $n\geq 2$ admitting a point whose cut locus consists of a single point, and $\langle\,\cdot\,,\cdot\,\rangle$ a Riemannian metric of $M_{k}$. Note that $M_{k}$ is homeomorphic to a sphere $S^{n}$ of dimension $n$. We take any point $p_{k}\in M_{k}$ satisfying $\mathop{\mathrm{Cut}}\nolimits(p_{k})=\{q_{k}\}$ where $q_{k}\in M_{k}$, and fix it. Here $\mathop{\mathrm{Cut}}\nolimits(p_{k})$ denotes the cut locus of $p_{k}$. Normalizing the metric, we can assume $d_{M_{k}}(p_{k},q_{k})=\pi$ where $d_{M_{k}}$ denotes the distance function of $M_{k}$. Set $\mathbb{S}_{p_{k}}^{n-1}:=\{u\in T_{p_{k}}M_{k}\,|\,\|u\|=1\}$ where $T_{p_{k}}M_{k}$ is the tangent space to $M_{k}$ at $p_{k}$. For each $u_{k}\in\mathbb{S}_{p_{k}}^{n-1}$, let $\tau_{u_{k}}:[0,\pi]\longrightarrow M_{k}$ denote a geodesic segment emanating from $p_{k}=\tau_{u_{k}}(0)$ to $q_{k}=\tau_{u_{k}}(\pi)$ in the direction $u_{k}=\dot{\tau}_{u_{k}}(0):=(d\tau_{u_{k}}/dt)(0)$, i.e.,

[TABLE]

for all $t\in[0,\pi]$.

Fix $u_{1}\in\mathbb{S}^{n-1}_{p_{1}}$. Let $I_{p_{1}}:T_{p_{1}}M_{1}\longrightarrow T_{p_{2}}M_{2}$ be a linear isometry. Set $u_{2}:=I_{p_{1}}(u_{1})$. For each $t\in[0,\pi]$ let $P^{(u_{k})}_{t}$ $(k=1,2)$ be the parallel translation along the geodesic $\tau_{u_{k}}$ from $p_{k}$ to $\tau_{u_{k}}(t)$ where note that $\tau_{u_{2}}=\tau_{I_{p_{1}}(u_{1})}$. Define the linear isometry $\Psi^{(u_{1})}_{t}:T_{p_{1}}M_{1}\longrightarrow T_{\tau_{u_{2}}(t)}M_{2}$ by

[TABLE]

Moreover we define the function $\lambda:[0,\pi]\longrightarrow\mathbb{R}$ by

[TABLE]

for all $t\in[0,\pi]$ where $\mathbb{S}_{u_{1}}^{n-2}:=\{x\in T_{u_{1}}(\mathbb{S}_{p_{1}}^{n-1})\,|\,\|x\|=1\}$ and $K^{(k)}(x\wedge y)$ ($k=1,2$) is the sectional curvature of the plane spanned by two linearly independent tangent vectors $x$ and $y$ at a point on $M_{k}$, i.e.,

[TABLE]

In Eq. (1.4), $R^{(k)}$ denotes the curvature tensor of $M_{k}$ defined by

[TABLE]

for all vector fields $X,Y,Z$ on $M_{k}$, where $\nabla$ is the Levi–Civita connection on $M_{k}$. Note that the definition of $R^{(k)}$ differs from that of curvature tensor in literatures such as [8] and [24] by a sign.

With the these notations above our main theorem is stated as follows.

Theorem 1.2

(Main Theorem)* There exists a constant $\varepsilon_{n}(M_{1})>0$ depending on $n$ and $M_{1}$ such that if*

[TABLE]

then $M_{2}$ is diffeomorphic to $M_{1}$.

Remark 1.3

We give here several remarks on Theorem 1.2 and related results to it:

• •

Theorem 1.2 is the very Cartan–Ambrose–Hicks theorem if $\lambda(t)\equiv 0$ on $[0,\pi]$, and hence produces a weak version of the theorem. Note here that sectional curvature and curvature tensor are equivalent (see, e.g., Eq. (Proof.)). Moreover, since $M_{k}$ ($k=1,2$) admits a point whose cut locus consists of a single point, Theorem 1.2 is a kind of a weak version of the Blaschke conjecture for spheres proved by Berger [4], which states that if a Riemannian manifold $N$ of dimension $n\geq 2$ homeomorphic to $S^{n}$ has diameter equal to its injectivity radius, then $N$ is isometric to a standard sphere of constant curvature. In particular our theorem generalizes Cheeger’s theorem [7] (or see [8, Theorem 7.36]) in the case where underlying manifolds admit a point with a single cut point, because we do not assume either closeness of $\nabla R^{(1)}$ and $\nabla R^{(2)}$ along $\tau_{u_{1}}$ and $\tau_{u_{2}}$ or $\mathop{\mathrm{vol}}\nolimits(M_{2})>\nu$ for some $\nu>0$ where $\mathop{\mathrm{vol}}\nolimits(M_{2})$ denotes the volume of $M_{2}$, that additionally he assumed in his theorem; besides, we need to look around such manifolds only at their base points $p_{1}$ and $p_{2}$. Moreover it is apparent that our theorem extends and weakens [17, (iii) of Theorem 3] to a wider class of metrics than that of radially symmetric metrics in it.

• •

The constant $\varepsilon_{n}(M_{1})$ in Eq. (1.5) is obtained as the unique solution of the following equation

[TABLE]

for all $x\in[0,\infty)$ where $\eta_{n}(M_{1})$, depending on $n$ and $M_{1}$, denotes some positive constant concerning with Jacobi fields along $\tau_{u_{1}}$ (see Eq. (2.24) for more details). The constant $1+\{(8/\pi)(n-1)\}^{-1/2}$ found in Eq. (1.6) is the same as Karcher [16] estimated in order to prove a sharper version of Shikata’s theorem in [25].

• •

The related results to Theorem 1.2 are the differentiable exotic sphere theorems I and II proved by authors [20]. In the theorem I, the hypothesis (1.5) can be replaced by either

[TABLE]

for all unit speed geodesic segments $\gamma([0,\pi])\subset\mathbb{S}_{q_{1}}^{n-1}:=\{v\in T_{q_{1}}M_{1}\,|\,\|v\|=1\}$, or $\mathop{\mathrm{Lip^{b}}}\nolimits(\sigma)^{2}\leq 1+\left\{(8/\pi)(n-1)\right\}^{-1/2}$ where $\sigma:\mathbb{S}^{n-1}_{q_{1}}\longrightarrow\mathbb{S}^{n-1}_{q_{2}}:=\{v\in T_{q_{2}}M_{2}\,|\,\|v\|=1\}$ is the diffeomorphism defined by Eq. (2.10), $\mathop{\mathrm{Lip^{b}}}\nolimits(\sigma)$ is the bi-Lipschitz constant of $\sigma$ defined by

[TABLE]

and $c_{\,\gamma}:=\sigma\circ\gamma:[0,\pi]\longrightarrow\mathbb{S}^{n-1}_{q_{2}}$ for each geodesic segment $\gamma([0,\pi])\subset\mathbb{S}_{q_{1}}^{n-1}$. In the theorem II, the hypothesis (1.7) is replaced by $\angle(\overline{c}_{\,\gamma}(t),c_{\,\gamma}(t))<\pi/2$ for all unit speed geodesic segments $\gamma([0,\pi])\subset\mathbb{S}_{q_{1}}^{n-1}$ where $\overline{c}_{\,\gamma}$ is the smooth curve on $T_{q_{2}}M_{2}$ given by $\overline{c}_{\,\gamma}(t):=c_{\,\gamma}(0)\cos t+(dc_{\,\gamma}/dt)(0)\sin t$ for all $t\in[0,\pi]$.

Acknowledgements.** **

In this work the first named author was supported by the JSPS KAKENHI Grant Numbers 17K05220, and partially 16K05133, 18K03280.

2

Key lemma

The aim of this section is to show Key Lemma (Lemma 2.2) that we shall apply to the proof of Theorem 1.2. The integral form of the Grönwall inequality [2, 10] plays an important role in the proof of Lemma 2.2.

Throughout this section, for each $k=1,2$ let $M_{k}$ be a closed manifold of dimension $n\geq 2$ admitting a point $p_{k}\in M_{k}$ such that $\mathop{\mathrm{Cut}}\nolimits(p_{k})=\{q_{k}\}$, and for any $u_{k}\in\mathbb{S}^{n-1}_{p_{1}}$ let $\tau_{u_{k}}:[0,\pi]\longrightarrow M_{k}$ be the geodesic segment emanating from $p_{k}$ to $q_{k}$ defined by Eq. (1.1). We assume $d_{M_{k}}(p_{k},q_{k}):=\pi$ by normalizing the metric where $d_{M_{k}}$ denotes the distance function on $M_{k}$. All other notations in the following are the same as those defined in Section 1.

Fix $u_{1}\in\mathbb{S}^{n-1}_{p_{1}}$. Choose an orthonormal basis $e_{1}^{(u_{1})},e_{2}^{(u_{1})},\dots,e_{n}^{(u_{1})}$ of $T_{p_{1}}M_{1}$ satisfying $e_{n}^{(u_{1})}=u_{1}$. Setting $u_{2}:=I_{p_{1}}(u_{1})$, we have the orthonormal basis $e_{1}^{(u_{2})},e_{2}^{(u_{2})},\dots,e_{n}^{(u_{2})}$ of $T_{p_{2}}M_{2}$ given by $e_{i}^{(u_{2})}:=I_{p_{1}}(e_{i}^{(u_{1})})$ for each $i=1,2,\ldots,n$, which satisfies $e_{n}^{(u_{2})}=u_{2}$. For each $k=1,2$ let

[TABLE]

$E_{1}^{(k)},E_{2}^{(k)},\dots,E_{n}^{(k)}$ are then the parallel orthonormal fields along $\tau_{u_{k}}$. In particular

[TABLE]

for each $i=1,2,\ldots,n-1$ and

[TABLE]

Moreover for each $i,j=1,2,\ldots,n-1$ let

[TABLE]

for all $t\in[0,\pi]$. Furthermore we define the square matrix $A(t\,;u_{k})$ of order $2(n-1)$ by

[TABLE]

where $I_{n-1}$ is the $(n-1)$-th unit matrix. Note that \big{(}a_{ij}^{(k)}(t)\big{)} is the symmetric matrix of order $n-1$.

Lemma 2.1

For any $t\in[0,\pi]$,

[TABLE]

holds where $\|\cdot\|$ denotes the linear operator norm. In particular

[TABLE]

for all $t\in[0,\pi]$ where $c_{1}(n,p_{1}):=\max\{\|A(t;u_{1})\|\,|\,u_{1}\in\mathbb{S}_{p_{1}}^{n-1},t\in[0,\pi]\}$.

Proof.** **

Fix $t\in[0,\pi]$. Since

[TABLE]

Eq. (1.4) gives

[TABLE]

Since $E_{i}^{(k)}(t)+E_{j}^{(k)}(t)$ is orthonormal to $\dot{\tau}_{u_{k}}(t)$, and since $\|E_{i}^{(k)}(t)+E_{j}^{(k)}(t)\|=\sqrt{2}$, combining Eqs. (1.4), (2.4), and (2.5) shows

[TABLE]

From Eqs. (2.5) and (2.6) we have

[TABLE]

Since $K^{(k)}(x\wedge y)$ does not depend on the choice of the spanning vectors, we see, by Eqs. (2.2), (2.3), (Proof.) and the triangle inequality, that

[TABLE]

We therefore see, by Eq. (2.8), that

[TABLE]

which is the first assertion. Since

[TABLE]

the second assertion follows from the first one. $\hfill\Box$

Let $\mathbb{S}^{n-1}_{q_{k}}:=\{v\in T_{q_{k}}M_{k}\,|\,\|v\|=1\}$ ($k=1,2$). Since $\mathop{\mathrm{Cut}}\nolimits(p_{k})=\{q_{k}\}$, we have the diffeomorphism $\sigma^{p_{k}}_{q_{k}}$ from $\mathbb{S}^{n-1}_{p_{k}}$ onto $\mathbb{S}^{n-1}_{q_{k}}$ given by

[TABLE]

for all $u_{k}\in\mathbb{S}^{n-1}_{p_{k}}$. The map $\sigma$ from $\mathbb{S}^{n-1}_{q_{1}}$ onto $\mathbb{S}^{n-1}_{q_{2}}$ defined by

[TABLE]

is thus a diffeomorphism where $\sigma^{q_{1}}_{p_{1}}:=(\sigma_{q_{1}}^{p_{1}})^{-1}$. Moreover for each $u_{1}\in\mathbb{S}^{n-1}_{p_{1}}$ let $I_{q_{1}}^{(u_{1})}:T_{q_{1}}M_{1}\longrightarrow T_{q_{2}}M_{2}$ denote the linear isometry given by

[TABLE]

Lemma 2.2

(Key Lemma)* Set*

[TABLE]

Then there is a constant $\varepsilon_{n}(M_{1})>0$ such that if

[TABLE]

then for any $u_{1}\in\mathbb{S}^{n-1}_{p_{1}}$ the differential $d\sigma_{v_{1}}$ of $\sigma$ at $v_{1}:=\sigma^{p_{1}}_{q_{1}}(u_{1})\in\mathbb{S}^{n-1}_{q_{1}}$ and $I^{(u_{1})}_{q_{1}}$ are $\delta_{n}/2$-close with respect to the linear operator norm.

Proof.** **

Fix $u_{1}\in\mathbb{S}^{n-1}_{p_{1}}$. Let $u_{2}:=I_{p_{1}}(u_{1})\in\mathbb{S}^{n-1}_{p_{2}}$, and $\mathbb{S}_{u_{2}}^{n-2}:=\{x\in T_{u_{2}}(\mathbb{S}_{p_{2}}^{n-1})\,|\,\|x\|=1\}$. For any fixed $x_{1}\in\mathbb{S}_{u_{1}}^{n-2}$ let $x_{2}:=I_{p_{1}}(x_{1})\in\mathbb{S}_{u_{2}}^{n-2}$ where we identify $T_{u_{k}}(T_{p_{k}}M_{k})$ with $T_{p_{k}}M_{k}$ ($k=1,2$). Let $J^{\,(x_{k})}_{u_{k}}$ be the Jacobi field along $\tau_{u_{k}}$ given by $J^{\,(x_{k})}_{u_{k}}(0)=0$ and $(DJ^{\,(x_{k})}_{u_{k}}/dt)(0)=x_{k}\in\mathbb{S}_{u_{k}}^{n-2}$ where $DJ^{\,(x_{k})}_{u_{k}}/dt$ denotes the covariant derivative of $J^{(x_{k})}_{u_{k}}$ along $\tau_{u_{k}}$. Setting $f^{\,(k)}_{i}(t):=\langle J^{\,(x_{k})}_{u_{k}}(t),E_{i}^{(k)}(t)\rangle$ for all $t\in[0,\pi]$, the definition of $J^{\,(x_{k})}_{u_{k}}$ and the Gauss lemma give $J^{\,(x_{k})}_{u_{k}}(t)=\sum_{i=1}^{n-1}f^{\,(k)}_{i}(t)E_{i}^{(k)}(t)$. For simplicity of notation we set

[TABLE]

for all $i=1,2,\ldots,n-1$. Since

[TABLE]

Eq. (2.10) shows

[TABLE]

From Eq. (2.1) and the second one in Eq. (2.2) we obtain

[TABLE]

We then have

[TABLE]

It follows from Eqs. (Proof.) and (2.14) that

[TABLE]

Let $\widetilde{J}^{\,(x_{k})}_{u_{k}}(t):={}^{t}(f^{\,(k)}_{1}(t),\ldots,f^{\,(k)}_{n-1}(t),\dot{f}^{\,(k)}_{1}(t),\ldots,\dot{f}^{\,(k)}_{n-1}(t))\in\mathbb{R}^{2(n-1)}$. Define the smooth function $\varphi:[0,\pi]\longrightarrow\mathbb{R}$ by $\varphi(t):=\|\widetilde{J}^{\,(x_{1})}_{u_{1}}(t)-\widetilde{J}^{\,(x_{2})}_{u_{2}}(t)\|$. In the case where $\varphi\equiv 0$ on $[0,\pi]$, $\dot{f}^{\,(1)}_{i}(\pi)=\dot{f}^{\,(2)}_{i}(\pi)$ holds for each $i=1,2,\ldots,n-1$, and hence Eq. (2.15) shows that $d\sigma_{v_{1}}$ and $I^{(u_{1})}_{q_{1}}$ are $\delta_{n}/2$-close. From this argument we next consider the case where there is an interval $[a,b)\subset[0,\pi]$ such that $\varphi(t)>0$ on $(a,b)$ with $\varphi(a)=0$: Since $J^{\,(x_{k})}_{u_{k}}$ satisfies the Jacobi equation

[TABLE]

we obtain

[TABLE]

for all $t\in[0,\pi]$ and $j=1,2,\ldots,n-1$. Substituting Eq. (2.16) for the following

[TABLE]

we have

[TABLE]

for all $t\in[0,\pi]$. Hence, applying the Cauchy–Schwarz inequality and the triangle one to $\varphi^{\prime}(t)$, we see that for any $t\in(a,b)$,

[TABLE]

Let

[TABLE]

for all $t\in(a,b)$. Since $\varphi(a)=0$, the integration of Eq. (Proof.) from $a$ to $t$ yields the inequality

[TABLE]

Since $\varphi(t)$, $h_{a}(t)$, and $\left\|A(t\,;u_{2})\right\|$ are continuous on $(a,b)$, and since $\left\|A(t\,;u_{2})\right\|\geq 0$ on $(a,b)$, the integral form of Grönwall’s inequality [10, 2] gives

[TABLE]

for all $t\in(a,b)$. Since $h_{a}$ is non-decreasing on $(a,b)$, we see, by Eq. (2.18), that

[TABLE]

Since the functions $\left\|A(t\,;u_{1})-A(t\,;u_{2})\right\|\cdot\|\widetilde{J}^{\,(x_{1})}_{u_{1}}(t)\|$ and $\|A(t\,;u_{2})\|$ are well-defined on $[0,b)$ and are integrable on $[0,b)$, it is clear that

[TABLE]

for all $t\in(a,b)$. Since the function t\longrightarrow h_{0}(t)\exp\big{(}\int_{0}^{t}\left\|A(r\,;u_{2})\right\|dr\big{)} is increasing on $[0,\pi]$, Eq. (2.20) has the form

[TABLE]

for all $t\in(a,b)$. Since Eq. (2.21) still holds for some $t_{0}\in[0,\pi]$ with $\varphi(t_{0})=0$, we get

[TABLE]

for all $t\in[0,\pi]$. Set $c_{2}(n,p_{1}):=\max\{\|\widetilde{J}^{\,(x_{1})}_{u_{1}}(t)\|\,|\,u_{1}\in\mathbb{S}_{p_{1}}^{n-1},\,x_{1}\in\mathbb{S}_{u_{1}}^{n-2},\,t\in[0,\pi]\}$. Applying Lemma 2.1 to Eq. (2.22), we have

[TABLE]

for all $t\in[0,\pi]$ where $c(n,M_{1}):=2(n-1)\cdot c_{2}(n,p_{1})\cdot\exp(\pi c_{1}(n,p_{1}))$. Let $\varepsilon_{n}(M_{1})>0$ be the unique solution of the following equation

[TABLE]

for all $x\in[0,\infty)$ where

[TABLE]

We now assume that

[TABLE]

Eq. (Proof.) then yields

[TABLE]

Combining Eqs. (2.15) and (2.25) shows

[TABLE]

From the arbitrariness of $x_{1}$, Eq. (Proof.) implies

[TABLE]

and hence $d\sigma_{v_{1}}$ and $I^{(u_{1})}_{q_{1}}$ are $\delta_{n}/2$-close. $\hfill\Box$

3

Proof of Theorem 1.2

The purpose of this section is to show Theorem 1.2. The idea of the proof is to construct a family of smooth immersions which approximates a bi-Lipschitz homeomorphism, defined by Eq. (3.1), between closed Riemannian manifolds admitting a point with a single cut point under the assumption of Theorem 1.2. Throughout the section, for each $k=1,2$ let $M_{k}$ be a closed manifold of dimension $n\geq 2$ admitting a point $p_{k}\in M_{k}$ such that $\mathop{\mathrm{Cut}}\nolimits(p_{k})=\{q_{k}\}$. Moreover we assume $d_{M_{k}}(p_{k},q_{k})=\pi$. Furthermore we will make the following assumption:

[TABLE]

where $\lambda(t)$ is the function defined by Eq. (1.3) and $\varepsilon_{n}(M_{1})>0$ is the unique solution of Eq. (2.24).

Choose a linear isometry $I_{p_{1}}:T_{p_{1}}M_{1}\longrightarrow T_{p_{2}}M_{2}$, and we define the bi-Lipschitz homeomorphism $F$ from $M_{1}$ onto $M_{2}$ by

[TABLE]

for all $(t,u_{1})\in[0,\pi]\times\mathbb{S}^{n-1}_{p_{1}}$. Since $\mathop{\mathrm{Cut}}\nolimits(p_{1})=\{q_{1}\}$, $F$ is not differentiable only at $q_{1}$, and $F|_{M_{1}\setminus\{q_{1}\}}$ is diffeomorphism. Let $\mathbb{B}_{\pi}(o_{q_{k}}):=\{x\in T_{q_{k}}M_{k}\,|\,\|x\|<\pi\}$ ($k=1,2$) where $o_{q_{k}}$ is the origin of $T_{q_{k}}M_{k}$. We define the map $\widetilde{F}:\mathbb{B}_{\pi}(o_{q_{1}})\longrightarrow\mathbb{B}_{\pi}(o_{q_{2}})$ by

[TABLE]

We then see, by the very same argument as [20, Section 3.3], that

[TABLE]

where $\sigma:\mathbb{S}^{n-1}_{q_{1}}\longrightarrow\mathbb{S}^{n-1}_{q_{2}}$ is the diffeomorphism defined by Eq. (2.10). Note that $\widetilde{F}$ is a bi-Lipschitz homeomorphism (see [20, Lemma 3.13] for more details).

Lemma 3.1

For any $x\in\mathbb{B}_{\pi}(o_{q_{1}})\setminus\{o_{q_{1}}\}$ and any $X\in T_{x}\mathbb{B}_{\pi}(o_{q_{1}})$ with $\|X\|=1$,

[TABLE]

holds where $\delta_{n}$ is the positive constant defined by Eq. (2.12).

Proof.** **

Fix $x\in\mathbb{B}_{\pi}(o_{q_{1}})\setminus\{o_{q_{1}}\}$. Then there is $v\in\mathbb{S}^{n-1}_{q_{1}}$ and $\ell>0$ such that $x=\ell v$. We then see, by the proof of [20, Lemma 3.7], that

[TABLE]

for all $a\in\mathbb{R}$, and

[TABLE]

for all $w\in\mathbb{S}^{n-2}_{v}:=\{w\in T_{v}(\mathbb{S}^{n-1}_{q_{1}})\,|\,\|w\|=1\}$. Since

[TABLE]

combining the triangle inequality and Lemma 2.2 gives

[TABLE]

for all $w\in\mathbb{S}^{n-2}_{v}$ where $I_{q_{1}}^{(\sigma^{q_{1}}_{p_{1}}(v))}:T_{q_{1}}M_{1}\longrightarrow T_{q_{2}}M_{2}$ is the linear isometry defined by Eq. (2.11), and $\sigma^{q_{1}}_{p_{1}}$ is the inverse of the diffeomorphism $\sigma^{p_{1}}_{q_{1}}:\mathbb{S}_{p_{1}}^{n-1}\longrightarrow\mathbb{S}_{q_{1}}^{n-1}$ defined by Eq. (2.9). Fix $X\in T_{x}(\mathbb{B}_{\pi}(o_{q_{1}}))$ with $\|X\|=1$. Then there are $w_{0}\in\mathbb{S}^{n-2}_{v}$ and $\alpha,\beta\in\mathbb{R}$ such that $X=\alpha v+\beta w_{0}$. Eqs. (3.3) and (3.4) imply

[TABLE]

Since $\alpha^{2}+\beta^{2}=1$ and $\delta_{n}\in(0,1)$, we see, by Eq. (3.5), that

[TABLE]

and that

[TABLE]

Substituting Eq. (3.6) for Eqs. (3.7) and (3.8), we get the assertion in this lemma. $\hfill\Box$

Lemma 3.2

For any $y,z\in\mathbb{B}_{\pi}(o_{q_{1}})$,

[TABLE]

holds. In particular

[TABLE]

holds where $\mathop{\mathrm{Lip^{b}}}\nolimits(\widetilde{F})$ denotes the bi-Lipschitz constant of $\widetilde{F}$ defined by

[TABLE]

Proof.** **

Fix $x\in\mathbb{B}_{\pi}(o_{q_{1}})\setminus\{o_{q_{1}}\}$. Let $Y\in T_{\widetilde{F}(x)}(\mathbb{B}_{\pi}(o_{q_{2}}))$ with $Y\not=0$. Since $d\widetilde{F}_{x}$ is bijective, there is $X\in T_{x}(\mathbb{B}_{\pi}(o_{q_{1}}))$ with $X\not=0$ satisfying $d\widetilde{F}_{x}(X)=Y$. Since $\|d\widetilde{F}_{x}(X)\|=\|Y\|$, the left side one in the inequality (3.2) gives $(1-\delta_{n}/2)\|X\|\leq\|d\widetilde{F}_{x}(X)\|=\|Y\|$, and hence

[TABLE]

holds. Since $(d\widetilde{F}_{x})^{-1}=(d\widetilde{F}^{-1})_{\widetilde{F}(x)}$, we have $(d\widetilde{F}^{-1})_{\widetilde{F}(x)}(Y)=X$. From Eq. (3.11) we thus have

[TABLE]

We first prove the left side one in the assertion (3.9). Fix $\tilde{y},\tilde{z}\in\mathbb{B}_{\pi}(o_{q_{2}})$. We can assume $\tilde{y}\not=\tilde{z}$ in this aim. Let $\tilde{v}:=(\tilde{z}-\tilde{y})/\|\tilde{z}-\tilde{y}\|$ and $a:=\|\tilde{z}-\tilde{y}\|$. The geodesic segment $\widetilde{\gamma}:[0,a]\longrightarrow\mathbb{B}_{\pi}(o_{q_{2}})$ emanating from $\tilde{y}$ to $\tilde{z}$ is then given by $\widetilde{\gamma}(t):=\tilde{y}+t\tilde{v}$. Since $\widetilde{F}^{-1}$ is Lipschitzian, and since $\dot{\widetilde{\gamma}}(t)=\tilde{v}$, we see, by Eq. (3.12), that

[TABLE]

Set $y:=\widetilde{F}^{-1}(\tilde{y})$ and $z:=\widetilde{F}^{-1}(\tilde{z})$. Eq. (Proof.) then shows the desired inequality. An analogous argument gives the right side one in the inequality (3.9).

We finally prove Eq. (3.10). Since $\delta_{n}\in(0,1)$, we have

[TABLE]

and hence $(1+\delta_{n})^{-1}<(1-\delta_{n}/2)$. Since $1+\delta_{n}/2<1+\delta_{n}$, it follows from Eq. (3.9) that

[TABLE]

We therefore obtain Eq. (3.10) from Eq. (3.14).$\hfill\Box$

By applying the Nash embedding theorem [22] to $M_{2}$, let $M_{2}$ be isometrically embedded into the Euclidean space $\mathbb{R}^{m}$ where $m\geq n+1$. $F$ then is a Lipschitz map from $M_{1}$ to $M_{2}\subset\mathbb{R}^{m}$.

For any $\varepsilon>0$ sufficiently small let $\widetilde{F}_{\varepsilon}$ be the standard convolution of $\widetilde{F}$, i.e., $\widetilde{F}_{\varepsilon}(x):=\int_{\mathbb{R}^{n}}\widetilde{F}(y)\rho_{\varepsilon}(x-y)dy$ where the mollifier $\rho_{\varepsilon}$ near $o_{q_{1}}$, and we identify $T_{q_{1}}M_{1}$ with $\mathbb{R}^{n}$. Substituting Eq. (2.12) for $\delta_{n}$ in Eq. (3.10), we have

[TABLE]

In virtue of Eq. (3.15) we can apply the proof of [16, Theorem 5.1] to $\widetilde{F}_{\varepsilon}$, and hence we see, by the proof, that $\widetilde{F}_{\varepsilon}$ is an immersion from some open ball $\mathbb{B}_{a}(o_{q_{1}})\subset\mathbb{B}_{\pi}(o_{q_{1}})$ into $\mathbb{B}_{\pi}(o_{q_{2}})$. Let $B_{a}(q_{1}):=\exp_{q_{1}}\mathbb{B}_{a}(o_{q_{1}})$. Define the map $F_{\varepsilon}^{(q_{1})}$ from $B_{a}(q_{1})$ into $M_{2}$ by $F_{\varepsilon}^{(q_{1})}:=\exp_{q_{2}}\circ\widetilde{F}_{\varepsilon}\circ\{\exp_{q_{1}}^{-1}|_{B_{a}(q_{1})}\}$. From the definition of $\widetilde{F}$, we see that $F_{\varepsilon}^{(q_{1})}$ is a smooth approximation of $F$ on $B_{a}(q_{1})$. It is clear that $F_{\varepsilon}^{(q_{1})}$ is an immersion on $B_{a}(q_{1})$.

Let $g:M_{1}\longrightarrow\mathbb{R}$ be a smooth function satisfying $0\leq g\leq 1$ on $M_{1}$, $g\equiv 1$ on $\overline{B_{r}(q_{1})}$, and $\mathop{\mathrm{supp}}\nolimits g\subset B_{R}(q_{1})$ where $0<r<R<a$. Define the map $F_{\varepsilon}:M_{1}\longrightarrow\mathbb{R}^{m}$ by $F_{\varepsilon}:=(1-g)F+gF_{\varepsilon}^{(q_{1})}$.

Lemma 3.3

For any $x_{1}\in M_{1}$, $(dF_{\varepsilon})_{x}$ is injective for an $\varepsilon>0$ sufficiently small.

Proof.** **

From the definition of $F_{\varepsilon}$ we see $F_{\varepsilon}=F_{\varepsilon}^{(q_{1})}$ on $\overline{B_{r}(q_{1})}$ and $F_{\varepsilon}=F$ on $M_{1}\setminus\mathop{\mathrm{supp}}\nolimits g$, and hence $F_{\varepsilon}$ is a local diffeomorphism on $\overline{B_{r}(q_{1})}\cup(M_{1}\setminus\mathop{\mathrm{supp}}\nolimits g)$. Since $\widetilde{F}$ is smooth on $\mathbb{B}_{\pi}(o_{q_{1}})\setminus\{o_{q_{1}}\}$, the definition of the differential of a smooth map shows that $\widetilde{F}_{\varepsilon}$ uniformly converges to $\widetilde{F}$ on $\overline{\mathbb{B}_{R}(o_{q_{1}})}\setminus\mathbb{B}_{r}(o_{q_{1}})$ in the $C^{1}$-topology by letting $\varepsilon\downarrow 0$. Since $dF_{\varepsilon}^{(q_{1})}=d\exp_{q_{2}}\circ d\widetilde{F}_{\varepsilon}\circ d\exp_{q_{1}}^{-1}$ and $dF=d\exp_{q_{2}}\circ d\widetilde{F}\circ d\exp_{q_{1}}^{-1}$ where note that $F$ is differentiable on $M_{1}\setminus\{q_{1}\}$, we see, by the argument above, that $F_{\varepsilon}^{(q_{1})}$ uniformly converges to $F$ on $\overline{B_{R}(q_{1})}\setminus B_{r}(q_{1})$ in the $C^{1}$-topology by letting $\varepsilon\downarrow 0$. Since

[TABLE]

on $M_{1}$, and since

[TABLE]

for all $v\in T_{x}M_{1}$ ($x\in M_{1}\setminus\{q_{1}\}$), the second argument above shows that $F_{\varepsilon}$ uniformly converges to $F$ on $\overline{B_{R}(q_{1})}\setminus B_{r}(q_{1})$ in the $C^{1}$-topology by letting $\varepsilon\downarrow 0$. Since $F$ is diffeomorphic on $M_{1}\setminus\{q_{1}\}$, the third argument above implies that for any $x\in\overline{B_{R}(q_{1})}\setminus B_{r}(q_{1})$, $(dF_{\varepsilon})_{x}$ is injective for an $\varepsilon>0$ sufficiently small, and hence for any $x\in M_{1}$, $(dF_{\varepsilon})_{x}$ is too for such an $\varepsilon>0$. $\hfill\Box$

Since $M_{2}$ is isometrically embedded into $\mathbb{R}^{m}$, the tubular neighborhood theorem (cf. [14], [21]) via the normal exponential map $\exp^{\perp}:TM_{2}^{\perp}\longrightarrow\mathbb{R}^{m}$ shows that there is a constant $\mu>0$ such that $\exp^{\perp}$ is a diffeomorphism from an open neighborhood $\mathcal{U}_{\mu}(O(TM_{2}^{\perp}))$ of the zero section $O(TM_{2}^{\perp})$ onto an $\mathcal{U}_{\mu}(M_{2})$ of $M_{2}$ in $\mathbb{R}^{m}$ where the two sets are given by $\mathcal{U}_{\mu}(O(TM_{2}^{\perp})):=\{X\in TM_{2}^{\perp}\,|\,\|X\|<\mu\}$ and $\mathcal{U}_{\mu}(M_{2}):=\exp^{\perp}[\mathcal{U}_{\mu}(O(TM_{2}^{\perp}))]$. Since $\exp^{\perp}|_{\mathcal{U}_{\mu}(O(TM_{2}^{\perp}))}$ is bijective, for any $y\in\mathcal{U}_{\mu}(M_{2})$ there is a unique point $(x,v)\in\mathcal{U}_{\mu}(O(TM_{2}^{\perp}))$ such that $y=\exp^{\perp}(x,v)$. For such a pair $(y,(x,v))$ we thus have the smooth projection $\pi_{M_{2}}:\mathcal{U}_{\mu}(M_{2})\longrightarrow M_{2}$ given by $\pi_{M_{2}}(y)=\pi_{M_{2}}(\exp^{\perp}(x,v)):=x$. Since $M_{1}$ is compact, we see, by the definition of $F_{\varepsilon}$ and the proof of Lemma 3.3, that $\lim_{\varepsilon\downarrow 0}\|F_{\varepsilon}(p)-F(p)\|=0$ for all $p\in M_{1}$, which implies $F_{\varepsilon}(M_{1})\subset\mathcal{U}_{\mu}(M_{2})$ for an $\varepsilon>0$ sufficiently small. We can now define the smooth map $\psi_{\varepsilon}:M_{1}\longrightarrow M_{2}$ by

[TABLE]

for an $\varepsilon>0$ sufficiently small.

Lemma 3.4

$\psi_{\varepsilon_{0}}$* is a smooth immersion for an $\varepsilon_{0}>0$ sufficiently small.*

Proof.** **

Fix a sufficiently small $\varepsilon>0$ so that $\psi_{\varepsilon}$ is defined. Since $\dim{\rm Im}(dF_{\varepsilon})_{p}=n$ ($p\in M_{1}$) by Lemma 3.3, we shall show below that for each $p\in M_{1}$,

[TABLE]

As we have noted in the proof of Lemma 3.3, $F_{\varepsilon}=F_{\varepsilon}^{(q_{1})}$ on $\overline{B_{r}(q_{1})}$ and $F_{\varepsilon}=F$ on $M_{1}\setminus\mathop{\mathrm{supp}}\nolimits g$, and hence we have $\psi_{\varepsilon}(\overline{B_{r}(q_{1})})=F_{\varepsilon}(\overline{B_{r}(q_{1})})\subset M_{2}$ and $\psi_{\varepsilon}(M_{1}\setminus\mathop{\mathrm{supp}}\nolimits g)=F_{\varepsilon}(M_{1}\setminus\mathop{\mathrm{supp}}\nolimits g)\subset M_{2}$. So it is sufficient to show that Eq. (3.16) holds for all $p\in\overline{B_{R}(q_{1})}\setminus B_{r}(q_{1})$: We first see, by the definition of $\pi_{M_{2}}$, that

[TABLE]

for all $p\in\overline{B_{R}(q_{1})}\setminus B_{r}(q_{1})$ where $o_{F(p)}$ denotes the origin of $T_{F(p)}\mathbb{R}^{m}$. As we have seen in the proof of Lemma 3.3, $F_{\varepsilon}$ uniformly converges to $F$ on $\overline{B_{R}(q_{1})}\setminus B_{r}(q_{1})$ in the $C^{1}$-topology by letting $\varepsilon\downarrow 0$. From Eq. (3.17) and the similar argument as in [19, Section 5.2] we see that for an $\varepsilon_{0}\in(0,\varepsilon]$ sufficiently small,

[TABLE]

for all $p\in\overline{B_{R}(q_{1})}\setminus B_{r}(q_{1})$ where $o_{F_{\varepsilon_{0}}(p)}$ denotes the origin of $T_{F_{\varepsilon_{0}}(p)}\mathbb{R}^{m}$. Eq. (3.18) shows that Eq. (3.16) holds for all $p\in\overline{B_{R}(q_{1})}\setminus B_{r}(q_{1})$ and an $\varepsilon_{0}>0$ sufficiently small, which completes the proof.$\hfill\Box$

Finally we will show that $\psi_{\varepsilon_{0}}$ is a global diffeomorphism from $M_{1}$ onto $M_{2}$ for an $\varepsilon_{0}>0$ sufficiently small: Fix ${\varepsilon_{0}}>0$ sufficiently small so that $\psi_{\varepsilon_{0}}$ is a smooth immersion. Since $\psi_{\varepsilon_{0}}(M_{1})\subset M_{2}$ is compact, and since $M_{2}$ is Hausdorff, $\psi_{\varepsilon_{0}}(M_{1})$ is closed in $M_{2}$. Since $\psi_{\varepsilon_{0}}$ is a local homeomorphism on $M_{1}$ by Lemma 3.4, $\psi_{\varepsilon_{0}}(M_{1})$ is open in $M_{2}$. $\psi_{\varepsilon_{0}}(M_{1})$ is now open and closed in $M_{2}$, and hence $\psi_{\varepsilon_{0}}(M_{1})=M_{2}$, i.e., $\psi_{\varepsilon_{0}}$ is surjective. Since $\psi_{\varepsilon_{0}}^{-1}(V)$ is closed in $M_{1}$ for all closed sets $V$ in $M_{2}$, the compactness of $M_{i}$ ($i=1,2$) shows that $\psi_{\varepsilon_{0}}^{-1}(V)\subset M_{1}$ and $V\subset M_{2}$ are compact, which implies that $\psi_{\varepsilon_{0}}$ is a proper map, in particular, is a covering map. Since $M_{2}$ is simply connected, $\psi_{\varepsilon_{0}}$ is injective. Therefore, $\psi_{\varepsilon_{0}}$ is a global diffeomorphism from $M_{1}$ onto $M_{2}$.$\hfill\Box$