1 Introduction

Beginning with the Gauss-Bonnet theorem, sphere theorems show how geometry can be used to decide the topology of a manifold. By asking the 1/4-pinching question, which is now a theorem, Hopf opened a door to the study of sphere theorems. More previous results and history can be found in the survey papers [1,2,3] etc.

Continuing the paradigm, we will add two kinds of sphere theorems to enrich the subject. Details about the conditions will be given later.

Theorem A. Let (M, g) be an orientable closed Riemannian n-manifold with scalar curvature\(\ge n(n-1)\). Suppose that there exists a \((1, \wedge ^1)\)-contracting map \(f: M \rightarrow S^n\) of non-zero degree and that the map f is harmonic with condition \(C\le 0\), then f is an isometric map.

Remark 1

The idea of the proof is based on Chern-Goldberg’s argument. Their argument originated in S.S. Chern’s results that generalize Schwartz lemma to Hermitian manifolds in [6] and was used to form the Riemannian version of Schwarz lemma in [5]. In fact, the target of the map can be relaxed to general Einstein manifolds, which is shown by our proof.

Theorem B. Let \((X,d, \mathcal {H}^n)\) be a compact non-collapsed \(\mathrm {RCD}(n-1,n)\) space with full support and \(\mathrm {md}(X)\) be the mean distance of \((X,d, \mathcal {H}^n)\), then

  1. I.

    \(\mathrm {md}(X)\le \frac{\pi }{2}\).

  2. II.

    \(\mathrm {md}(X)= \frac{\pi }{2}\) if and only if X is isometric to the standard round n-sphere \(S^n\) and \(\mathcal {H}^n=a\mathrm {dVol}\) for some \(a>0\), where \(\mathrm {dVol}\) is the volume form of \(S^n\).

  3. III.

    there is an \(\epsilon (n)>0\) such that \(\mathrm {md}(X)\ge \frac{\pi }{2}-\epsilon (n)\) implies that X is homeomorphic to \(S^n\).

Remark 2

Mean distance is an old and well-studied metric invariant in graph theory, see [11]. Note that mean distance is also called distance covariance in probability, see [26].

Remark 3

We will use Ketterer’s maximal diameter theorem [21, Theorem 1.4] and Honda-Mondello’s topological sphere theorem [20, Theorem A] (for the metric measure space) to prove Theorem B. We also define a whole family of variants of the concept of mean distance for metric measure spaces and prove sphere theorems for all of them.

The paper is organized as follows. In Sect. 2, we introduce the notions of harmonic map with condition C and prove Theorem A. In Sect. 3, we collect facts about \(\mathrm {RCD}(n-1,n)\) spaces and show Theorem B.

Finished in May 2021, this note was dedicated to the \(110^{\mathrm {th}}\) anniversary of the birth of Shiing-Shen Chern.

2 Harmonic maps with condition C

Let \(S^n\) or \((S^n,g_{st})\) be the standard round n-sphere, then one has many rigidity theorems about \(S^n\). We will focus on the rigidity theorem with scalar curvature bounded below.

Definition 4

Let M and N be two Riemannian manifolds of the same dimension n. A smooth map \(f: M\rightarrow N\) is said to be \(\epsilon \)-contracting if \(\Vert f_*\nu \Vert \le \epsilon \Vert \nu \Vert \) for all tangent vectors \(\nu \) on M.

A smooth map \(f:M\rightarrow N\) is said to be \((\epsilon , \wedge ^k)\)-contracting if \(\Vert f^*\varphi \Vert \le \epsilon \Vert \varphi \Vert \), for all k-forms \(\varphi \in \wedge ^k(N)\) and \(1\le k\le n\).

Notice that 1-contracting and volume contracting with contraction constant 1 means \((1, \wedge ^1)\)- and \((1, \wedge ^n)\)-contracting, respectively. If f is \((1, \wedge ^p)\)-contracting, then f is \((1, \wedge ^{p+k})\)-contracting for \(k\ge 1\).

Theorem 5

(Llarull [24]) For all n, k, D with \(n\ge k\ge 3\) and \(D>0\), there exists a Riemannian n-manifold \(M^n\) with scalar curvature\(\ge D\) and a \((1, \wedge ^k)\)-contracting map \(f:M^n\rightarrow S^n\) of degree 1.

In particular, a volume decreasing and non-zero degree map for \(M^n\) (with scalar curvature bounded below) into \(S^n\) generally will not be an isometric map.

By confirming and generalizing Gromov’s conjecture [17], Llarull showed the following rigidity theorem using index theoretical method.

Theorem 6

(Llarull [25]) Let \((M^n,g)\) be a closed, connected, Riemannian spin n-manifold with scalar curvature\(\ge n(n-1)\). Assume that there exists a 1-contracting map or \((1, \wedge ^2)\)-contracting map \(f: M^n \rightarrow S^n\) of non-zero degree, then f is an isometric map.

Note that the spin condition on \(M^n\) can be relaxed to require that the map f is the spin map. \(S^n\) can be replaced by the Riemannian manifold, which satisfies a non-negative curvature operator and certain harmonic spinor conditions [19, Lemma 1.1]. Llarull’s rigidity theorem can also be generalized to the weighted rigidity theorem on weighted Riemannian manifold with positive weighted scalar curvature in [8, Proposition 4.21].

Remark 7

It is natural to appeal to use Schoen-Yau’s minimal surface method [31] to give another proof of Llarull’s rigidity theorem. Since Gromov-Lawson’s index theoretical approach [16] and Schoen-Yau’s minimal surface method are two of the fundamental methods of studying scalar curvature, one can try to apply one approach to give a proof of results that has been showed by the other.

Here we will use harmonic maps to approach the rigidity problem on scalar curvature. Furthermore, our argument can be applied to the rigidity question of the arbitrary Einstein manifolds.

We follow the setup and notations in [15]. Let M and N be Riemannian n-manifolds. Let \(ds^2_M\) and \(ds^2_N\) be the Riemannian metrics of M and N, respectively. Then we can write, locally,

$$\begin{aligned} ds^2_M=\omega ^2_1+ \cdots + \omega ^2_n, \quad ds^2_N= \omega ^{*2}_1+ \cdots +\omega ^{*2}_n, \end{aligned}$$

where \(\omega _i\) \((1\le i\le n)\) and \(\omega ^*_a\) \((1\le a \le n)\) are linear differential forms in M and N, respectively. The structure equations in M are

$$\begin{aligned}&d\omega _i=\sum \limits _{j} \omega _j\wedge \omega _{ji} \\&d\omega _{ij}= \sum \limits _{k}\omega _{ik}\wedge \omega _{kj} - \frac{1}{2}\sum \limits _{k,l}R_{ijkl}\omega _k\wedge \omega _l. \end{aligned}$$

The Ricci tensor \(R_{ij}\) is defined as

$$\begin{aligned} R_{ij}=\sum \limits _{k}R_{ikjk} \end{aligned}$$

and the scalar curvature is defined as

$$\begin{aligned} R=\sum \limits _{i}R_{ii}. \end{aligned}$$

Similar equations are valid in N and the corresponding quantities are denoted in the same notation with asterisks.

Let f be a \(C^{\infty }\)-mapping of M into N and

$$\begin{aligned} f^*\omega ^*_a=\sum \limits _{i}A^a_i\omega _i. \end{aligned}$$

Later, we will drop \(f^*\) from such formulas when its presence is clear in the context.

Let \(e_i\) (resp. \(e^*_a\)) be a frame that is dual to the coframe \(\omega _i\) (resp. \(\omega ^*_a\)), then we have

$$\begin{aligned} f_*e_i=\sum \limits _{a}A^a_ie^*_a. \end{aligned}$$

The covariant differential of \(A^a_i\) is defined by

$$\begin{aligned} DA^a_i:= dA^a_i+ \sum \limits _{j}A^a_j\omega _{ji}+\sum \limits _{b}A^b_i\omega ^*_{ba}:= \sum \limits _jA^a_{ij}\omega _j \end{aligned}$$
(1)

with

$$\begin{aligned} A^a_{ij}=A^a_{ji} \end{aligned}$$
(2)

The following geometrical interpretation of \(A^a_{ij}\) was given by Chern-Goldberg [5, P.136]: Let \(x\in M\) and let \(T_x\) and \(T_{f(x)}\) be the tangent spaces at x and f(x), respectively. The mapping

$$\begin{aligned} f_{**}: T_x\rightarrow T_{f(x)} \end{aligned}$$

defined by

$$\begin{aligned} f_{**}(\nu )=\sum \limits _{a,i,j}A^a_{ij}\lambda _i\lambda _je^*_a, \quad \nu =\sum \limits _{i}\lambda _ie_i \end{aligned}$$

is quadratic. The mapping has the property that if \(\nu \) is a unit vector, \(f_{**}(\nu )\) is the acceleration vector of \(f(\gamma )\) at f(x), where \(\gamma \) is the geodesic tangent to \(\nu \) at x.

Definition 8

(Harmonic maps) The mapping f is called harmonic if

$$\begin{aligned} \sum \limits _{i}A^a_{ii}=0. \end{aligned}$$

Notice that the tensor field with the components \(\sum \limits _{i}A^a_{ii}\) is the tension vector field of Eells-Sampson [13]. Eells and Sampson proved that the mapping f is harmonic (in Eells-Sampson’s sense) if and only if \(\sum \limits _{i}A^a_{ii}=0\).

Differentiating the Eq. (1) and using the structure equations in M and N, we get

$$\begin{aligned} \sum \limits _{j}DA^a_{ij}\wedge \omega _j=-\frac{1}{2}\sum _{j,k,l}A^a_lR_{jikl}\omega _k\wedge \omega _l - \frac{1}{2}\sum _{b,c,d}A^b_iR^*_{bacd}\omega ^*_c\wedge \omega ^*_d, \end{aligned}$$
(3)

where

$$\begin{aligned} DA^a_{ij}:= dA^a_{ij}+\sum _bA^b_{ij}\omega ^*_{ba}+\sum \limits _kA^a_{kj}\omega _{ki}+ \sum \limits _{k}A^a_{ik}\omega _{kj}:= \sum \limits _kA^a_{ijk}\omega _k. \end{aligned}$$

Thus, we have

$$\begin{aligned} A^a_{ijk}-A^a_{ikj}=-\sum \limits _{l}A^a_lR_{likj}-\sum \limits _{b,c,d}A^b_iA^c_kA^d_jR^*_{bacd}. \end{aligned}$$
(4)

According to (2) and (4), we can calculate the Laplacian

$$\begin{aligned} \Delta A^a_i:=\sum \limits _kA^a_{ikk}=\sum \limits _kA^a_{kik}=\sum \limits _kA^a_{kki}+\sum \limits _lA^a_lR_{li}-\sum \limits _{b,c,d,k}R^*_{bacd}A^b_kA^c_kA^d_i. \end{aligned}$$
(5)

The ration \( A:=\frac{f^*d\nu _N}{d\nu _M}\) of volume elements has the expression \(A=\mathrm {det}(A^a_i)\). Let \((B^i_a)\) be the adjoint of \((A^a_j)\), i.e., \(\sum \limits _aB^i_aA^a_j=\delta ^i_jA\). Then, by (1),

$$\begin{aligned} dA=\sum \limits _{i,a}B^i_adA^a_i=\sum \limits _{i,j,a}B^i_adA^a_{ij}\omega _j =: \sum \limits _j \mathbb {A}_j\omega _j \end{aligned}$$

Let \(V=A^2\), then we will compute the Laplacian of V and use it to prove Theorem A.

First, let

$$\begin{aligned} dV=\sum \limits _kV_k\omega _k, \end{aligned}$$
(6)

where

$$\begin{aligned} V_k=2A\sum \limits _{i,a}B^i_aA^a_{ik}. \end{aligned}$$
(7)

Second, exterior differentiation of (6) gives

$$\begin{aligned} \sum \limits _k(dV_k-\sum \limits _i V_i \omega _{ki})\wedge \omega _k =0. \end{aligned}$$

Hence we can write

$$\begin{aligned} dV_k-\sum \limits _i V_i\omega _{ki} =\sum \limits _jV_{kj}\omega _j, \end{aligned}$$
(8)

where

$$\begin{aligned} V_{jk}=V_{kj}. \end{aligned}$$

Then the Laplacian of V is by definition equal to

$$\begin{aligned} \Delta V=\sum \limits _kV_{kk}. \end{aligned}$$

Third, by differentiating (7), using (8), and simplifying, we get

$$\begin{aligned} \frac{1}{2}V_{kj}=2\mathbb {A}_j\mathbb {A}_k-\sum \limits _{i,l,a,b}B_a^iB_b^lA_{lj}^aA_{ik}^b+A\sum \limits _{i,a}B_a^iA_{ikj}^a. \end{aligned}$$

Using (5), the Laplacian of V can be given by

$$\begin{aligned} \frac{1}{2}\Delta V= 2\sum \limits _j(\mathbb {A}_j)^2+V(R-\sum \limits _{b,c,j}R^*_{b,c}A^b_jA^c_j)-C+A\sum \limits _{a,i,j}B^i_aA^a_{jji}, \end{aligned}$$
(9)

where C is a scalar invariant of the mapping given by

$$\begin{aligned} C=\sum \limits _{\mathop {a,b}\limits _{i,j,k}}B^i_aB^k_bA^a_{kj}A^b_{ij}. \end{aligned}$$
(10)

If f is harmonic, then the last term of (9) vanishes. That is, see [5, P.141, formula (57)],

$$\begin{aligned} \begin{aligned}&\text {(for a harmonic map)}\\ \frac{1}{2}\Delta V= 2\sum \limits _j(\mathbb {A}_j)^2+V(R-\sum \limits _{b,c,j}R^*_{b,c}A^b_jA^c_j)-C. \end{aligned} \end{aligned}$$
(11)

The geometric meaning of Condition C (10) was also given by Chern-Goldberg [5, P.142, Remark] as follows: the scalar C may be interpreted geometrically as a weighted measure of the deviation of the square length of the tensor \(C_{ijk}\) from the square length of its symmetric part, where \(C_{ijk}\) is the pullback of \(A^a_{ij}\) under f.

Proof of Theorem A

From the fact that the round sphere \(S^n\) is an Einstein manifold, we get

$$\begin{aligned} \sum \limits _{b,c,j}R^*_{b,c}A^b_jA^c_j=\frac{R^*}{n}\sum \limits _{a,i}(A^a_i)^2 \end{aligned}$$

Since M is compact and f is non-zero degree, V does attain its maximum at the point x in M. Then \(V(x)>0\) and \(\Delta V(x)\le 0\). Notice that V(x) is independent of the choice of the frame and coframe. At the point x, we choose a local g-orthonormal frame \(e_1, \dots , e_n\) on \(T_xM\) and a local \(g_{st}\)-orthonormal frame \(e^*_1,\dots , e^*_n\) on \(T_{f(x)}S^n\), such that there exists \(\lambda _1\ge \lambda _2\ge \cdots \ge \lambda _n>0\), with \(f_*e_i=\lambda _ie^*_i\). This can be done by diagonalizing \(f^*g_{st}\) with respect to the metric g. As f is \((1, \wedge ^1)\)-contracting, we have \(\lambda _i\le 1\) for all \(1\le i\le n\). That means

$$\begin{aligned} \sum \limits _{b,c,j}R^*_{b,c}A^b_jA^c_j=\frac{R^*}{n}\sum \limits _{a,i}(A^a_i)^2\le R^*. \end{aligned}$$
(12)

On account of \(R^*=n(n-1)\), \(R\ge n(n-1)\), and \(C\le 0\), we have

$$\begin{aligned} \frac{1}{2}\Delta V(x)\ge V(x)(R-n(n-1))\ge 0. \end{aligned}$$

Then by combining \(V(x)>0\) and \(\Delta V(x)\le 0\), we get \(R=n(n-1)\) and \(\lambda _i=1\) for all \(1\le i\le n\). That means f is an isometric map. \(\square \)

In fact, the target of the map can be relaxed to an arbitrary Einstein manifold as (12) shows.

Corollary 9

Let (N, g) be an orientable closed (Riemannian) Einstein n-manifold with scalar curvature \(R^*\) and (M, g) be an orientable closed Riemannian n-manifold with scalar curvature R.

Suppose \(R\ge R^*\), there exists a \((1, \wedge ^1)\)-contracting map \(f: M \rightarrow N\) of non-zero degree, and the map f is harmonic with condition \(C\le 0\), then f is a locally isometric map. If the fundamental group of N is trivial, then f is an isometric map.

Proof

As N is an Einstein manifold, then the inequality (12) and \(R\ge R^*\) implies f is a locally isometric map. \(\square \)

It is not clear whether the method of the poof in this section can be generalized to weighted Riemannian manifold with positive weighted scalar curvature in [8].

Conjecture 10

Let (M, g) be an orientable closed Riemannian n-manifold with scalar curvature\(\ge n(n-1)\). Suppose that there exists a \((1, \wedge ^1)\)-contracting map \(f: M \rightarrow S^n\) of non-zero degree, then f is an isometric map.

Remark 11

If Conjecture 10 is confirmed, then Theorem B in [9] holds for all dimensions. And then we can get a new obstruction of the existence of a Riemannian metric with positive scalar curvature on an arbitrary closed smooth manifold.

3 Non-collapsed RCD spaces and mean distance

The framework of Riemannian manifolds with Ricci curvature bounded below is generalized to the metric measure space \((X,d,\mu )\), which satisfies Riemannian curvature-dimension condition. And the metric measure spaces are called \(\mathrm {RCD}(K,N)\) spaces.

Definition 12

(Non-collapsed \(\mathrm {RCD}(K,n)\) spaces) Let \(K\in \mathbb {R}\), \(n\in \mathbb {N}\) and \(n\ge 1\). The metric measure space \((X,d,\mu )\) is called a non-collapsed \(\mathrm {RCD}(K,n)\) space, if

  1. 1.

    \((X,d,\mu )\) is an \(\mathrm {RCD}(K,n)\) space.

  2. 2.

    [Non-collapsed condition] \(\mu =\mathcal {H}^n\), where \(\mathcal {H}^n\) is the n-dimensional Hausdorff measure with respect to the metric d.

The concept of non-collapsed \(\mathrm {RCD}(K,N)\) space was defined by De Philippis and Gigli [12, Definition 1.1]. De Philippis-Gigli showed that N must be an integer. The proof of our theorem below is mainly based on De Philippis and Gigli’s results.

Definition 13

(Mean distance) Let \((X,d, \mu )\) be a compact metric measure space with full support, then the mean distance of X is defined as

$$\begin{aligned} \mathrm {md}(X):=\int \limits _{X\times X} \frac{ d(\cdot , \cdot )}{\mu (X)\times \mu (X)} \mu \otimes \mu \end{aligned}$$

Using Cheng’s maximal diameter theorem and Cheeger-Colding’s differentiable sphere theorem [4], Kokkendorff [22, Theorem 4] showed the following theorem.

Theorem 14

(Kokkendorff) Let \((M, g, d\mathrm {Vol}_g)\) be a Riemannian n-manifold \((n\ge 2)\) with Ricci curvature\(\ge n-1\), where \( d\mathrm {Vol}_g\) is the Riemannian volume, then

  1. 1.

    \(\mathrm {md}(M)\le \frac{\pi }{2}\)

  2. 2.

    \(\mathrm {md}(M)= \frac{\pi }{2}\) if and only if M is isometric to the standard round n-sphere \((S^n, d_{S^n})\).

  3. 3.

    there is an \(\epsilon '(n)>0\) such that \(\mathrm {md}(M)\ge \frac{\pi }{2}-\epsilon '(n)\) implies that M is diffeomorphic to \(S^n\).

We generalize Kokkendorff’s theorem to compact non-collapsed \(\mathrm {RCD}(n-1,n)\) spaces.

The strategy of the proof of Theorem B is the same as Kokkendorff’s proof, but we need to replace Cheng’s maximal diameter theorem and Cheeger-Colding’s differentiable sphere theorem (for the Riemannian manifold) with Ketterer’s maximal diameter theorem [21, Theorem 1.4] and Honda-Mondello’s topological sphere theorem [20, Theorem A] (for metric measure spaces).

The radius \(\mathrm {rad}(X,d)\) of a metric space (X, d) is defined as

$$\begin{aligned} \mathrm {rad}(X,d):=\inf \limits _{x\in X}\sup \limits _{y\in X}d(x,y). \end{aligned}$$

Before giving the proof of Theorem B, let us collect the relevant properties of a non-collapsed \(\mathrm {RCD}(n-1,n)\) space with full support so that we can use them in the proof.

  1. (1).

    The diameter \(\mathrm {diam}(X,d)\) is at most \(\pi \) (in particular \(\mathrm {rad}(X,d)\le \pi \)) [21, Theorem 4.8].

  2. (2).

    \((X,d, \mathcal {H}^n)\) satisfies the generalized Bishop-Gromov inequality:

    $$\begin{aligned} \frac{ \mathcal {H}^n(B_r(x))}{ \mathcal {H}^n(B_R(x))}\ge \frac{\mathrm {Vol}_{S^n}(B_r)}{\mathrm {Vol}_{S^n}(B_R)} \end{aligned}$$

    for any \(x\in X\) and \(0\le r \le R \le \pi \), where \(B_r(x)\) is the closed r-ball with center x in X, \(B_s\) is a closed s-ball in the sphere \(S^n\), and \(\mathrm {Vol}\) is the volume on the \(S^n\). Equivalently

    $$\begin{aligned} \frac{ \mathcal {H}^n(B_r(x))}{ \mathcal {H}^n(B_R(x))}\ge \frac{\int _0^r[\sin (t)]^{n-1}dt}{\int _0^R[\sin (t)]^{n-1}dt} \end{aligned}$$

    for any \(x\in X\) and \(0\le r \le R \le \pi \) [12].

  3. (3).

    [Ketterer’s maximal diameter theorem ] If \(\mathrm {rad}(X,d)=\pi \), then (X, d) is isometric to \((S^n, d_{S^n})\) [21, Theorem 1.4].

  4. (4).

    [Honda-Mondello’s topological sphere theorem] For all \(n\in \mathbb {N}_{\ge 2}\), there exists a positive constant \(\epsilon _1(n)>0\) such that if a compact metric space (X, d) satisfies that \(\mathrm {rad}(X,d)\ge \pi - \epsilon _1(n)\), and that \((X,d, \mu )\) is an \(\mathrm {RCD}(n-1,n)\) space for some Borel measure \(\mu \) on X with full support, then X is homeomorphic to the n-dimensional sphere [20, Theorem A].

Proof of Theorem B

Since the diameter of a non-collapsed \(\mathrm {RCD}(n-1,n)\) space is at most \(\pi \), we can apply the argument in the proof of Theorem 1.3 in [14] to prove I.

Let \(\mathrm {D}(x):=\sup \{d(x,y):y\in X\}\le \pi \) for a fixed \(x\in X\), \(h(y):=d(x,y)\), and \(\mu _0:=\frac{\mathcal {H}^n}{\mathcal {H}^n(X)}\), then we have

$$\begin{aligned} \mathrm {md}(x)&:= \int _Xh(y)\mu _0(dy)=\int _0^\infty \mu _0(\{h\ge s\})ds \end{aligned}$$
(13)
$$\begin{aligned}&=\int _0^{\mathrm {D}(x)}[1-\mu _0(h^{-1}([0,s))]ds=\int _0^{\mathrm {D}(x)} [1-\mu _0(B_s(x))]ds. \end{aligned}$$
(14)

We have the generalized Bishop-Gromov inequality for \(\mathrm {RCD}(n-1,n)\) space, i.e.,

$$\begin{aligned} \mu _0(B_s(x))\ge \frac{\mathrm {Vol}(B_s)}{\mathrm {Vol}(S^n)} \end{aligned}$$
(15)

for \(x\in X\) and \(0\le s\le \pi \). Hence, we get

$$\begin{aligned} \mathrm {md}(x)=\int _Xh(y)\mu _0(dy)\le \pi -\int _0^\pi \frac{\mathrm {Vol}(B_s)}{\mathrm {Vol}(S^n)}ds. \end{aligned}$$
(16)

The right hand side of the inequality (16) coincides with \(\mathrm {md}(p)\) for any \(p\in S^n\) by virtue of the formula in (14). As \(\mathrm {md}(S^n)=\frac{\pi }{2}\) was showed by Kokkendorff above, we have \(\mathrm {md}(x) \le \frac{\pi }{2}\). That means \(\mathrm {md}(X)\le \frac{\pi }{2}\).

If we have equality \(\mathrm {md}(X)=\frac{\pi }{2}\), then Bishop-Gromov inequality (15) must be equality for all \(0\le s\le \pi \). It implies that for \(\mu _0\)-a.e. point x, there must exist a point \(x'\) with \(d(x,x')=\pi \). And this must hold even for every \(x\in X\), because X is compact and \(\mu _0\) is full support. Therefore, \(\mathrm {rad}(X)=\pi \) and then (X, d) is isometric to \((S^n, d_{S^n})\) by Ketterer’s maximal diameter theorem.

We will prove III. by showing that \(\mathrm {md}(X)\) close to \(\frac{\pi }{2}\) implies that \(\mathrm {rad}(X,d)\) is closed to \(\pi \) and by applying Honda-Mondello’s topological sphere theorem.

We will prove the claim by contradiction. Suppose that there exists \(x_0\in X\) such that \(D(x_0)< \pi -\epsilon _1(n)\) and that the \(x_0\) realizes \(\mathrm {rad}(X,d)\). Here we take \(\epsilon _1(n)\) from Honda-Mondello’s topological sphere theorem. Then \(\mu _0(B_s(x_0))\) will achieve its maximum value 1 for \( \pi -\epsilon _1(n)\). Hence, we get an estimate

$$\begin{aligned} \mathrm {md}(x_0)&=\int _0^{\mathrm {D}(x)}[1-\mu _0(B_s(x))]ds\le \int _0^{\pi -\epsilon _1(n)}\left[ 1-\frac{\mathrm {Vol}(B_s)}{\mathrm {Vol}(S^n)}\right] ds\\&= \int _0^{\pi }\left[ 1-\frac{\mathrm {Vol}(B_s)}{\mathrm {Vol}(S^n)}\right] ds- \int _{\pi -\epsilon _1(n)}^\pi \left[ 1-\frac{\mathrm {Vol}(B_s)}{\mathrm {Vol}(S^n)}\right] ds\\&=:\frac{\pi }{2}- \delta (\epsilon _1(n),n). \end{aligned}$$

Since \(\mathrm {D}\) is 1-Lipschitz, we have

$$\begin{aligned} \mathrm {D}(y)< \pi - \frac{\epsilon _1(n)}{2} \end{aligned}$$

for any \(y\in B_{ \frac{\epsilon _1(n)}{2}}(x_0)\). This implies that we have the estimate

$$\begin{aligned} \mathrm {md}(y)<\frac{\pi }{2}- \delta \left( \frac{\epsilon _1(n)}{2},n\right) \end{aligned}$$

for any \(y\in B_{\frac{\epsilon _1(n)}{2}}(x_0)\). Finally, we have

$$\begin{aligned} \mathrm {md}(X)&=\int _X \mathrm {md}(x)\mu _0(dx)=\int _{X \setminus B_{ \frac{\epsilon _1(n)}{2}}(x_0)}\mathrm {md}(x)\mu _0(dx)\\&\quad + \int _{ B_{ \frac{\epsilon _1(n)}{2}}(x_0)}\mathrm {md}(x)\mu _0(dx)\\&< \frac{\pi }{2}- \mu _0(B_{\frac{\epsilon _1(n)}{2}}(x_0))\delta (\frac{\epsilon _1(n)}{2},n)\le \frac{\pi }{2}-\frac{\mathrm {Vol}(B_\frac{\epsilon _1(n)}{2})}{\mathrm {Vol}(S^n)}\delta (\frac{\epsilon _1(n)}{2},n). \end{aligned}$$

Now we take

$$\begin{aligned} \epsilon (n)=\frac{\mathrm {Vol}(B_\frac{\epsilon _1(n)}{2})}{\mathrm {Vol}(S^n)}\delta (\frac{\epsilon _1(n)}{2},n)>0. \end{aligned}$$

Then \(\mathrm {md}(X)\ge \frac{\pi }{2}-\epsilon (n)\) implies \(\mathrm {rad}(X,d)\ge \pi - \epsilon _1(n)\). Otherwise, it would be a contradiction. Then X is homeomorphic to the n-dimensional sphere by Honda-Mondello’s topological sphere theorem. \(\square \)

Remark 15

The definition of metric measure spaces with the measure contraction property \(\mathrm {MCP}(K, n)\) was given independently by Ohta [27] and Sturm [30]. On general metric measure spaces, the two definitions slightly differ, but on essentially non-branching spaces they coincide [7, Appendix A]. \(\mathrm {MCP}(K, n)\) spaces are another kind of generalization of Riemannian manifolds with Ricci curvature lower bound, but a measure contraction property is weaker than the usual curvature dimension conditions [30, Remark 5.6].

However, \(\mathrm {MCP}(K, n)\) spaces also satisfy generalized Bishop–Gromov inequality and the generalized Bonnet-Myers theorem [30, Remark 5.2] [27, Theorem 4.3; Theorem 5.1]. Therefore, Theorem B. I. also holds for a compact non-collapsed \(\mathrm {MCP}(n-1, n)\) space with full support.

Ketterer and Rajala show that the non-branching assumption is essential in Ohta’s topological rigidity [28, Theorem 5.5] in [23, Theorem 2]. It is not clear (for the author) whether the maximal diameter (rigidity) theorem holds for a compact non-collapsed and non-branching \(\mathrm {MCP}(n-1, n)\) space with full support.

Remark 16

A notion for lower bounds of Ricci curvature on Alexandrov spaces was introduced by Zhang-Zhu in [32, 33]. Bisho-Gromov inequality and maximal diameter theorem hold for Zhang-Zhu’s spaces [32, Corollary 5.1]. Thus, one can show that Theorem B. I. and II. hold for an n-dimensional Alexandrov space without boundary, and with full support and Ricci curvature\(\ge n-1\) in Zhang-Zhu’s sense.

Inspired by the concept of mean distance, one can define similar metric invariants. For a compact metric measure space \((X,d, \mu )\) with full support, given a continuous function \(\alpha :[0,\infty )\rightarrow \mathbb {R}\), we can define the metric invariant \(\mathrm {M_{\alpha }}(X)\) as

$$\begin{aligned} \mathrm {M_{\alpha }}(X):=\int \limits _{X\times X} \frac{\alpha ( d(\cdot , \cdot ))}{\mu (X)\times \mu (X)} \mu \otimes \mu \end{aligned}$$

And then one can try to generalize Theorem B to \(\mathrm {M_{\alpha }}(X)\) case.

In particular, Erbar and Sturm [14] defines

$$\begin{aligned} \mathrm {M_{f}}(X)&:=\int \limits _{X\times X} \frac{f( d(\cdot , \cdot ))}{\mu (X)\times \mu (X)} \mu \otimes \mu , \\ \mathrm {M_{f, n}^*}&:= \frac{\int _0^\pi f(r)[\sin (r)]^{n-1}dr }{\int _0^\pi [\sin (r)]^{n-1}dr}, \end{aligned}$$

where \(f:[0,\pi ]\rightarrow \mathbb {R}\) is a continuous and strictly increasing function.

Theorem 17

(Erbar-Sturm) Let \((X,d,\mu )\) be a compact non-collapsed \(\mathrm {RCD}(n-1,n)\) space with \(n\ge 1\), then \(\mathrm {M_{f}}(X)\le \mathrm {M_{f, n}^*}\) and \(\mathrm {M_{f}}(X)= \mathrm {M_{f, n}^*}\) if and only if X is isometric to the standard round n-sphere \(S^n\).

Erbar-Sturm also noticed that an analogous statement (with \(\mathrm {M_{f}}(X)\ge \mathrm {M_{f, n}^*}\) in place of \(\mathrm {M_{f}}(X)\le \mathrm {M_{f, n}^*}\)) holds for strictly decreasing f. Without loss of generality, let us assume that f is continuous and strictly increasing. Then we have the following corollary by combining Erbar-Sturm’s theorem and the proof of Theorem B.

Corollary 18

Let \((X,d, \mathcal {H}^n)\) be a compact non-collapsed \(\mathrm {RCD}(n-1,n)\) space with full support, then there is an \(\epsilon _2(f,n)>0\) such that \(\mathrm {M_{f}}(X)\ge \mathrm {M_{f, n}^*}-\epsilon _2(f,n)\) implies that X is homeomorphic to \(S^n\).

Proof

We only need to show that \(\mathrm {M_{f}}(X)\) close to \(\mathrm {M_{f, n}^*}\) implies that \(\mathrm {rad}(X,d)\) is closed to \(\pi \). We will prove the claim by contradiction.

Suppose that there exists \(x_0\in X\) such that \(D(x_0)< \pi -\epsilon _1(n)\) and the \(x_0\) realizes \(\mathrm {rad}(X,d)\). Here we take \(\epsilon _1(n)\) from Honda-Mondello’s topological sphere theorem. Let \(g(y):=f(d(x_0,y))\), \(\mu _0:=\frac{\mathcal {H}^n}{\mathcal {H}^n(X)}\), and \(\mathrm {M_{f}}(x_0):=\int _X g(y)\mu _0(dy)\), then \(\mu _0(B_s(x_0))\) will achieve its maximum value 1 for \( \pi -\epsilon _1(n)\).

Hence, we get an estimate

$$\begin{aligned} \mathrm {M_{f}}(x_0)< & {} \mathrm {M_{f, n}^*} - \int _{f(\pi -\epsilon _1(n))}^{f(\pi )}\left[ 1- \frac{\int _0^{f^{-1}(s)}[\sin (t)]^{n-1}dt}{\int _0^{\pi }[\sin (t)]^{n-1}dt}\right] ds\\:= & {} \mathrm {M_{f, n}^*}- \delta _1(f,\epsilon _1(n),n). \end{aligned}$$

Since \(\mathrm {D}\) is 1-Lipschitz and f is continuous and strictly increasing, we have

$$\begin{aligned} f(\mathrm {D}(y))< f\left( \pi - \frac{\epsilon _1(n)}{2}\right) \end{aligned}$$

for any \(y\in B_{ \frac{\epsilon _1(n)}{2}}(x_0)\). It implies that we have the estimate

$$\begin{aligned} \mathrm {M_{f}}(y)<\mathrm {M_{f, n}^*}- \delta _1\left( f, \frac{\epsilon _1(n)}{2}),n\right) \end{aligned}$$

for any \(y\in B_{\frac{\epsilon _1(n)}{2}}(x_0)\). Finally, we have

$$\begin{aligned} \mathrm {M_{f}}(X)&=\int _X \mathrm {M_{f}}(x)\mu _0(dx)=\int _{X \setminus B_{ \frac{\epsilon _1(n)}{2}}(x_0)} \mathrm {M_{f}}(x)\mu _0(dx)\\&\quad + \int _{ B_{ \frac{\epsilon _1(n)}{2}}(x_0)} \mathrm {M_{f}}(x)\mu _0(dx)\\&< \mathrm {M_{f, n}^*}- \mu _0(B_{\frac{\epsilon _1(n)}{2}}(x_0))\delta _1(f,\frac{\epsilon _1(n)}{2},n)\\&\le \mathrm {M_{f, n}^*}-\frac{\mathrm {Vol}(B_\frac{\epsilon _1(n)}{2})}{\mathrm {Vol}(S^n)}\delta _1(f, \frac{\epsilon _1(n)}{2},n). \end{aligned}$$

Now we take

$$\begin{aligned} \epsilon _2(f,n)=\frac{\mathrm {Vol}(B_\frac{\epsilon _1(n)}{2})}{\mathrm {Vol}(S^n)}\delta _1\left( f, \frac{\epsilon _1(n)}{2},n)\right) >0. \end{aligned}$$

Then \(\mathrm {M_{f}}(X)\ge \mathrm {M_{f, n}^*}-\epsilon (n)\) implies \(\mathrm {rad}(X,d)\ge \pi - \epsilon _1(n)\). Otherwise, it would be a contradiction. Then X is homeomorphic to the n-dimensional sphere by Honda-Mondello’s topological sphere theorem. \(\square \)

Remark 19

The result of Corollary 18 is even new (for \(f\ne \mathrm {id}\)) for a smooth Riemannian n-manifold with Ricci curvature bounded below by \((n-1)\) and X is diffeomorphic to \(S^n\) by Cheeger-Colding’s differentiable sphere theorem.

Gromov defines the observable diameter \(\mathrm {ObsDiam}(X;-\kappa )\) for metric measure spaces \((X,d,\mu )\) in [18] and shows the following theorem. The detailed proof can be found in [29, Theorem 2.29].

Theorem 20

(Gromov) Let X be a closed n-dimensional Riemannian manifold with Ricci curvature\(\ge (n-1)\). Then, for any \(\kappa \) with \(0<\kappa \le 1\), we have

$$\begin{aligned} \mathrm {ObsDiam}(X;-\kappa )\le \mathrm {ObsDiam}(S^n;-\kappa )=\pi -2v^{-1}(\frac{\kappa }{2}) \end{aligned}$$

where

$$\begin{aligned} v(r):=\frac{\int _0^r[\sin (t)]^{n-1}dt}{\int _0^{\pi }[\sin (t)]^{n-1}dt}. \end{aligned}$$

It is not clear whether Gromov’s theorem can be generalized to non-collapse \(\mathrm {RCD}(n-1,n)\) spaces. If it can, can we show the sphere theorem about it?

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