Some rigidity theorems on Finsler manifolds

1.   Introduction

In Riemannian geometry, Myers's theorem proves that if $(M, g)$ is a complete connected Riemannian $n$-manifold such that $\text{Ric}\geq (n-1)k$ for some positive number $k$, then it is compact and $Diam(M)\leq\frac{\pi}{\sqrt{k}}$. Further, Cheng's maximum diam theorem states that if its diameter attains its maximal value, then the manifold is isometric to the Euclidean sphere $\mathbb{S}^n(\frac{1}{\sqrt{k}})$. Under the same curvature condition, Lichnerowicz estimate indicates that the first closed eigenvalue of the Laplacian is not less than $nk$, while Obata rigidity theorem shows if the first closed eigenvalue attains its lower bound, then it is isometric to $\mathbb{S}^n(\frac{1}{\sqrt{k}})$. By the same token, Bishop-Gromov comparison theorem also demonstrates that if $\text{Vol}(M) = \text{Vol}(\mathbb{S}^n(\frac{1}{\sqrt{k}}))$, then it is isometric to $\mathbb{S}^n(\frac{1}{\sqrt{k}})$. Therefore, on a complete connected Riemannian $n$-manifold with $\text{Ric}\geq (n-1)k$ for some positive $k$, the following conditions are equivalent:

● $(M, g)$ is the Euclidean sphere $\mathbb{S}^n(\frac{1}{\sqrt{k}})$;

● $Diam(M) = \frac{\pi}{\sqrt{k}}$;

● the first eigenvalue of Laplacian is $\lambda_1(M) = nk$;

● $\text{Vol}(M) = \text{Vol} (\mathbb{S}^n(\frac{1}{\sqrt{k}}))$.

It is natural to ask:

In the Finsler setting, can we also characterize and determine Finsler spheres by the diam, the first eigenvalue and the volume?

In Finsler geometry, Cheng's maximum diam theorem is studied in ^[2] for reversible Finsler manifolds and in ^[10] for the general case. Lichnerowicz estimate and Obata rigidity theorem in Finsler situation are also considered in ^[8,9,10]. Along this line, the authors give a positive answer to the problem above. As to Bishop-Gromov comparison theorem, there are several generalized results established in ^[4,6,7,11], respectively. However, they (see ^[4,6,7]) do not study the rigidity phenomenon when the volume reaches its maximum value. In ^[11], Zhao-Shen study the rigidity problem and give some characterizations by using constant radial flag curvature and constant radial $S$ curvature. Yet, there is still much to be desired.

Let $\mathfrak{S}^n(\frac{1}{\sqrt{k}})$ denote a Finsler sphere which has Busemann Hausdorff volume form, constant flag curvature $k$ and vanishing $S$ curvature. When the Finsler metric is a Randers metric, the sphere is called a Randers sphere denoted by $\mathcal{S}^n(\frac{1}{\sqrt{k}})$. There are infinitely many Randers spheres, and if the metric is reversible, the sphere is just the Euclidean sphere (see ^[10]). For more details, we refer to Section 2 below.

In this paper, we characterize Finsler spheres and obtain some rigidity results in the following.

Theorem 1.1. Let $(M, F, d\mu)$ be a complete connected Finsler $n$-manifold with Busemann-Hausdorff volume form. If the weighted Ricci curvature ${\rm{Ric}}_n\geq (n-1)k > 0$, then the following conditions are equivalent:

(1) $(M, F, d\mu)$ is a Finsler sphere $\mathfrak{S}^n(\frac{1}{\sqrt{k}})$;

(2) $Diam(M) = \frac{\pi}{\sqrt{k}}$;

(3) the first eigenvalue of Finsler-Laplacian is $\lambda_1 = nk$;

(4) ${\rm{vol}}_F^{d\mu}(M) = {\rm{vol}}(\mathbb{S}^n(\frac{1}{\sqrt{k}}))$.

Since the classification for Finsler metrics with constant flag curvature is not solved, we can not determine all Finsler sphere metrics. However, for a Randers sphere $\mathcal{S}^n(\frac{1}{\sqrt{k}})$, the metric $F$ can be expressed by navigation data $(\mathfrak{g}, W)$, where $\mathfrak{g}$ is the standard sphere metric and $W$ is a Killing vector. Therefore, Randers spheres are of most importance among Finsler spheres. By narrowing the scope in Theorem 1.1, we have

Theorem 1.2. Let $(M, F, d\mu)$ be a complete connected Randers $n$-manifold with Busemann-Hausdorff volume form. If the weighted Ricci curvature ${\rm{Ric}}_n\geq (n-1)k > 0$, then the following conditions are equivalent:

(1) $(M, F, d\mu)$ is a Randers sphere $\mathcal{S}^n(\frac{1}{\sqrt{k}})$;

(2) $Diam(M) = \frac{\pi}{\sqrt{k}}$;

(3) the first eigenvalue of Finsler-Laplacian is $\lambda_1 = nk$;

(4) ${\rm{vol}}_F^{d\mu}(M) = {\rm{vol}}(\mathbb{S}^n(\frac{1}{\sqrt{k}}))$.

Remark 1.3. Theorems 1.1 and 1.2 show that, apart from the Euclid sphere, the maximum diameter, the maximum volume and the lower bound of the first eigenvalue can be attained on countless Finsler (especially Randers) spheres.

The paper is organized as follows. In Section 2, some fundamental concepts and formulas which are necessary for the present paper are given, and some lemmas are contained. The volume comparison theorem and Theorem 1.1 are then proved in Sections 3 and 4, repectively.

2.   Preliminaries

Let $M$ be an $n$-dimensional smooth manifold and $\pi :TM\to M$ be the natural projection from the tangent bundle $TM$. Let $(x, y)$ be a point of $TM$ with $x\in M$, $y\in T_xM$, and let $(x^i, y^i)$ be the local coordinates on $TM$ with $y = y^i\partial /\partial x^i$. A Finsler metric on $M$ is a function $F:TM\to [0, +\infty)$ satisfying the following properties:

(ⅰ) Regularity: $F(x, y)$ is smooth in $TM\setminus 0$;

(ⅱ) Positive homogeneity: $F(x, \lambda y) = \lambda F(x, y)$ for $\lambda > 0$;

(ⅲ) Strong convexity: The fundamental quadratic form

is positively definite.

Let $X = X^{i}\frac{\partial}{\partial x^{i}}$ be a differentiable vector field. Then the covariant derivative of $X$ by $v\in T_{x}M$ with reference vector $w\in T_{x}M\backslash 0$ is defined by

where $\Gamma^{i}_{jk}$ denote the coefficients of the Chern connection.

Given two linearly independent vectors $V, W\in T_{x}M\backslash0$, the flag curvature is defined by

where $R^{V}$ is the Chern curvature:

Then the Ricci curvature for $(M, F)$ is defined as

where $e_{1}, \cdots, e_{n-1}, \frac{V}{F(V)}$ form an orthonormal basis of $T_{x}M$ with respect to $g_{V}$.

Let $(M, F, d\mu)$ be a Finsler $n$-manifold with $d\mu = \sigma(x)dx^1\wedge\cdots \wedge dx^n$. The distortion is given by

For the vector $V\in T_xM$, let $\gamma :(-\varepsilon, \varepsilon)\to M$ be a geodesic with $\gamma (0) = x, \; \dot\gamma (0) = V$. Then the $S$-curvature measures the rate of changes of the distortion along geodesics

Define

Then the weighted Ricci curvature of $(M, F, d\mu)$ is defined by (see ^[3])

For a smooth function $u$, the gradient vector of $u$ at $x$ is defined by $\nabla u(x): = \mathcal{L}^{-1}(du)$, where $\mathcal{L}:T_xM\to T_x^*M$ is the Legendre transform. Let $V = V^{i}\frac{\partial}{\partial x^{i}}$ be a smooth vector field on $M$. The divergence of $V$ with respect to an arbitrary volume form $d\mu$ is defined by

Then the Finsler-Laplacian of $u$ can be defined by

The equality is in the weak $W^{1, 2}(M)$ sense. Namely, for any $\varphi\in C^{\infty}_0(M)$, we have

Recall that Bao-Shen ^[1] found a family Randers sphere metrics on $\mathbb{S}^3$, which shows that the maximum diam can be achieved in non-Riemannian case.

Example 2.1. ^[1] View $\mathbb{S}^3$ as a compact Lie group. Let $\zeta^1, \zeta^2, \zeta^3$ be the standard right invariant 1-form on $\mathbb{S}^3$ satisfying

For $k\geq1$, define

Then $F_k = \alpha_k+\beta_k$ is a Randers metric on $\mathbb{S}^3$ satisfying

Inspired by Bao-Shen's example, we give the following definition.

Definition 2.2. A Finsler manifold with Busemann-Hausdorff volume form is said to be a Finsler sphere if it has positively constant flag curvature and vanishing $S$-curvature. In particular, if the metric is a Randers metric, we call it a Randers sphere.

Generally, we are not able to identify a Finsler sphere metric. However, for a Randers sphere metric, $F$ can be expressed by (see ^[10])

where $\mathfrak{g}$ is the standard sphere metric, and $W$ is a Killing vector field on $\mathbb{S}^n(\frac{1}{\sqrt{k}})$.

To prove our results, we further introduce the following lemmas.

Lemma 2.3. ^[4] Let $(M, F, d\mu)$ be a Finsler n manifold. If the weighted Ricci curvature satisfies ${\rm{Ric}}_{N}\geq (N-1)k, N\in[n, \infty)$, then the Laplacian of the distance function $r(x) = d_{F}(p, x)$ from any given point $p\in M$ can be estimated as follows:

pointwise on $M\backslash (\{p\}\cup \text{Cut}(p))$ and in the sense of distributions on $M\backslash \{p\}$, where

Lemma 2.4. ^[3] Let $(M, F, d\mu)$ be a Finsler n manifold. If the weighted Ricci curvature satisfies ${\rm{Ric}}_{N}\geq (N-1)k, N\in[n, \infty)$, then for any $0 < r < R$ ($\leq\frac{\pi}{\sqrt{k}}$ if $k > 0$), it holds that

3.   Volume comparison theorems

In ^[6,7], the volume comparison theorems are established on Finsler manifolds satisfying $\text{Ric}\geq(n-1)k$ and some $S$ curvature condition. Later, Zhao-Shen ^[11] generalize them and further obtain the rigidity result. Using the weighted Ricci curvature condition, Ohta ^[3] obtain another version of the relative volume comparison (see Lemma 2.4 above). However, the problem about the rigidity result remains open. The main obstacle is that, for any volume form, the limit

does not necessarily exists. Therefore, to obtain the rigidity result, it is suitable to give a restriction on the volume form.

Let $(M, F, d\mu)$ be a Finsler $n$-manifold. Fix a point $p\in M$. Then on $T_pM$ the Finsler metric $F(p, y)$ induces a Riemannian metric $g_p(y): = g_{ij}(p, y)dy^i\otimes dy^j$, which also induces a Riemannian metric $\dot{g}_p$ on $S_pM: = \{y|y\in T_pM, F(y) = 1\}$. Let $(r, \theta)$ be the polar coordinate around $p$ and write the volume form by $d\mu = \sigma_p(r, \theta)drd\theta$. Then we can give the volume comparison theorem as follows:

Theorem 3.1. Let $(M, F, d\mu)$ be a forward complete Finsler n-manifold with arbitrary volume form. If the weighted Ricci curvature ${\rm{Ric}}_n\geq(n-1)k$, then for some positive number $C_p: = \lim\limits_{r\to0}\int_{S_pM}\frac{\sigma_p(r, \theta)}{r^{n-1}}d\theta$,

where $B_p^+(r)$ denotes the forward geodesic ball centered at $p$ of radius $r$, and $\textbf{i}_p$ is the cut value of $p$. Moreover, the equality holds for $r_0 > 0$ if and only if for $\forall y\in S_pM$,

where $\gamma_y(t)$ is the geodesic satisfying $\gamma_y(0) = p, \dot{\gamma}_y(0) = y$. In this case, under the polar coordinate $(r, \theta)$ of $p$, we have

where $\dot{g}_p$ denotes the Riemannian metric on $S_pM$.

Proof. From Lemma 2.4, we have

is monotone decreasing on $r$. By

we obtain (3.1).

Let $r(x) = d_F(p, x)$ be the distance function from $p$. Since $\text{Ric}_n\geq(n-1)k$, by Laplacian comparison theorem (Lemma 2.3), we have

which yields

where $\tilde{\sigma}: = \textbf{s}_k(r)^{n-1}$. Define $f(r) = \frac{\sigma_p(r, \theta)}{\tilde{\sigma}(r)}$. Then

Hence, $f(r)$ is monotone decreasing on $r$. As a result,

Assume that the equality holds in (3.1). That is,

Differentiating it with respect to $r$ on both sides gives

By the monotonicity of $f(r)$, we deduce that

which implies

Thus the equality holds in (3.2), which gives

Let $S_{p}(r(x))$ be the forward geodesic sphere of radius $r(x)$ centered at $p$. Choosing the local $g_{\nabla r}$-orthonormal frame $E_{1}, \cdots, E_{n-1}$ of $S_{p}(r(x))$ near $x$, we get local vector fields $E_{1}, \cdots, E_{n-1}, E_{n} = \nabla r$ by parallel transport along geodesic rays. Thus, it follows from ^[7] that

where $H(r)$ is the Hessian of the distance function $r$. On the other hand, we also have (see ^[7])

Therefore, by (3.4) and (3.5), we obtain

It follows from (3.6) that

which means

where $\textbf{ct}_k(r): = \frac{\textbf{s}_k'(r)}{\textbf{s}_k(r)}$. Next we shall compute the flag curvature. From (3.7), we know that $\{E_i\}_{i = 1}^{n-1}$ are $(n-1)$ eigenvectors of $\nabla^2r$. That is,

Noticed that $\nabla r$ is a geodesic field of $(M, F)$. Therefore, the flag curvature $K(\nabla r; \cdot)$ equals to the sectional curvature of the weighted Riemannian manifold $(M, g_{\nabla r})$. Note that $\{E_i\}_{i = 1}^{n-1}$ are $(n-1)$ eigenvectors of $\nabla^2r$ and parallel along the geodesic ray. By a straightforward computation, we get, for $1\leq i\leq n-1$,

We are now to prove that if $K(\dot{\gamma}_y(t); \cdot) = k$, then the equality holds in (3.1). Under the polar coordinate of $p$, we have $(r, \theta) = (r(q), \theta^1(q), \cdots, \theta^{n-1}(q))$ for $q\in \mathcal{D}_p\backslash\{p\}$, where

Then

So, for $y\in S_pM$, $\frac{\partial}{\partial\theta^{\alpha}}$ can be viewed as a Jacobi field on $\gamma_y(t)$, and

Since $K(\nabla r; \cdot) = k$, $J_{\alpha}(t) = \frac{\partial}{\partial\theta^{\alpha}}|_{(t, y)} = \textbf{s}_k(t)E_{\alpha}(t)$ is the Jacobi field satisfying $J(0) = 0$, where $E_{\alpha}(t)$ is a parallel vector field on $\gamma_y(t)$, and $E_{\alpha}(0) = \frac{\partial}{\partial\bar{\theta}^{\alpha}}|_y.$ By Gauss lemma, $g_{\nabla r}(\nabla r, \frac{\partial}{\partial\theta^{\alpha}}) = 0$. Therefore,

Since $S(\dot{\gamma}_y(t)) = 0$, we have

which means that

Thus, we obtain

That is,

From Theorem 3.1, it is easy to obtain the following:

Corollary 3.2. Let $(M, F, d\mu)$ be a forward complete Finsler n-manifold with arbitrary volume form. If the weighted Ricci curvature ${\rm{Ric}}_n\geq(n-1)k$, and there exists some positive number $C$ such that $\lim\limits_{r\to0}\int_{S_pM}\frac{\sigma_p(r, \theta)}{r^{n-1}}d\theta = C$ for $\forall p\in M$, then

if and only if $K\equiv k$.

4.   Rigidity results on Finsler manifolds

Recall that, for a Randers sphere $(\mathcal{S}^n(\frac{1}{\sqrt{k}}))$, the metric $F$ is expressed by (see ^[10])

where $\mathfrak{g}$ is the standard sphere metric, $W$ is a Killing vector field on $\mathbb{S}^n(\frac{1}{\sqrt{k}})$. Moreover, we have (see ^[10])

In what follows, we show that the properties above still hold for a general Finsler sphere.

Proposition 4.1. On a Finsler sphere $\mathfrak{S}^n(\frac{1}{\sqrt{k}})$, we have

(1) ${\rm{vol}}^{d\mu}_F(\mathfrak{S}^n(\frac{1}{\sqrt{k}})) = vol_\mathfrak{g}(\mathbb{S}^n(\frac{1}{\sqrt{k}}))$;

(2) $Diam(\mathfrak{S}^n(\frac{1}{\sqrt{k}})) = \frac{\pi}{\sqrt{k}}$.

Proof. Since $d\mu$ is the Busemann-Hausdorff volume form, the constant $C_p$ in (3.1) is

Thus, from Corollary 3.2, we have

Now fix $p\in \mathfrak{S}^n(\frac{1}{\sqrt{k}})$. Using $K = k$ and Theorem 3.1 in ^[5], there exists $q\in \mathfrak{S}^n(\frac{1}{\sqrt{k}})$ such that

where $S_p(\mathfrak{S}^n(\frac{1}{\sqrt{k}})): = \{v|v\in T_p(\mathfrak{S}^n(\frac{1}{\sqrt{k}})), F(v) = 1\}$. From the proof of the volume comparison theorem (Theorem 3.1),

where $\sigma_n(r)$ denotes the volume of the metric ball of radius $r$ in $\mathbb{S}^n(\frac{1}{\sqrt{k}})$. The equality holds if and only if $B^+_p(r)\subset \mathcal{D}_p$, i.e., ${\bf{i}}_p\geq r$. By the Bonnet-Myers theorem, $Diam(\mathfrak{S}^n(\frac{1}{\sqrt{k}}))\leq\frac{\pi}{\sqrt{k}}$, which means $\overline{B^+_p(\frac{\pi}{\sqrt{k}})} = \mathfrak{S}^n(\frac{1}{\sqrt{k}})$. Therefore,

We deduce that ${\bf{i}}_p\geq \frac{\pi}{\sqrt{k}}$, which yields $d_F(p, q) = \frac{\pi}{\sqrt{k}}$.

Theorem 4.2. Let $(M, F, d\mu)$ be a complete connected Finsler n-manifold with Busemann-Hausdorff volume form. If the weighted Ricci curvature ${\rm{Ric}}_n\geq(n-1)k > 0$, and ${\rm{vol}}_F^{d\mu}(M) = vol_{\mathfrak{g}}(\mathbb{S}^n(\frac{1}{\sqrt{k}}))$, then $(M, F)$ is isometric to a Finsler sphere.

Proof. Since $d\mu$ is the Busemann-Hausdorff volume form, the constant $C_p$ in (3.1) is $C_p = \text{vol}(\mathbb{S}^{n-1}).$ Then it follows Theorem 3.1 that, for any $p\in M$,

where $\sigma_n(r)$ denotes the volume of the metric ball of radius $r$ in $\mathbb{S}^n(\frac{1}{\sqrt{k}})$. The equality holds if and only if $B^+_p(r)\subset \mathcal{D}_p$. That is, ${\bf{i}}_p\geq r$. Since

we deduce that, for $\forall p\in M$ and $\forall r\leq{\bf{i}}_p$,

Then, from Corollary 3.2, we have

This completes the proof.

To prove Theorem 1.1, we further need the following theorems.

Theorem 4.3. ^[10] Let $(M, F, d\mu)$ be a complete connected Finsler n-manifold with the Busemann-Hausdorff volume form. If the weighted Ricci curvature satisfies ${\rm{Ric}}_n\geq(n-1)k > 0$, and $Diam(M) = \frac{\pi}{\sqrt{k}}$, then $(M, F)$ is isometric to a Finsler sphere.

Theorem 4.4. ^[10] Let $(M, F, d\mu)$ be a complete connected Finsler n-manifold with the Busemann-Hausdorff volume form. If the weighted Ricci curvature satisfies ${\rm{Ric}}_n\geq(n-1)k > 0$, then the first eigenvalue of Finsler-Laplacian $\lambda_1 = nk$ if and only if $(M, F)$ is isometric to a Finsler sphere.

Proof of Theorem 1.1.

It follows from Proposition 4.1, Theorems 4.2–4.4 directly.

From the proof of Theorem 3.1, we know that the key step is

Therefore, we get another equivalent condition.

Theorem 4.5. Let $(M, F, d\mu)$ be a complete connected Finsler $n$-manifold with Busemann-Hausdorff volume form. If the weighted Ricci curvature ${\rm{Ric}}_n\geq (n-1)k > 0$, then the following conditions are equivalent:

(1) $(M, F, d\mu)$ is a Finsler sphere $\mathfrak{S}^n(\frac{1}{\sqrt{k}})$;

(2) $\Delta r = (n-1)\cot r$ for any distance function $r(x) = d_F(p, x), \forall p\in M$.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 11971253).

Conflict of interest

The author declares that they have no conflicts of interest.