Preserving cultural landscapes in the face of globalization. The musealization of Sicilian heritage

Emanuela Caravello; Emanuela Caravello

doi:10.3934/geosci.2023037

AIMS Geosciences

2023, Volume 9, Issue 4: 697-712. doi: 10.3934/geosci.2023037

Previous Article Next Article

Research article Special Issues

Preserving cultural landscapes in the face of globalization. The musealization of Sicilian heritage

Emanuela Caravello ^,

Department of Psychology, Educational Science and Human Movement (SPPEFF), University of Palermo, Palermo, Italy

Received: 17 October 2023 Revised: 06 November 2023 Accepted: 06 November 2023 Published: 13 November 2023

This contribution critically reflects the musealization of landscapes as an effective response to the rapid transformations brought about by globalization. Focusing on the case of Sicilian heritage, we examine how the conservation and representation of traditional landscapes in museums serve as a defensive reaction to the perceived threats of homogenization and cultural loss caused by global processes. This article fits into the debate on cultural landscapes and outlines the protection policies implemented by UNESCO while delving into the role of museum collections. In this specific context of preserving the tangible and intangible components of cultural heritage, the landscape is intertwined with the role of local communities in a changing world. We also explore the concept of authenticity in landscapes and its significance in preserving cultural identities. Through qualitative methodology involving critical analysis of literature and document examination, the research illustrates how the musealization of Sicilian landscapes has aimed to safeguard collective memory and cultural heritage. However, this article also highlights potential risks associated with this process, such as the static representation of dynamic cultures and the selective nature of museum curation. Ultimately, this study advocates for transparent and multifaceted interpretations of cultural landscapes to avoid the creation of artificial myths and to preserve the polysemy of the landscape's meaning. By critically examining the context through significant examples, this paper contributes to a deeper understanding of the role of musealization in conserving and representing cultural landscapes in the face of globalization's challenges.

Keywords:

Citation: Emanuela Caravello. Preserving cultural landscapes in the face of globalization. The musealization of Sicilian heritage[J]. AIMS Geosciences, 2023, 9(4): 697-712. doi: 10.3934/geosci.2023037

Related Papers:

[1]	Jin Liu, Botao Wang . A rigidity result for $2$ -dimensional $\lambda$ -translators. AIMS Mathematics, 2023, 8(10): 24947-24956. doi: 10.3934/math.20231272
[2]	Derya Sağlam, Cumali Sunar . Translation hypersurfaces of semi-Euclidean spaces with constant scalar curvature. AIMS Mathematics, 2023, 8(2): 5036-5048. doi: 10.3934/math.2023252
[3]	Kunquan Li . Analytical solutions and asymptotic behaviors to the vacuum free boundary problem for 2D Navier-Stokes equations with degenerate viscosity. AIMS Mathematics, 2024, 9(5): 12412-12432. doi: 10.3934/math.2024607
[4]	Rajesh Kumar, Sameh Shenawy, Lalnunenga Colney, Nasser Bin Turki . Certain results on tangent bundle endowed with generalized Tanaka Webster connection (GTWC) on Kenmotsu manifolds. AIMS Mathematics, 2024, 9(11): 30364-30383. doi: 10.3934/math.20241465
[5]	Tong Wu, Yong Wang . Super warped products with a semi-symmetric non-metric connection. AIMS Mathematics, 2022, 7(6): 10534-10553. doi: 10.3934/math.2022587
[6]	Yayun Chen, Tongzhu Li . Classification of spacelike conformal Einstein hypersurfaces in Lorentzian space $\mathbb{R}^{n+1}_1$ . AIMS Mathematics, 2023, 8(10): 23247-23271. doi: 10.3934/math.20231182
[7]	Mingyu Zhang . On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity. AIMS Mathematics, 2024, 9(9): 23313-23330. doi: 10.3934/math.20241133
[8]	Wenjing Song, Haiyun Deng, Ganshan Yang . Two-uniqueness of rational ghost soliton solution and well-posedness of perturbed Einstein-Yang-Mills equations. AIMS Mathematics, 2021, 6(11): 12065-12076. doi: 10.3934/math.2021699
[9]	Yanlin Li, Nasser Bin Turki, Sharief Deshmukh, Olga Belova . Euclidean hypersurfaces isometric to spheres. AIMS Mathematics, 2024, 9(10): 28306-28319. doi: 10.3934/math.20241373
[10]	Yongkun Li, Xiaoli Huang, Xiaohui Wang . Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays. AIMS Mathematics, 2022, 7(4): 4861-4886. doi: 10.3934/math.2022271

Abstract

1. Introduction

Let $(M^{n}, g)$ be an $n$ -dimensional Riemannian manifold with the metric $g$ and the dimension $n\geq3$ . If there exists a non-constant smooth function $f$ such that

$\begin{equation} f_{ij} = f\Big(R_{ij}-\frac{1}{n-1}R g_{ij}\Big), \end{equation}$

(1.1)

then $(M^{n}, g, f)$ is called a vacuum static space (for more backgrounds, see ^[8,10,19,23]). Here $f_{ij}$ , $R_{ij}$ and $R$ denote components of the Hessian of $f$ , the Ricci curvature tensor and the scalar curvature, respectively. In ^[8], Fischer-Marsden proposed the following conjecture: The standard spheres are the only $n$ -dimensional compact vacuum static spaces. In ^[18], Kobayashi gave a classification for $n$ -dimensional complete vacuum static spaces that are locally conformally flat. On the other hand, he and Lafontaine ^[20] also provided some counterexamples for the above conjecture.

In fact, according to the second Bianchi identity, any vacuum static space has constant scalar curvature. Moreover, Bourguignon ^[2] and Fischer-Marsden ^[8] have proved that the set $f^{-1}(0)$ has the measure zero and the set $f^{-1}(0)$ is a totally geodesic regular hypersurface.

Let $\mathring{R}_{ij} = R_{ij}-\frac{R}{n} g_{ij}$ be the trace-free Ricci curvature, then (1.1) can be written as

$\begin{equation} f_{ij} = f\mathring{R}_{ij}-\frac{R}{n(n-1)} fg_{ij}, \end{equation}$

(1.2)

which gives

$\Delta f = -\frac{R}{n-1}f.$

It is well known that the Weyl curvature tensor $W$ and the Riemannian curvature tensor is related by

$\begin{aligned} R_{ijkl} = W_{ijkl}+\frac{1}{n-2}(R_{ik}g_{jl}-R_{il}g_{jk} +R_{jl}g_{ik}-R_{jk}g_{il}) -\frac{R}{(n-1)(n-2)}(g_{ik}g_{jl}-g_{il}g_{jk}). \end{aligned}$

In this paper, we consider rigidity results for closed vacuum static spaces. By using the maximum principle, some rigidity theorems are obtained under some pointwise inequalities and show that the squared norm of the Ricci curvature tensor is discrete.

Theorem 1.1. Let $(M^n, g, f)$ be a closed vacuum static space with the positive scalar curvature and $f_{l}W_{lijk} = 0$ (that is, zero radial Weyl curvature), where $n\geq4$ . If

$\begin{equation} \frac{(n-1)(n-2)}{2}|W|^2+n(n-1)|\mathring{R}_{ij}|^2\leq R^2, \end{equation}$

(1.3)

then it must be of Einstein as long as there exists a point such that the inequality in (1.3) is strict.

Next, by substituting (1.3) with a stronger condition, we can obtain the following characterizations:

Theorem 1.2. Let $(M^n, g, f)$ be a closed vacuum static space with the positive scalar curvature and $f_{l}W_{lijk} = 0$ (that is, zero radial Weyl curvature), where $n\geq4$ . If

$\begin{equation} \sqrt{\frac{(n-1)(n-2)}{2}}|W|+\sqrt{n(n-1)}|\mathring{R}_{ij}|\leq R, \end{equation}$

(1.4)

then it must be of Einstein or a Riemannian product $\mathbb{S}^1\times \mathbb{S}^{n-1}$ . In particular, it must be of Einstein as long as there exists a point such that the inequality in (1.4) is strict.

When $W = 0$ , the formula (2.1) shows that the Einstein metric with the positive scalar curvature must be of positive constant sectional curvature. Hence, Theorem 1.2 gives the following:

Corollary 1.3. Let $(M^n, g, f)$ be a closed vacuum static space with the positive scalar curvature and $W = 0$ . If

$\begin{equation} |\mathring{R}_{ij}|\leq \frac{R}{\sqrt{n(n-1)}}, \end{equation}$

(1.5)

then it must be of either $\mathbb{S}^n$ with positive constant sectional curvature or a Riemannian product $\mathbb{S}^1\times \mathbb{S}^{n-1}$ .

In particular, when $n = 3$ , we have $W = 0$ automatically and Corollary 1.3 yields the following result (which has been proved by Ambrozio in [1, Theorem A]) immediately:

Corollary 1.4. Let $(M^3, g, f)$ be a closed vacuum static space with the positive scalar curvature. If

$\begin{equation} |\mathring{R}_{ij}|\leq\frac{1}{\sqrt{6}}R, \end{equation}$

(1.6)

then it must be of either $\mathbb{S}^3$ with positive constant sectional curvature or a Riemannian product $\mathbb{S}^1\times \mathbb{S}^{2}$ .

Remark 1.1. It is easy to see that the condition (1.4) is stronger than (1.3). On the other hand, one can check that when $M^n = \mathbb{S}^1\times \mathbb{S}^{n-1}$ , we have $|\mathring{R}_{ij}| = \frac{R}{\sqrt{n(n-1)}}$ , and when $M^n = \mathbb{S}^n$ , we have $|\mathring{R}_{ij}| = 0$ . Hence, for closed vacuum static spaces with $W = 0$ , Corollary 1.3 gives the following pinching results: If $0\leq|\mathring{R}_{ij}|\leq \frac{R}{\sqrt{n(n-1)}}$ , then $|\mathring{R}_{ij}| = 0$ or $|\mathring{R}_{ij}| = \frac{R}{\sqrt{n(n-1)}}$ . That is, the value of $|\mathring{R}_{ij}|$ is discrete.

Remark 1.2. Recently, by a generalized maximum principle, Cheng and Wei ^[6] considered the classifications for three-dimensional complete vacuum static spaces with constant squared norm of Ricci curvature tensor. For the classifications for closed cases, see ^{[17,24,25,26]} and the references therein.

2. Some necessary lemmas

It is well known that the Weyl curvature tensor and the Cotton tensor are defined respectively as follows:

$\begin{align} R_{ijkl}& = W_{ijkl}+\frac{1}{n-2}(R_{ik}g_{jl}-R_{il}g_{jk} +R_{jl}g_{ik}-R_{jk}g_{il})\\ &-\frac{R}{(n-1)(n-2)}(g_{ik}g_{jl}-g_{il}g_{jk})\\ & = W_{ijkl}+\frac{1}{n-2}(\mathring{R}_{ik}g_{jl}-\mathring{R}_{il}g_{jk} +\mathring{R}_{jl}g_{ik}-\mathring{R}_{jk}g_{il})\\ &+\frac{R}{n(n-1)}(g_{ik}g_{jl}-g_{il}g_{jk}) \end{align}$

(2.1)

and

$\begin{align} C_{ijk} = \mathring{R}_{ij,k}-\mathring{R}_{ik,j}+\frac{n-2}{2n(n-1)}(R_{,k}g_{ij}-R_{,j}g_{ki}). \end{align}$

(2.2)

From (2.2), it is easy to see that $C_{ijk}$ is skew-symmetric with respect to the last two indices; that is, $C_{ijk} = -C_{ikj}$ and is trace-free in any two indices:

$\begin{align} C_{iik} = 0 = C_{iji}. \end{align}$

(2.3)

In addition,

$\begin{align} C_{ijk}+C_{jki}+C_{kij} = 0, \end{align}$

(2.4)

and in using the Ricci identity, one has

$\begin{align} C_{ilk,l} = C_{kli,l},\ \ \ C_{ijl,l} = C_{jil,l},\ \ \ C_{lij,l} = 0. \end{align}$

(2.5)

Associated to (1.1), there is a (0.3)-tensor $T_{ijk}$ , which can be written as

$\begin{align} T_{ijk} = \frac{n-1}{n-2}(\mathring{R}_{ik}f_j-\mathring{R}_{ij}f_k) +\frac{1}{n-2}(g_{ik}\mathring{R}_{jl}-g_{ij}\mathring{R}_{kl})f_l. \end{align}$

(2.6)

A direct calculation enables us to observe that $T$ satisfies the following properties:

$T_{ijk} = -T_{ikj},\ \ \ \ T_{iik} = 0 = T_{iji},$

$T_{ijk}+T_{jki}+T_{kij} = 0.$

Moreover, the tensor $C_{ijk}$ is related to $T$ by (see ^{[3,4,11,15,25]}):

$\begin{align} fC_{ijk} = T_{ijk}+f_{l}W_{lijk}. \end{align}$

(2.7)

Lemma 2.1. Let $(M^n, g, f)$ be a vacuum static space with $f$ satisfying (1.2). We have

$\begin{align} \Delta f_{ij}& = 2f\mathring{R}_{mk}W_{mijk}+\frac{2n}{n-2}f\mathring{R}_{im}\mathring{R}_{mj} +\frac{R^2}{n(n-1)^2}fg_{ij}\\ &-\frac{2}{n-2}f|\mathring{R}_{kl}|^2g_{ij}+\frac{1}{n-1}R f\mathring{R}_{ij}+f_{m}C_{jmi}+f_m \mathring{R}_{mi,j} \end{align}$

(2.8)

and

$\begin{align} f\Delta \mathring{R}_{ij}& = 2f\mathring{R}_{mk}W_{mijk}+\frac{2n}{n-2}f\mathring{R}_{im}\mathring{R}_{mj}-\frac{2}{n-2}f|\mathring{R}_{kl}|^2g_{ij} \\ &+f_{m}(C_{jmi}+C_{imj})+\frac{2R}{n-1}f\mathring{R}_{ij}-f_k\mathring{R}_{ij,k}. \end{align}$

(2.9)

Proof. By the Ricci identity, we have

$\begin{aligned} f_{ij,kl} = &f_{ik,jl}+(f_m R_{mijk})_{,l}\\ = &f_{ik,jl}+f_{ml}R_{mijk}+f_{m}R_{mijk,l}\\ = &f_{ik,lj}+f_{mk}R_{mijl}+f_{im}R_{mkjl}+f_{ml}R_{mijk}+f_{m}R_{mijk,l}\\ = &f_{kl,ij}+(f_mR_{mkil})_{,j}+f_{mk}R_{mijl}\\ +&f_{im}R_{mkjl}+f_{ml}R_{mijk}+f_{m}R_{mijk,l}\\ = &f_{kl,ij}+f_{mj} R_{mkil}+f_{mk}R_{mijl}+f_{im}R_{mkjl}\\ +&f_{ml}R_{mijk}+f_{m}R_{mijk,l}+f_m R_{mkil,j}, \end{aligned}$

which gives

$\begin{align} \Delta f_{ij} = f_{ij,kk} = (\Delta f)_{,ij}+f_{mj} R_{mi}+2f_{mk}R_{mijk}+f_{im}R_{mj} +f_{m}R_{mijk,k}+f_m R_{mi,j}. \end{align}$

(2.10)

Since the scalar curvature $R$ is constant, then

$(\Delta f)_{,ij} = -\frac{1}{n-1}R f\Big[\mathring{R}_{ij}-\frac{R}{n(n-1)}g_{ij}\Big],$

$\begin{aligned} f_{mj} R_{mi} = &\Big[f\mathring{R}_{mj}-\frac{ R}{n(n-1)}fg_{mj}\Big]\Big(\mathring{R}_{mi}+\frac{R}{n} g_{mi}\Big)\\ = &f\mathring{R}_{im}\mathring{R}_{mj}+\frac{n-2}{n(n-1)}R f\mathring{R}_{ij}-\frac{R^2}{n^2(n-1)}fg_{ij}, \end{aligned}$

which is equivalent to

$f_{mj}\mathring{R}_{mi} = f\mathring{R}_{im}\mathring{R}_{mj}-\frac{R}{n(n-1)} f\mathring{R}_{ij},$

$\begin{aligned} f_{mk}R_{mijk} = &f_{mk}\Big[W_{mijk}+\frac{1}{n-2} (\mathring{R}_{mj}g_{ik}-\mathring{R}_{mk}g_{ij} +\mathring{R}_{ik}g_{mj}-\mathring{R}_{ij}g_{mk})\notag\\ +&\frac{R}{n(n-1)}(g_{mj}g_{ik}-g_{mk}g_{ij})\Big]\notag\\ = &f\mathring{R}_{mk}W_{mijk}+\frac{1}{n-2}\Big[f_{ik}\mathring{R}_{kj} +f_{jk}\mathring{R}_{ki} -f_{mk}\mathring{R}_{mk}g_{ij}\notag\\ -&(\Delta f)\mathring{R}_{ij}\Big]+\frac{R}{n(n-1)}[f_{ij}-(\Delta f)g_{ij}]\\ = &f\mathring{R}_{mk}W_{mijk}+\frac{1}{n-2}\Big[2f\mathring{R}_{im}\mathring{R}_{mj} -\frac{2 R}{n(n-1)}f\mathring{R}_{ij}\notag\\ -&f|\mathring{R}_{kl}|^2g_{ij}+\frac{R}{n-1}f\mathring{R}_{ij}\Big]+\frac{R}{n(n-1)}\Big[f\mathring{R}_{ij}+\frac{R}{n} fg_{ij}\Big]. \end{aligned}$

In particular, by virtue of the second Bianchi identity, we have

$R_{jkim,m} = R_{ij,k}-R_{ik,j} = C_{ijk},$

where, in the last equality, we used the formula (2.2) since the scalar curvature $R$ is constant. Thus, we obtain

$\begin{align} \Delta f_{ij} = &-\frac{1}{n-1}R\Big[f\mathring{R}_{ij}-\frac{R}{n(n-1)}fg_{ij}\Big]+2f\mathring{R}_{im}\mathring{R}_{mj}\\ &+\frac{2(n-2)}{n(n-1)}R f\mathring{R}_{ij}-\frac{2R^2}{n^2(n-1)}fg_{ij}+2f\mathring{R}_{mk}W_{mijk}\\ &+\frac{2}{n-2}\Big[2f\mathring{R}_{im}\mathring{R}_{mj} -\frac{2}{n(n-1)}R f\mathring{R}_{ij}-f|\mathring{R}_{kl}|^2g_{ij}+\frac{R}{n-1} f\mathring{R}_{ij}\Big]\\ &+\frac{2R}{n(n-1)}\Big[f\mathring{R}_{ij}+\frac{R}{n}fg_{ij}\Big]+f_{m}C_{jmi}+f_m \mathring{R}_{mi,j}\\ = &2f\mathring{R}_{mk}W_{mijk}+\frac{2n}{n-2}f\mathring{R}_{im}\mathring{R}_{mj} +\frac{R^2}{n(n-1)^2}fg_{ij}-\frac{2}{n-2}f|\mathring{R}_{kl}|^2g_{ij}\\ &+\frac{1}{n-1}R f\mathring{R}_{ij}+f_{m}C_{jmi}+f_m \mathring{R}_{mi,j}, \end{align}$

(2.11)

and the formula (2.8) is achieved.

From (1.2), we have

$\begin{equation} f\mathring{R}_{ij,k} = f_{ij,k}-f_k\mathring{R}_{ij}+\frac{R}{n(n-1)}f_kg_{ij}, \end{equation}$

(2.12)

$\begin{equation} f_l\mathring{R}_{ij,k}+f\mathring{R}_{ij,kl} = f_{ij,kl}-f_{kl}\mathring{R}_{ij} -f_k\mathring{R}_{ij,l}+\frac{R}{n(n-1)}f_{kl}g_{ij}. \end{equation}$

(2.13)

Therefore,

$\begin{align} f\Delta\mathring{R}_{ij} = &f\mathring{R}_{ij,kk} = \Delta f_{ij}-(\Delta f)\mathring{R}_{ij} -2f_k\mathring{R}_{ij,k}+\frac{R}{n(n-1)}(\Delta f)g_{ij}\\ = &2f\mathring{R}_{mk}W_{mijk}+\frac{2n}{n-2}f\mathring{R}_{im}\mathring{R}_{mj}-\frac{2}{n-2}f|\mathring{R}_{kl}|^2g_{ij} \\ +&f_{m}(C_{jmi}+C_{imj})+\frac{2R}{n-1}f\mathring{R}_{ij}-f_k\mathring{R}_{ij,k}. \end{align}$

(2.14)

The proof of Lemma 2.1 is completed. □

Lemma 2.2. Let $(M^n, g, f)$ be a vacuum static space with $f$ satisfying (1.2). If $f_{l}W_{lijk} = 0$ (that is, zero radial Weyl curvature), then

$\begin{align} \frac{1}{2}f\Delta |\mathring{R}_{ij}|^2+\frac{1}{2}\nabla f\nabla|\mathring{R}_{ij}|^2 = &f\mathring{R}_{ij,k}^2+2f W_{mijk}\mathring{R}_{ij}\mathring{R}_{mk}+\frac{2n}{n-2}f\mathring{R}_{im}\mathring{R}_{mj}\mathring{R}_{ji} \\ +&\frac{n-2}{n-1}f|C_{ijk}|^2+\frac{2R}{n-1}f|\mathring{R}_{ij}|^2. \end{align}$

(2.15)

Proof Using (2.9), we have

$\begin{align} \frac{1}{2}f\Delta |\mathring{R}_{ij}|^2+\frac{1}{2}\nabla f\nabla|\mathring{R}_{ij}|^2 = &f\mathring{R}_{ij,k}^2 +f\mathring{R}_{ij}\Delta\mathring{R}_{ij}+f_k\mathring{R}_{ij}\mathring{R}_{ij,k}\\ = &f\mathring{R}_{ij,k}^2+2f W_{mijk}\mathring{R}_{ij}\mathring{R}_{mk}+\frac{2n}{n-2}f\mathring{R}_{im}\mathring{R}_{mj}\mathring{R}_{ij} \\ +&(C_{jmi}+C_{imj})\mathring{R}_{ij}f_{m}+\frac{2R}{n-1}f|\mathring{R}_{ij}|^2\\ = &f\mathring{R}_{ij,k}^2+2f W_{mijk}\mathring{R}_{ij}\mathring{R}_{mk}+\frac{2n}{n-2}f\mathring{R}_{im}\mathring{R}_{mj}\mathring{R}_{ji} \\ -&2C_{ijk}\mathring{R}_{ij}f_{k}+\frac{2R}{n-1}f|\mathring{R}_{ij}|^2. \end{align}$

(2.16)

Since $f_{l}W_{lijk} = 0$ , then (2.7) gives

$fC_{ijk} = T_{ijk}$

and

$\begin{align} fC_{ijk}\mathring{R}_{ij}f_{k} = &T_{ijk}\mathring{R}_{ij}f_{k}\\ = &\Big[\frac{n-1}{n-2}(\mathring{R}_{ik}f_j-\mathring{R}_{ij}f_k) +\frac{1}{n-2}(g_{ik}\mathring{R}_{jl}-g_{ij}\mathring{R}_{kl})f_l\Big]\mathring{R}_{ij}f_{k}\\ = &\frac{n}{n-2}\mathring{R}_{ki}\mathring{R}_{kj}f_{i}f_{j}-\frac{n-1}{n-2}|\mathring{R}_{ij}|^2|\nabla f|^2. \end{align}$

(2.17)

On the other hand,

$\begin{align} f^2|C_{ijk}|^2 = &|T_{ijk}|^2\\ = &\Big|\frac{n-1}{n-2}(\mathring{R}_{ik}f_j-\mathring{R}_{ij}f_k) +\frac{1}{n-2}(g_{ik}\mathring{R}_{jl}-g_{ij}\mathring{R}_{kl})f_l\Big|^2\\ = &-\frac{2n(n-1)}{(n-2)^{2}}\mathring{R}_{ki}\mathring{R}_{kj}f_{i}f_{j}+\frac{2(n-1)^2}{(n-2)^{2}}|\mathring{R}_{ij}|^2|\nabla f|^2. \end{align}$

(2.18)

Combining (2.17) and (2.18), we achieve

$-2(n-1)C_{ijk}\mathring{R}_{ij}f_{k} = (n-2)f|C_{ijk}|^2.$

Thus, (2.16) becomes

(2.19)

and the formula (2.15) is attained. □

We also need the following lemma (see ^[9,13,14,21]):

Lemma 2.3. For any $\rho\in \mathbb{R}$ , the following estimate holds:

$\begin{align} \Big|-W_{ijkl}\mathring{R}_{jl}\mathring{R}_{ik} +\frac{\rho}{n-2}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ki}\Big| \leq \sqrt{\frac{n-2}{2(n-1)}}\Big(|W|^2+\frac{2\rho^2}{n(n-2)}|\mathring{R}_{ij}|^2 \Big)^{\frac{1}{2}}|\mathring{R}_{ij}|^2. \end{align}$

(2.20)

3. Proof of results

3.1. Proof of Theorem 1.1

Multiplying both sides of (2.15) with $f$ , we have

$\begin{align} \frac{1}{2}f^2\Delta |\mathring{R}_{ij}|^2+\frac{1}{2}f\nabla f\nabla|\mathring{R}_{ij}|^2 & = f^2\mathring{R}_{ij,k}^2+2 f^2W_{mijk}\mathring{R}_{ij}\mathring{R}_{mk}+\frac{2n}{n-2}f^2\mathring{R}_{im}\mathring{R}_{mj}\mathring{R}_{ji} \\ &+\frac{n-2}{n-1}f^2|C_{ijk}|^2+\frac{2R}{n-1}f^2|\mathring{R}_{ij}|^2. \end{align}$

(3.1)

Since the manifold is closed, then (3.1) together with (2.20) yields

$\begin{align} \frac{1}{2}f^2\Delta |\mathring{R}_{ij}|^2+\frac{1}{2}f\nabla f\nabla|\mathring{R}_{ij}|^2 &\geq f^2\Big(\mathring{R}_{ij,k}^2+\frac{n-2}{n-1}|C_{ijk}|^2\Big)\\ &+2f^2\Big[\frac{R}{n-1} -\sqrt{\frac{n-2}{2(n-1)}}\Big(|W|^2+\frac{2n}{n-2}|\mathring{R}_{ij}|^2 \Big)^{\frac{1}{2}}\Big]|\mathring{R}_{ij}|^2. \end{align}$

(3.2)

Therefore, under the assumption (1.3), it follows from (3.2) that

$\begin{align} \frac{1}{2}f^2\Delta |\mathring{R}_{ij}|^2+\frac{1}{2}f\nabla f\nabla|\mathring{R}_{ij}|^2 \geq&f^2\Big(\mathring{R}_{ij,k}^2+\frac{n-2}{n-1}|C_{ijk}|^2\Big)\\ +&2f^2\Big[\frac{R}{n-1} -\sqrt{\frac{n-2}{2(n-1)}}\Big(|W|^2+\frac{2n}{n-2}|\mathring{R}_{ij}|^2 \Big)^{\frac{1}{2}}\Big]|\mathring{R}_{ij}|^2\\ \geq&0, \end{align}$

(3.3)

which shows that $|\mathring{\rm R}_{ij}|^2$ is subharmonic on $M^n$ . Using the maximum principle, we obtain that $|\mathring{\rm R}_{ij}|$ is constant and $\mathring{R}_{ij, k} = 0$ . In this case, (3.3) becomes

$\begin{align} \Big[\frac{R}{n-1} -\sqrt{\frac{n-2}{2(n-1)}}\Big(|W|^2+\frac{2n}{n-2}|\mathring{R}_{ij}|^2 \Big)^{\frac{1}{2}}\Big]|\mathring{R}_{ij}|^2 = 0. \end{align}$

(3.4)

If there exists a point $x_0$ such that (1.3) is strict, then from (3.4) we have $|\mathring{R}_{ij}|(x_0) = 0$ , which with $|\mathring{\rm R}_{ij}|$ constant shows that $\mathring{R}_{ij}\equiv0$ . That is, the metric is Einstein and the proof of Theorem 1.1 is completed.

3.2. Proof of Theorem 1.2

We recall the following inequality, which was first proved by Huisken (cf. [16, Lemma 3.4]):

$\begin{equation} |W_{ikjl}\mathring{R}_{ij}\mathring{R}_{kl}|\leq\sqrt{\frac{n-2}{2(n-1)}}|W||\mathring{R}_{ij}|^2 \end{equation}$

(3.5)

and

$\begin{equation} |\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ki}|\leq\frac{n-2}{\sqrt{n(n-1)}}|\mathring{R}_{ij}|^3, \end{equation}$

(3.6)

with the equality in (3.6) at some point $p\in M$ if, and only if, $\mathring{R}_{ij}$ can be diagonalized at $p$ and the eigenvalue multiplicity of $\mathring{R}_{ij}$ is at least $n-1$ ^[12,22]. Thus, from (2.15), we obtain

$\begin{aligned} \frac{1}{2}f^2\Delta|\mathring{R}_{ij}|^2+\frac{1}{2}f\nabla f\nabla|\mathring{R}_{ij}|^2 \geq&f^2\Big(\mathring{R}_{ij,k}^2+\frac{n-2}{n-1}|C_{ijk}|^2-\sqrt{\frac{2(n-2)}{n-1}}|W||\mathring{R}_{ij}|^2\\ -&2\sqrt{\frac{n}{n-1}}|\mathring{R}_{ij}|^3+\frac{2R}{n-1}|\mathring{R}_{ij}|^2\Big)\\ = &f^2\Big(\mathring{R}_{ij,k}^2+\frac{n-2}{n-1}|C_{ijk}|^2\Big)+2f^2\Big(\frac{R}{n-1}\\ -&\sqrt{\frac{n-2}{2(n-1)}}|W| -\sqrt{\frac{n}{n-1}}|\mathring{R}_{ij}|\Big)|\mathring{R}_{ij}|^2. \end{aligned}$

Similarly, under the assumption (1.4), we obtain

$\begin{align} \frac{1}{2}f^2\Delta|\mathring{R}_{ij}|^2+\frac{1}{2}f\nabla f\nabla|\mathring{R}_{ij}|^2 \geq&f^2\Big(\mathring{R}_{ij,k}^2+\frac{n-2}{n-1}|C_{ijk}|^2\Big)+2f^2\Big(\frac{R}{n-1}\\ -&\sqrt{\frac{n-2}{2(n-1)}}|W| -\sqrt{\frac{n}{n-1}}|\mathring{R}_{ij}|\Big)|\mathring{R}_{ij}|^2\\ \geq&0, \end{align}$

(3.7)

which shows that $|\mathring{\rm R}_{ij}|^2$ is subharmonic on $M^n$ . Using the maximum principle again, we obtain that $|\mathring{\rm R}_{ij}|$ is constant and $\mathring{R}_{ij, k} = 0$ . In this case, (3.7) becomes

$\begin{align} \Big(\frac{R}{n-1}-\sqrt{\frac{n-2}{2(n-1)}}|W| -\sqrt{\frac{n}{n-1}}|\mathring{R}_{ij}|\Big)|\mathring{R}_{ij}|^2 = 0 \end{align}$

(3.8)

and the equalities in (3.5) and (3.6) occur.

In particular, writing $\mathring{R}_{ij} = ag_{ij}+bv_iv_j$ at $p$ with some scalars $a, b$ and a vector $v$ , we see that the left hand side of (3.5) is zero ^[12] at every point $p$ . As (3.5) is an equality and, according to ^[7], $g$ is real-analytic, the metric $g$ must be conformally flat or Einstein.

If there exists a point $x_0$ such that (1.4) is strict, then from (3.8) we have $|\mathring{R}_{ij}|(x_0) = 0$ . Which with $|\mathring{\rm R}_{ij}|$ constant shows that $\mathring{R}_{ij}\equiv0$ and the metric is Einstein. Otherwise, we have that the equality in (1.4) occurs and

$\begin{equation} \sqrt{\frac{(n-1)(n-2)}{2}}|W|+\sqrt{n(n-1)}|\mathring{R}_{ij}| = R. \end{equation}$

(3.9)

In this case, we have $W = 0$ and (3.9) becomes $|\mathring{R}_{ij}| = \frac{R}{\sqrt{n(n-1)}}$ , and then $M^n = \mathbb{S}^1\times \mathbb{S}^{n-1}$ ^[5].

Therefore, we complete the proof of Theorem 1.2.

4. Conclusions

The aim of this paper is to study rigidity results for closed vacuum static spaces. The main tool is to apply the maximum principle to the function $|\mathring{R}_{ij}|^2$ since the manifolds are closed. More precisely, we obtain rigidity theorems by establishing some pointwise inequalities and applying the maximum principle, which further proves that the squared norm of the Ricci curvature tensor is discrete.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the referee for valuable suggestions, which made the paper more readable. The research of the authors is supported by NSFC(No. 11971153) and Key Scientific Research Project for Colleges and Universities in Henan Province (No. 23A110007).

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	Bowen G (2009) Document Analysis as a Qualitative Research Method. Qual Res J 9: 27–40. https://doi.org/10.3316/QRJ0902027 doi: 10.3316/QRJ0902027
[2]	Pu H, Wang X (2023) The impact of environment on cultural relics. Sci Cult 9: 49–62. https://doi.org/10.5281/zenodo.7918252 doi: 10.5281/zenodo.7918252
[3]	Stoffle R, Arnold R, Van Vlack K (2022) Landscape Is Alive: Nuwuvi Pilgrimage and Power Places in Nevada. Land 11: 1208. https://doi.org/10.3390/land11081208 doi: 10.3390/land11081208
[4]	UNESCO (2008) Operational Guidelines for the Implementation of the World Heritage Convention, Annex Ⅲ. Paris: World Heritage Centre.
[5]	Rössler M (2006) World Heritage cultural landscapes: A UNESCO flagship programme 1992–2006. Landscape Res 31: 333–353. https://doi.org/10.1080/01426390601004210 doi: 10.1080/01426390601004210
[6]	Padiglione V (2008) Poetiche dal museo etnografico. Spezie morali e kit di sopravvivenza. Imola: La Mandragora.
[7]	Harrison R (2012) Heritage: critical approaches. Routledge. https://doi.org/10.4324/9780203108857
[8]	Smith MK, Robinson M (2006) Cultural Tourism in a Changing World: Politics, Partecipation and (Re)Presentation. Clevedon and Buffalo: Channel View Pubblication. https://doi.org/10.21832/9781845410452
[9]	Council of Europe (2005) Convention on the Value of Cultural Heritage for Society - Faro Convention. Strasbourg.
[10]	Davison G (2008) Heritage: From Pastiche to Patrimony. In: Fairclough G, Harrison R, Jameson JH Jr, Schofield J (eds.), The Heritage Reader. Abingdon and New York: Routledge, 31–41.
[11]	Beck U (1992) Risk Society: Towards a New Modernity. London: Sage.
[12]	Giddens A (1990) Consequences of Modernity. Cambridge: Polity Press.
[13]	Harvey D (1990) The Condition of Postmodernity. Oxford: Blackwell.
[14]	Allen J, Hamnett C (1995) A Shrinking World? Global Unevenness and Inequality. Oxford: Oxford University Press.
[15]	Cairncross F (1997) The death of distance: How the communication revolution will change our lives. New York: McGraw-Hill.
[16]	Badie F (1995) La fin des territoires. Paris: Fayard.
[17]	Castells M (1996) The rise of the network society. Oxford: Blackwell.
[18]	Relph E (1987) The Modern Urban Landscape. London: Croom Helm.
[19]	Daniels S (1993) Fields of vision: landscape imagery and national identity in England and the United States. Princeton, NJ: Princeton University Press.
[20]	Massey D, Jess P (2001) Luoghi, culture e globalizzazione. Torino: UTET.
[21]	Said EW (2000) Invention, memory, and place. Crit Inquiry 26: 175–192. https://doi.org/10.1086/448963 doi: 10.1086/448963
[22]	Cosgrove D (1984) Social Formation and Symbolic Landscape. London: Croom Helm Ltd.
[23]	Cosgrove D, Daniels S (1988) The Iconography of Landscape. Cambridge: Cambridge University Press.
[24]	Whatmore S (2002) From farming to agribusiness: the global agro-food system. In: Johnston RJ, Taylor P, Watts M (eds.), Geographies of global change, 2nd edn. Oxford: Blackwell, 57–67.
[25]	Labadi S (2010) World Heritage, authenticity and post-authenticity. In: Labadi S, Long C (eds.) Heritage and Globalisation, London: Routledge, 66–84.
[26]	Livingstone DN (2010) Putting Science in Its Place: Geographies of Scientific Knowledge. Chicago: University of Chicago Press.
[27]	Ionesov VI, Kurulenko EA (2015) Things as characters of culture: symbolic nature and meanings of material objects in changing world. Sci Cult 1: 47–50. https://doi.org/10.5281/zenodo.18450 doi: 10.5281/zenodo.18450
[28]	Buttitta A (1984) Dove fiorisce il limone. Palermo: Sellerio.
[29]	de Spuches G (2003) Luoghi comuni e mezze verità. Sguardi europei e vedute della Sicilia. In: Cusimano G (ed.) Scritture di paesaggio, Bologna: Pàtron Editore, 155–170.
[30]	Cusimano G (1999) La costruzione del paesaggio siciliano: geografi e scrittori a confronto. Annali della Facoltà di Lettere e Filosofia dell'Università di Palermo – La Memoria 12.
[31]	Rose G (1995) Place and identity: a sense of place. In Massey DB, Jess P (eds.), A place in the world? Places, Cultures and Globalization, Oxford: Open University, 87–132.
[32]	Clemente P (1989) Museografia, estetica, comunicazione, Nuove Effemeridi Ⅱ: 97–102.
[33]	Broccolini A, Clemente P, Giancristofaro L (2021) Patrimonio in comunicazione. Nuove sfide per i Musei Demoetnoantropologici, Quaderni di Antropologia museale 4. Palermo: Edizioni Museo Pasqualino.
[34]	D'Agostino G (2020) Postfazione. La museografia etnoantropologica in Italia e i processi di patrimonializzazione. In: R. Harrison Il patrimonio culturale. Un approccio critico, Milano-Torino: Pearson, 211–249.
[35]	Cocchiara G (1938) La vita e l'arte del popolo siciliano nel Museo Pitrè. Palermo: Ciuni.
[36]	Palumbo B (2003) L'Unesco e il campanile. Antropologia, politica e beni culturali in Sicilia orientale. Roma: Meltemi.
[37]	De Varine H (2005) Le radici del futuro. Il patrimonio culturale al servizio dello sviluppo locale. Bologna: Clueb.
[38]	Vecchio B (2002) Riflettendo sul paesaggio chiuso in un museo. In: de Spuches G (ed.), Atlante virtuale, vol. Ⅱ, Palermo: Laboratorio Geografico, 45–52.
[39]	Simmel G (1985) Il volto e il ritratto. Saggi sull'arte. Bologna: Il Mulino.
[40]	Daniels S (2011) Geographical imagination. Transactions of the Institute of British Geographers 36: 182–187. https://doi.org/10.1111/j.1475-5661.2011.00440.x doi: 10.1111/j.1475-5661.2011.00440.x

This article has been cited by:

Guangyue Huang, Qi Guo, Bingqing Ma, Rigidity of closed vacuum static spaces, 2025, 98, 09262245, 102217, 10.1016/j.difgeo.2024.102217

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Geosciences

0.8

Metrics

Article views(3542) PDF downloads(139) Cited by(2)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(4) / Tables(2)

AIMS Geosciences

Preserving cultural landscapes in the face of globalization. The musealization of Sicilian heritage

Related Papers:

Abstract

1. Introduction

2. Some necessary lemmas

3. Proof of results

3.1. Proof of Theorem 1.1

3.2. Proof of Theorem 1.2

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Geosciences

Preserving cultural landscapes in the face of globalization. The musealization of Sicilian heritage

Related Papers:

Abstract

1. Introduction

2. Some necessary lemmas

3. Proof of results

3.1. Proof of Theorem 1.1

3.2. Proof of Theorem 1.2

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog