Space Form Research Articles

The Sociality of Form: Camillo Sitte's Urban Morphologies

AbstractThis article examines a text by a lesser‐known figure of fin‐de‐siècle Vienna, Camillo Sitte's Der Städtebau nach seinen künstlerischen Grundsätzen (1889), tracing the relationship between urban form and social structure in Sitte's treatise. It identifies the key points of this relationship in terms of causality: the form of public spaces can determine the form of social life; surrogacy: the throngs of the places’ inanimate inhabitants, the statues and monuments, are placeholders for the missing human gatherings; and finally displacement: the thick sociality of community life is transferred onto a heightened relationality with urban forms. Sitte's text sublimates the social; it diffuses abstract social structures—density, relationality, connection—onto the spatial morphologies of the city. Ultimately, this diffusion results in a capacious yet weakened model of sociality.

On a stratification of positive scalar curvature compact manifolds

For a compact PSC Riemannian [Formula: see text]-manifold [Formula: see text], the metric constant [Formula: see text] is defined to be the infinimum over [Formula: see text] of the spectral scalar curvature [Formula: see text] of [Formula: see text], where [Formula: see text] are the eigenvalues of the curvature operator of [Formula: see text] and [Formula: see text] is the maximal eigenvalue. The functional [Formula: see text] is continuous, re-scale invariant and defines a stratification of the space of PSC metrics over [Formula: see text]. We introduce as well the smooth constant [Formula: see text], which is the supremum of [Formula: see text] over the set of all PSC Riemannian metrics [Formula: see text] on [Formula: see text]. In this paper, we show that in the top layer, compact manifolds with [Formula: see text] are positive space forms. No manifolds have their [Formula: see text] in the interval [Formula: see text]. The manifold [Formula: see text] and arbitrary connected sums of copies of it with connected sums of positive space forms all have [Formula: see text]. For [Formula: see text], we prove that the manifolds [Formula: see text] take the intermediate values [Formula: see text]. From the bottom, we prove that simply connected (resp. [Formula: see text]-connected, [Formula: see text]-connected and non-string) compact manifolds of dimension [Formula: see text] (resp. [Formula: see text], [Formula: see text]) have [Formula: see text] (resp. [Formula: see text], [Formula: see text]). The proof of these last three results is based on surgery. In fact, we prove that the smooth [Formula: see text] constant doesn’t decrease after a surgery on the manifold with adequate codimension.

The quantization of Maxwell theory in the Cauchy radiation gauge: Hodge decomposition and Hadamard states

AbstractThe aim of this paper is to prove the existence of Hadamard states for the Maxwell equations on any globally hyperbolic spacetime. This will be achieved by introducing a new gauge fixing condition, the Cauchy radiation gauge, that will allow to suppress all the unphysical degrees of freedom. The key ingredient for achieving this gauge is a new Hodge decomposition for differential ‐forms in Sobolev spaces on complete (possibly noncompact) Riemannian manifolds.

Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures

The purpose of this paper is to generalize definitions of Bertrand and Mannheim curves to non-null framed curves and to non-flat three-dimensional (Riemannian or Lorentzian) space forms. Denote by $\mathbb{M}_q^n(c)$ the $n$-dimensional space form of index $q=0,1$ and constant curvature $c\neq 0$. We introduce two types of framed Bertrand curves and framed Mannheim curves in $\mathbb{M}_q^3(c)$ by using two different moving frames: the general moving frame and the Frenet-type frame. We investigate geometric properties of these framed Bertrand and framed Mannheim curves in $\mathbb{M}_q^3(c)$ that may have singularities. We then give characterizations for a non-null framed curve to be a framed Bertrand curve or to be a framed Mannheim curve. We show that in special cases these characterizations reduce to the well-known classical formulas: $\lambda \kappa+\mu \tau=1$ for Bertrand curves and $\lambda(\kappa^2+\tau^2)=\kappa$ for Mannheim curves. We provide several examples to support our results, and we visualize these examples by using the Hopf map, the hyperbolic Hopf map, and the spherical projection.

Homogeneous Riemannian Structures in Thurston Geometries and Contact Riemannian Geometries

We give explicit parametrizations for all the homogeneous Riemannian structures on model spaces of Thurston geometry. As an application, we give all the homogeneous contact metric structures on $3$-dimensional Sasakian space forms.

New curvature characterizations for spherical space forms and complex projective spaces

In this paper, we introduce a new positivity notion for curvature of Riemannian manifolds and obtain characterizations for spherical space forms and the complex projective space C P n \mathbb {C}\mathbb {P}^n .

Gender as biology, discourse, and construct: The form of space in The Tenant of Wildfell Hall as a subversion of the fairy tale “Blue-Beard”

The gendered prescriptions that permeate fairy tales can shape societal structures and perspectives. However, just as literature plays a role in reinforcing societal constructs, it can be used to challenge and subvert them. This paper looks at the form of space in The Tenant of Wildfell Hall by Anne Brontë (1848) as a subversion of “Blue-Beard” by Charles Perrault (1697) and its relation to understanding gender as biology, discourse, and construct.

Submanifolds with constant scalar curvature in space forms

Let [Formula: see text] be an [Formula: see text]-dimensional oriented compact submanifold with constant normalized scalar curvature [Formula: see text] in the space form [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the mean curvature and the Ricci curvature of [Formula: see text] respectively. By applying Cheng-Yau’s self-adjoint operator, we first prove that if [Formula: see text] is a hypersurface in a unit sphere, and [Formula: see text], then [Formula: see text] is totally umbilical. Furthermore, we investigate the submanifolds in [Formula: see text] with flat normal bundle satisfying [Formula: see text], and obtain a complete classification theorem.

Ricci flow of Kaehlerian slant submanifolds in complex space forms and its applications

AbstractThe normalized Ricci flow converges to a constant curvature metric for a connected Kaehlerian slant submanifold in a complex space form if the squared norm of the second fundamental form satisfies certain upper bounds. These bounds include the constant sectional curvature, the slant angle, and the squared norm of the mean curvature vector. Additionally, we demonstrate that the submanifold is diffeomorphic to the sphere $$\mathbb {S}^{n_1}$$ S n 1 under some restriction on the mean curvature. We claim that some of our previous results are rare cases.

Rigidity for inscribed radius estimate of asymptotically hyperbolic Einstein manifold

The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper, we generalize this result to asymptotically hyperbolic (AH) Einstein manifold. We get an upper bound of the relative volume of AH manifold and if we combine it with the recent work of Wang and Zhou, then the rigidity is obtained.

Local rigidity of constant mean curvature hypersurfaces in space forms

In this paper, we study the local rigidity of constant mean curvature (CMC) hypersurfaces. Let x:Mn→Mn+1(c),n≥4, be a piece of immersed constant mean curvature hypersurface in the (n+1)-dimensional space form Mn+1(c). We prove that if the scalar curvature R is constant and the number g of the distinct principal curvatures satisfies g≤3, then Mn is an isoparametric hypersurface. Further, if Mn is a minimal hypersurface, then Mn is a totally geodesic hypersurface for c≤0, and Mn is either a Cartan minimal hypersurface, a Clifford minimal hypersurface, or a totally geodesic hypersurface for c>0, which solves the high dimensional version of Bryant Conjecture.

Mineral effects on methane hydrate formation and distribution in sand sediments

Methane hydrate is a promising energy resource widely occurring in the world, but the formation and distribution of methane hydrate in marine sediments with complex minerals remains unclear. Due to the small pore space and fine-grained size of clay minerals, the contact relationship between methane hydrate and clay mineral surface under excess water has not been characterized fully, which is required to investigate the mineral effect on methane hydrate formation. In this study, cryo-SEM was applied to observe the distribution of methane hydrate synthesized in pure quartz sands and the quartz sands mixed with montmorillonite or illite separately. According to the results of the research, methane hydrate dispersedly forms in the pore space away from and on the surface of quartz under a local excess water condition. In addition, methane hydrate occurs away from montmorillonite in pores with excess water and contacts the edge of montmorillonite under a local excess-gas condition. Moreover, methane hydrate was only observed to form away from illite aggregates in the pore with residual water. Montmorillonite and illite influence the preferable distribution of methane hydrate to occur away from mineral surface in excess water condition. In the sediments composed of different minerals, methane hydrate forms stochastically in the pore space and grows up individually. As a result, the findings of this study significantly expand the understanding of the formation and distribution of methane hydrate in marine sediments and the prediction of physical properties of hydrate-bearing sediments, and provide insights into natural gas hydrate exploration and production.

‘Skateboarding is not a sport’: Creativity at the margins of capitalism

One might expect skateboarders to be jubilant that the Olympics recognized theirs as an Olympic sport, but their response is ambivalent. Alexis Sablone, a women's street competition participant, does not consider skateboarding ‘a sport’. What is it if not a sport? This article argues that it is a fugitive, non‐teleological, open‐source practice for creating new ‘existential territories’ – new forms of personhood, public space and social relationships. Skateboarders’ resistance to centralization and their emphasis on DIY creativity suggests skateboarding involves challenging culturally specific and local norms. It rests on creative remakings of derelict and unevenly developed urban spaces and the construction of new forms of identity and social purpose. The argument is based on fieldwork with the Osaka Daggers.

Chen-Type Inequality for Generic Submanifolds of Quaternionic Space Form and Its Application

In 1993, the theory of Chen invariants started when Chen wrote basic inequalities for submanifolds in space forms. This inequality is called Chen’s first inequality. Afterward, many geometers studied many papers dealing with this new inequality. The present paper aims to establish a Chen inequality for quaternionic generic submanifolds in a quaternionic space form and obtain this inequality for real hypersurfaces.

Hybrid Copresence: Issues of Re-Spatialization of Remote Work in Coworking Spaces

The re-spatialization of work in coworking spaces alters the social experience of teleworkers by involving two copresences associated with distinct organizational and spatiotemporal contexts: a remote copresence with the company and a physical copresence in the coworking space. Each can be experienced to different degrees between social isolation and perceived proximity. Nevertheless, the current literature does not provide an opportunity to make sense of the combination of copresences. From this perspective, data collected during an organizational ethnography conducted within two networks of coworking spaces were analyzed. The results show that the re-spatialization of work in coworking spaces generates a hybrid copresence that can be experienced in four ways: (reinforced) isolation, ubiquity, (guilty) compensation, and (frustrated) refocusing. The characteristics of these distinct situations and the possible evolutions from one to the other are explored and explained. The results are discussed in relation to the evolution of work in a post-COVID-19 era that seems to encourage the development of more hybrid practices and spatialities. Contributions to three fields of organizational literature are made: organizational space, telecommuting and distantiated forms of work, and coworking spaces. Managerial implications are also discussed around reflections on a more hybrid and multi-spatialized organization of work practices in organizations.

$$\mathcal {D}$$-Parallel structure Lie operators on real hypersurfaces in nonflat complex space forms

$$\mathcal {D}$$-Parallel structure Lie operators on real hypersurfaces in nonflat complex space forms

Bi-$f$-Harmonic Legendre Curves on $(\alpha,\beta)$-Trans-Sasakian Generalized Sasakian Space Forms

In this study, we consider bi-$f$-harmonic Legendre curves on $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form. We provide the necessary and sufficient conditions for a Legendre curve to be bi-$f$-harmonic on $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form without any restrictions by a main theorem. Afterward, we investigate these conditions under nine different cases. As a result of these investigations, we obtain the original theorems and corollaries as well as the nonexistence theorems. We perform these investigations according to the $\rho_{2}$ and $\rho_{3}$ functions from the curvature tensor of the $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form, the curvature and torsion of the bi-$f$-harmonic Legendre curve, and finally, the positions of the basis vectors relative to each other.

Geometry of Statistical Sequential Warped Product Manifolds.

The main purpose of the present study is to generalize the dualistic structures, specially the statistical structures on warped product manifolds to statistical structures on sequential warped product manifolds. On the one hand we proved that the statistical structure on sequential warped product manifold induces statistical on the factors and conversely. On the other hand we proved that if the statistical warped product is dually at space, then the base manifold is also dually at and ber is a space form, i.e. of constant sectional curvature.

Simulation of CO2 hydrate formation in porous medium and comparison with laboratory trial data

The storage of CO2 in gas hydrate form in the pore space in depleted natural gas reservoirs has been considered a method for greenhouse gas control. The formation of CO2 hydrate in porous medium is a strongly coupled Thermal-Hydraulic-Chemical (THC) problem under certain thermodynamic conditions. In this study, laboratory data on CO2 hydrate formation in silica sand, including the profiles of pressure, temperature, the amount of CO2, H2O, and CO2 hydrate, etc., were compared with results from numerical simulations. The minimized deviations between the simulation and experimental results, including the pressure drop, the significant temperature increase caused by hydrate formation, and the amount of CO2, water, and hydrate, were all less than 10 % in the numerical simulations. The sensitivities of the deviations between numerical simulations and experimental data to the domain discretization, thermal conductivity of the silica sand, absolute permeability, and kinetics of CO2 hydrate formation were analyzed. One of the major findings is that CO2 hydrate formation in the porous medium in this study is dominated by the kinetics of chemical reaction, rather than the heat or mass transfer. Another key finding of this study is the acquisition of the modeling parameters of the CO2 storage process in the laboratory-scale sand reservoir, including the thermal conductivity of the silica sand λs = 2.2 W/m/K, the absolute permeability k0 = 3.0 × 10−11 m2, and the kinetic constant Kf0 = 8.4 × 1011 kg/m2/Pa/s and the reduction exponent β = 5.3 in the kinetic model of CO2 hydrate formation. It is noteworthy that the mathematical models and the parameters faithfully fit three independent experiments of Run 1–3. The results of the experiments and corresponding numerical simulations provide a reliable method to evaluate the capacity, technical and commercial feasibility of CO2 storage in marine and permafrost reservoirs.

The Effects of Public Spaces on People’s Experiences and Satisfaction in Taif City: A Cross-Sectional Study

Public places play a central role in shaping positive and negative human experiences. They have a profound impact on people’s lives, affecting their sense of place, well-being, and overall quality of life. While public spaces have such an impact, little is known about their effects on the overall experiences of people in Saudi cities. Particularly, in Saudi Arabia, the form of public spaces has rapidly evolved and changed in recent years. Therefore, this study examined the impact of urban public spaces on people’s experiences, aiming to assess satisfaction levels within the broader context of urban life in Saudi Arabia. Specifically, it investigated the correlation between the public spaces and individuals’ perceptions and feelings toward the environment of Taif city. Taif is one of the key cities in Saudi Arabia targeted for significant improvements in quality of life as part of the 2030 vision. Therefore, it is important to study the effects on the residents there. The method used in the study involved quantitative data collected through a survey of 384 participants from Taif, which included questions measuring multiple aspects of the quality of urban public life, including the safety, walkability, social interaction, and functionality of public spaces. The results indicate that people in Taif would face a jeopardised level of satisfaction, as not all elements of the public spaces meet the desired standards. Therefore, their experiences would be impacted negatively if the noted concerns are not addressed. By identifying key urban design elements that impact people’s experiences and overall well-being, this research provides a foundation for planners and policymakers to work towards achieving the Saudi vision of creating more liveable and enduring urban environments in Saudi Arabian cities.