Warped Space Research Articles

MGluR5 and GABAA receptor-specific parametric PET atlas construction-PET/MR data processing pipeline, validation, and application.

The glutamate and γ‐aminobutyric acid neuroreceptor subtypes mGluR5 and GABAA are hypothesized to be involved in the development of a variety of psychiatric diseases. However, detailed information relating to their in vivo distribution is generally unavailable. Maps of such distributions could potentially aid clinical studies by providing a reference for the normal distribution of neuroreceptors and may also be useful as covariates in advanced functional magnetic resonance imaging (MR) studies. In this study, we propose a comprehensive processing pipeline for the construction of standard space, in vivo distributions of non‐displaceable binding potential (BP ND), and total distribution volume (V T) based on simultaneously acquired bolus‐infusion positron emission tomography (PET) and MR data. The pipeline was applied to [11C]ABP688‐PET/MR (13 healthy male non‐smokers, 26.6 ± 7.0 years) and [11C]Flumazenil‐PET/MR (10 healthy males, 25.8 ± 3.0 years) data. Activity concentration templates, as well as V T and BP ND atlases of mGluR5 and GABAA, were generated from these data. The maps were validated by assessing the percent error δ from warped space to native space in a selection of brain regions. We verified that the average δABP = 3.0 ± 1.0% and δFMZ = 3.8 ± 1.4% were lower than the expected variabilities σ of the tracers (σABP = 4.0%–16.0%, σFMZ = 3.9%–9.5%). An evaluation of PET‐to‐PET registrations based on the new maps showed higher registration accuracy compared to registrations based on the commonly used [15O]H2O‐template distributed with SPM12. Thus, we conclude that the resulting maps can be used for further research and the proposed pipeline is a viable tool for the construction of standardized PET data distributions.

Super warped products with a semi-symmetric non-metric connection

<abstract><p>In this paper, we define a semi-symmetric non-metric connection on super Riemannian manifolds. And we compute the curvature tensor and the Ricci tensor of a semi-symmetric non-metric connection on super warped product spaces. Next, we introduce two kinds of super warped product spaces with a semi-symmetric non-metric connection and give the conditions that two super warped product spaces with a semi-symmetric non-metric connection are the Einstein super spaces with a semi-symmetric non-metric connection.</p></abstract>

On Riemannian Poisson warped product space

A formal treatment of Killing 1-form and 2-Killing 1-form on Riemannian Poisson manifold, Riemannian Poisson warped product space are presented. In this way, we obtain Bochner type results on compact Riemannian Poisson manifold, compact Riemannian Poisson warped product space for Killing 1-form and 2-Killing 1-form. Finally, we give the characterization of a 2-Killing 1-form on (R2, 1,?).

Model Compression via Position-Based Scaled Gradient

We propose the position-based scaled gradient (PSG) that scales the gradient depending on the position of a weight vector to make it more compression-friendly. First, we theoretically show that applying PSG to the standard gradient descent (GD), which is called PSGD, is equivalent to the GD in the warped weight space, a space made by warping the original weight space via an appropriately designed invertible function. Second, we empirically show that PSG acting as a regularizer to the weight vectors is favorable for model compression domains such as quantization, pruning, and knowledge distillation. PSG reduces the gap between the weight distributions of a full-precision model and its compressed counterpart. This enables the versatile deployment of a model either as an uncompressed mode or as a compressed mode depending on the availability of resources. The experimental results on CIFAR-10/100 and ImageNet datasets show the effectiveness of the proposed PSG in model compression including an iterative pruning method and the knowledge distillation.

Global Strichartz estimates for the Dirac equation on symmetric spaces

Abstract In this paper, we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq 3$ . In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds.

On Einstein warped product space with respect to semi symmetric metric connection

In this paper, we study Einstein warped product space with respect to semi symmetric metric connection. During this study we establish some results on curvature, Ricci and scalar tensors with respect to semi symmetric metric connection and second order semi symmetric metric connection. In the last section, we investigate under what conditions, if $M$ is an Einstein warped space with nonpositive scalar curvature and compact base with respect to semi symmetric metric connection then $M$ is simply a Riemannian product space.

Implications of the spectrum of dynamically generated string tension theories

The string tension does not have to be put in by hand, it can be dynamically generated, as in the case when we formulate string theory in the modified measure formalism, and other formulations as well. Then string tension appears, but as an additional dynamical degree of freedom. It can be seen however that this string tension is not universal, but rather each string generates its own string tension, which can have a different value for each string. We also define a new Tension scalar background field which change locally the value of the string tension along the world sheets of the strings. When there are many strings with different string tensions this Tension field can be determined from the requirement of world sheet conformal invariance and for two types of string tensions depending on the relative sign of the tensions we obtain nonsingular cosmologies and warp space scenarios and when the two string tensions are positive, we obtain scenarios where the Hagedorn temperature is avoided in the early universe or in regions of warped space time where the string tensions become very big.

On Surfaces Properties of Doubly Warped product Manifolds hI×fB in generalized Robertson-Walker spacetimes

This paper concerns the properties of submanifolds of doubly warped product space. For Riemannian manifold B and interval I, we investigate some surfaces properties of doubly warped product manifolds hI×fB, and obtain the scalar curvature for the doubly warped product space.

Stability of mean curvature flow solitons in warped product spaces

In this paper we establish a natural framework for the stability of mean curvature flow solitons in warped product spaces. These solitons are regarded as stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. We also prove some non-existence results for solitons as applications of a comparison principle which suits well the structure of the diffusion elliptic operator associated to the weighted measures we are considering.

Super quasi-Einstein warped products with affine connections

In this paper, we study super quasi-Einstein warped product spaces with semi-symmetric non-metric connection and quarter symmetric connection. Then we give the expressions of the Ricci tensors and scalar curvatures for the bases and fibers with these affine connections. Next we find the obstructions to the existence of super quasi-Einstein warped products with respect to affine connections. In the last section, we obtain an example of super quasi-Einstein space time.

Embeddedness, convexity, and rigidity of hypersurfaces in product spaces

We establish the following Hadamard–Stoker-type theorem: Let \(f:M^n\rightarrow \mathscr{H} ^{\,\,\, n}\times \mathbb{R}\) be a complete connected hypersurface with positive definite second fundamental form, where \(\mathscr{H} ^{\,\,\, n}\) is a Hadamard manifold. If the height function of f has a critical point, then it is an embedding and M is homeomorphic to \(\mathbb{S}^n\) or \(\mathbb{R}^n.\) Furthermore, f(M) bounds a convex set in \(\mathscr{H} ^{\,\,\, n}\times \mathbb{R}.\) In addition, it is shown that, except for the assumption on convexity, this result is valid for hypersurfaces in \(\mathbb{S}^n\times \mathbb{R}\) as well. We apply these theorems to show that a compact connected hypersurface in \(\mathbb{Q}_{\epsilon}^{n}\times \mathbb{R}\) (\(\epsilon =\pm 1\)) is a rotational sphere, provided it has either constant mean curvature and positive-definite second fundamental form or constant sectional curvature greater than \((\epsilon +1)/2.\) We also prove that, for \(\bar{M}=\mathscr{H} ^{\,\,\, n} \,{\mathrm{or}} \,\, \mathbb{S}^n,\) any connected proper hypersurface \(f:M^n\rightarrow \bar{M}^n \times \mathbb{R}\) with positive semi-definite second fundamental form and height function with no critical points is embedded and isometric to \(\Sigma ^{n-1}\times \mathbb{R},\) where \(\Sigma ^{n-1}\subset \bar{M}^n\) is convex and homeomorphic to \(\mathbb{S}^{n-1}\) (for \(\bar{M}^n=\mathscr{H} ^{\,\,\, n}\) we assume further that f is cylindrically bounded). Analogous theorems for hypersurfaces in warped product spaces \(\mathbb{R}\times _\varrho \mathscr{H} ^{n}\) and \(\mathbb{R}\times _\varrho \mathbb{S}^{n}\) are obtained. In all of these results, the manifold \(M^{n}\) is assumed to have dimension \(n\ge 3.\)

Einstein Poisson warped product space

In this paper, we provide some results on Poisson manifold (M, Π) with contravariant Levi–Civita connection associated to pair (Π, g). We introduce the notion of Einstein Poisson warped product space (M = B × f F, Π, g f ) (where Π = Π1 + Π2). Moreover, we show that if M is an Einstein Poisson warped product space with nonpositive scalar curvature and compact base B, J 1 is a field endomorphism on T*B satisfies , then M is simply a Riemannian Poisson product. For a contravariant Lorentzian Poisson warped space (M = B × f F, g, Π) (where ) one can determine contravariant Einstein equations and the cosmological constant Λ corresponding to the contravariant Einstein equation G = −Λg. Moreover, it is shown that Einstein equation G = −Λg, induces the contravariant Einstein equation with cosmological constant Λ F on fiber space (F, g F , Π F ).

Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifold M n of Sasakian space forms M 2 m + 1 ε . As Chen–Ricci inequality applications, we found the characterization of the base of the warped product M n via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base B of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere S p .

Convergence result and blow-up examples for the Guan–Li mean curvature flow on warped product spaces

We examine the question of convergence of solutions to a geometric flow which was introduced by Guan and Li for starshaped hypersurfaces in space forms and generalized by Guan, Li, and Wang to the case of warped product spaces. We obtain a convergence result under a condition on the optimal modulus of continuity of the initial data. Moreover we show by examples that this condition is optimal at least in the one-dimensional case.

Recent rigidity results for graphs with prescribed mean curvature

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $\Omega \subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.

Ricci almost soliton and almost Yamabe soliton on Kenmotsu manifold

We prove that a Ricci almost soliton on a Kenmotsu manifold of dimension [Formula: see text] reduces to an expanding Ricci soliton satifying certain condition on the potential vector field or on the soliton function. Next, we show that any Ricci almost soliton on a Kenmotsu manifold is trivial (Einstein) if the soliton vector leaves the contact form [Formula: see text] invariant. Finally, we classify (locally) a Kenmotsu manifold admitting an almost Yamabe soliton. Some examples have been constructed of almost Yamabe solitons on different class of warped product spaces.

Warped flatland

We study warped flat geometries in three-dimensional topologically massive gravity. They are quotients of global warped flat spacetime, whose isometries are given by the 2-dimensional centrally extended Poincaré algebra. The latter can be obtained as a certain scaling limit of Warped AdS3 space with a positive cosmological constant. We discuss the causal structure of the resulting spacetimes using projection diagrams. We study their charges and thermodynamics, together with asymptotic Killing vectors preserving a consistent set of boundary conditions including them. The asymptotic symmetry group is given by a Warped CFT algebra, with a vanishing current level. A generalization of the derivation of the Warped CFT Cardy formula applies in this case, reproducing the entropy of the warped flat cosmological spacetimes.

Minkowski inequalities and constrained inverse curvature flows in warped spaces

Abstract This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows, we prove long-time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases, new Minkowski type inequalities are the consequence. In higher dimensions, we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

Statistical submanifolds from a viewpoint of the Euler inequality

We generalize the Euler inequality for statistical submanifolds. Several basic examples of doubly autoparallel statistical submanifolds in warped product spaces are described, for which the equality holds at each point. Besides, doubly totally-umbilical submanifolds are also illustrated.

Application of Hyper-generalized Quasi-Einstein Spacetimes in General Relativity

In this paper, we study the hyper-generalized quasi-Einstein (HGQE) warped product spaces with non-positive scalar curvature. This note deals with investigating of some geometric and physical properties of $$(HGQE)_{n}$$ manifolds. Next, we study the general relativistic viscous fluid $$(HGQE)_{4}$$ spacetimes with some physical applications. Lastly, we show the existence of $$(HGQE)_{4}$$ spacetimes by constructing a non-trivial example.