Showing papers on "Riemann curvature tensor published in 2019"
TL;DR: In this article, an n-dimensional generalized Robertson-Walker space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress energy tensor for any f(R) gravity model.
Abstract: We show that an n-dimensional generalized Robertson–Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress–energy tensor for any f(R) gravity model....
129 citations
TL;DR: In this article, the effects of f (G, T) terms on the study of structure scalars and their influence on the formulation of the Raychaudhuri, shear and Weyl scalar equations are explored.
Abstract: This work is devoted to exploring the effects of f (G, T) terms on the study of structure scalars and their influence on the formulation of the Raychaudhuri, shear and Weyl scalar equations. For this purpose, we have assumed non-static spherically symmetric geometry coupled with shearing viscous locally anisotropic dissipative matter content. We have developed relations among Misner-Sharp mass, Weyl scalar, matter and structure variables. We have also formulated a set of f (G, T) structure scalars after orthogonally breaking down the Riemann curvature tensor. The influences of these scalar functions on the modeling of relativistic radiating spheres are also studied. The factor involved in the emergence of inhomogeneities is also explored for the constant and varying modified curvature corrections. We inferred that f (G, T) structure scalars could provide an effective tool to study the Penrose-Hawking singularity theorems and the Newman-Penrose formalism.
73 citations
TL;DR: In this paper, the authors extended the complexity of spherically symmetric fluid distributions to axially symmetric static sources, defined in terms of three structure scalars obtained from the orthogonal splitting of the Riemann tensor.
Abstract: A previously found definition of complexity for spherically symmetric fluid distributions [1], is extended to axially symmetric static sources. In this case there are three different complexity factors, defined in terms of three structure scalars obtained from the orthogonal splitting of the Riemann tensor. All these three factors vanish, for what we consider the simplest fluid distribution, i.e a fluid spheroid with isotropic pressure and homogeneous energy density. However, as in the spherically symmetric case, they can also vanish for a variety of configurations, provided the energy density inhomogeneity terms cancel the pressure anisotropic ones in the expressions for the complexity factors. Some exact analytical solutions of this type are found and analyzed. At the light of the obtained results, some conclusions about the correlation (the lack of it) between symmetry and complexity, are put forward.
57 citations
TL;DR: In this paper, the authors construct cubic gravity and its $f(P)$ extension and investigate their early and late-time cosmological applications, showing that at early times one can obtain inflationary, de Sitter solutions, which are driven by an effective cosmology constant constructed purely from the cubic terms of the simple cubic or$f(p)$ gravity.
Abstract: We construct cubic gravity and its $f(P)$ extension and we investigate their early- and late-time cosmological applications. Cubic gravity is based on a particular invariant $P$, constructed from cubic contractions of the Riemann tensor, under three requirements: (i) the resulting theory possesses a spectrum identical to that of general relativity, (ii) it is neither topological nor trivial in four dimensions, and (iii) it is defined such that it is independent of the dimensions. Relaxing the last condition and restricting the parameters of cubic gravity we can obtain second-order field equations in a cosmological background. We show that at early times one can obtain inflationary, de Sitter solutions, which are driven by an effective cosmological constant constructed purely from the cubic terms of the simple cubic or $f(P)$ gravity. Concerning late-time evolution, the new terms constitute an effective dark-energy sector and we show that the Universe experiences the usual thermal history and the onset of late-time acceleration. In the case of $f(P)$ gravity, depending on the choice of parameters, we find that the dark-energy equation-of-state parameter can be quintessencelike or phantomlike or it can experience the phantom-divide crossing during the evolution, even if an explicit cosmological constant is absent.
53 citations
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TL;DR: In this article, the affine connection in non-Riemannian geometries has been studied for the first time in the literature, and the generalized Raychaudhuri equation in the presence of both torsion and non-metricity has been derived.
Abstract: This Thesis is devoted to the study of Metric-Affine Theories of Gravity and Applications to Cosmology. The thesis is organized as follows. In the first Chapter we define the various geometrical quantities that characterize a non-Riemannian geometry. In the second Chapter we explore the MAG model building. In Chapter 3 we use a well known procedure to excite torsional degrees of freedom by coupling surface terms to scalars. Then, in Chapter 4 which seems to be the most important Chapter of the thesis, at least with regards to its use in applications, we present a step by step way to solve for the affine connection in non-Riemannian geometries, for the first time in the literature. A peculiar f(R) case is studied in Chapter 5. This is the conformally (as well as projective invariant) invariant theory f(R)=a R^{2} which contains an undetermined scalar degree of freedom. We then turn our attention to Cosmology with torsion and non-metricity (Chapter 6). In Chapter 7, we formulate the necessary setup for the $1+3$ splitting of the generalized spacetime. Having clarified the subtle points (that generally stem from non-metricity) in the aforementioned formulation we carefully derive the generalized Raychaudhuri equation in the presence of both torsion and non-metricity (along with curvature). This, as it stands, is the most general form of the Raychaudhuri equation that exists in the literature. We close this Thesis by considering three possible scale transformations that one can consider in Metric-Affine Geometry.
47 citations
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TL;DR: In this paper, it was shown that the Strominger connection of a compact Hermitian manifold is K\"ahler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a K''ahler manifold, and therefore the metric must be pluriclosed.
Abstract: In this paper, we prove a conjecture raised by Angella, Otal, Ugarte, and Villacampa recently, which states that if the Strominger connection (also known as Bismut connection) of a compact Hermitian manifold is K\"ahler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a K\"ahler manifold, then the metric must be pluriclosed. Actually, we show that Strominger K\"ahler-like is equivalent to the pluriclosedness of the Hermitian metric plus the parallelness of the torsion, even without the compactness assumption.
28 citations
TL;DR: In this paper, it was shown that any gravitational effective action involving higher-curvature corrections is equivalent, via metric redefinitions, to a generalized quasi-topological gravities (GQTGs).
Abstract: Generalized quasi-topological gravities (GQTGs) are higher-curvature extensions of Einstein gravity characterized by the existence of non-hairy generalizations of the Schwarzschild black hole which satisfy $g_{tt}g_{rr}=-1$, as well as for having second-order linearized equations around maximally symmetric backgrounds. In this paper we provide strong evidence that any gravitational effective action involving higher-curvature corrections is equivalent, via metric redefinitions, to some GQTG. In the case of theories involving invariants constructed from contractions of the Riemann tensor and the metric, we show this claim to be true as long as (at least) one non-trivial GQTG invariant exists at each order in curvature ---and extremely conclusive evidence suggests this is the case in general dimensions. When covariant derivatives of the Riemann tensor are included, the evidence provided is not as definitive, but we still prove the claim explicitly for all theories including up to eight derivatives of the metric as well as for terms involving arbitrary contractions of two covariant derivatives of the Riemann tensor and any number of Riemann tensors. Our results suggest that the physics of generic higher-curvature gravity black holes is captured by their GQTG counterparts, dramatically easier to characterize and universal. As an example, we map the gravity sector of the Type-IIB string theory effective action in AdS$_5$ at order $\mathcal{O}({\alpha^{\prime}}^3)$ to a GQTG and show that the thermodynamic properties of black holes in both frames match.
27 citations
TL;DR: In this paper, it was shown that perfect fluid spacetimes with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensors are generalized Robertson Walker (GRW) spacetimits.
Abstract: We show that $n$-dimensional perfect fluid spacetimes with diver\-gen\-ce-free conformal curvature tensor and constant scalar curvature are generalized Robertson Walker (GRW) spacetimes; as a consequence a perfect fluid Yang pure space is a GRW spacetime. We also prove that perfect fluid spacetimes with harmonic generalized curvature tensor are, under certain conditions, GRW spacetimes. As particular cases, perfect fluids with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensor are GRW spacetimes. Finally, we explore some physical consequences of such results.
27 citations
TL;DR: In this article, it was shown that Generalized Quasi-Topological (GQTG) densities exist in general dimensions and at arbitrarily high curvature orders, and recursive formulas which allow for the systematic construction of $n$-th order densities from lower order ones, as well as explicit expressions valid at any order.
Abstract: A new class of higher-curvature modifications of $D(\geq 4$)-dimensional Einstein gravity has been recently identified. Densities belonging to this "Generalized quasi-topological" class (GQTGs) are characterized by possessing non-hairy generalizations of the Schwarzschild black hole satisfying $g_{tt}g_{rr}=-1$ and by having second-order equations of motion when linearized around maximally symmetric backgrounds. GQTGs for which the equation of the metric function $f(r)\equiv -g_{tt}$ is algebraic are called "Quasi-topological" and only exist for $D\geq 5$. In this paper we prove that GQTG and Quasi-topological densities exist in general dimensions and at arbitrarily high curvature orders. We present recursive formulas which allow for the systematic construction of $n$-th order densities of both types from lower order ones, as well as explicit expressions valid at any order. We also obtain the equation satisfied by $f(r)$ for general $D$ and $n$. Our results here tie up the remaining loose end in the proof presented in arXiv:1906.00987 that every gravitational effective action constructed from arbitrary contractions of the metric and the Riemann tensor is equivalent, through a metric redefinition, to some GQTG.
26 citations
TL;DR: In this article, it was shown that the physics of generic higher-curvature gravity black holes is captured by their generalized quasi-topological gravities (GQTGs) counterparts.
Abstract: Generalized quasi-topological gravities (GQTGs) are higher-curvature extensions of Einstein gravity characterized by the existence of non-hairy generalizations of the Schwarzschild black hole which satisfy g$_{tt}$g$_{rr}$ = –1, as well as for having second-order linearized equations around maximally symmetric backgrounds. In this paper we provide strong evidence that any gravitational effective action involving higher-curvature corrections is equivalent, via metric redefinitions, to some GQTG. In the case of theories involving invariants constructed from contractions of the Riemann tensor and the metric, we show this claim to be true as long as (at least) one non-trivial GQTG invariant exists at each order in curvature-and extremely conclusive evidence suggests this is the case in general dimensions. When covariant derivatives of the Riemann tensor are included, the evidence provided is not as definitive, but we still prove the claim explicitly for all theories including up to eight derivatives of the metric as well as for terms involving arbitrary contractions of two covariant derivatives of the Riemann tensor and any number of Riemann tensors. Our results suggest that the physics of generic higher-curvature gravity black holes is captured by their GQTG counterparts, dramatically easier to characterize and universal. As an example, we map the gravity sector of the Type-IIB string theory effective action in AdS$_{5}$ at order 𝒪 (α′$^{3}$) to a GQTG and show that the thermodynamic properties of black holes in both frames match.
25 citations
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TL;DR: A theory of Finite Element Systems is developed, for the purpose of discretizing sections of vector bundles, in particular those arizing in the theory of elasticity, and a discrete Bianchi identity is proved.
Abstract: We develop a theory of Finite Element Systems, for the purpose of discretizing sections of vector bundles, in particular those arizing in the theory of elasticity. In the presence of curvature we prove a discrete Bianchi identity. In the flat case we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress-displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes.
TL;DR: In this article, a generalized Higgs effective field theory (GHEFT) was proposed to describe nonminimal electroweak symmetry breaking models, which can be regarded as a nonlinear sigma model on a scalar manifold, and the scalar boson scattering amplitudes are described in terms of the Riemann curvature tensor (geometry) of the manifold and the covariant derivatives of the potential.
Abstract: We formulate a generalization of Higgs effective field theory (HEFT) including an arbitrary number of extra neutral and charged Higgs bosons---a generalized HEFT (GHEFT)---to describe nonminimal electroweak symmetry breaking models. Using the geometrical form of the GHEFT Lagrangian, which can be regarded as a nonlinear sigma model on a scalar manifold, it is shown that the scalar boson scattering amplitudes are described in terms of the Riemann curvature tensor (geometry) of the scalar manifold and the covariant derivatives of the potential. The coefficients of the one-loop divergent terms in the oblique correction parameters $S$ and $U$ can also be written in terms of the Killing vectors (symmetry) and the Riemann curvature tensor (geometry). It is found that the perturbative unitarity of the scattering amplitudes involving the Higgs bosons and the longitudinal gauge bosons demands that the scalar manifold be flat. The relationship between the finiteness of the electroweak oblique corrections and the perturbative unitarity of the scattering amplitudes is also clarified in this language: we verify that once the tree-level unitarity is ensured, the one-loop finiteness of the oblique correction parameters $S$ and $U$ is automatically guaranteed.
TL;DR: It is concluded that the complexity of the system increases with the inclusion of the scalar field and potential function.
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TL;DR: In this article, a special type of compact Hermitian manifolds that are Strominger K\"ahler-like, or SKL for short, were studied and a classification for compact non-K''ahler SKL manifolds in dimension $3 and those with degenerate torsion was given.
Abstract: In this paper, we study a special type of compact Hermitian manifolds that are Strominger K\"ahler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is K\"ahler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a K\"ahler manifold. Previously, we have shown that any SKL manifold $(M^n,g)$ is always pluriclosed, and when the manifold is compact and $g$ is not K\"ahler, it can not admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when $n=2$, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-K\"ahler SKL manifolds in dimension $3$ and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, for any compact non-K\"ahler SKL manifold, its K\"ahler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal K\"ahler when $n\geq 3$, and the manifold does not admit any Hermitian symplectic metric.
TL;DR: In this paper, the authors formulate the locally supersymmetric E7(7) exceptional field theory in a (4 + 56|32) dimensional superspace, corresponding to a 4D N = 8 "external" superspace augmented with an "internal" 56-dimensional space.
Abstract: We formulate the locally supersymmetric E7(7) exceptional field theory in a (4 + 56|32) dimensional superspace, corresponding to a 4D N = 8 “external” superspace augmented with an “internal” 56-dimensional space. This entails the unification of external diffeomorphisms and local supersymmetry transformations into superdiffeomorphisms. The solutions to the superspace Bianchi identities lead to on-shell duality equations for the p-form field strengths for p ≤ 4. The reduction to component fields provides a complete description of the on-shell supersymmetric theory. As an application of our results, we perform a generalized Scherk-Schwarz reduction and obtain the superspace formulation of maximal gauged supergravity in four dimensions parametrized by an embedding tensor.
TL;DR: In this paper, the authors proved uniform curvature estimates for homogeneous Ricci flows on homogeneous spaces, and showed that the norm of the curvature tensor at time t is bounded by the maximum of C(n)/t and C( n/(scal(g(t)) - scal(g (0))).
Abstract: We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on [0, t] the norm of the curvature tensor at time t is bounded by the maximum of C(n)/t and C(n)(scal(g(t)) - scal(g(0))). This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, with constants depending only on the dimension n. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on that space. The above curvature estimates follow from a gap theorem for Ricci-flatness on homogeneous spaces. This theorem is proved by contradiction, using a local W-2,W-p convergence result which holds without symmetry assumptions.
TL;DR: The objectives of this paper are to demonstrate the viability, and to validate, in part, a multiscale method for calculating curvature tensors on measured surface topographies with two different methods of specifying the scale.
Abstract: The objectives of this paper are to demonstrate the viability, and to validate, in part, a multiscale method for calculating curvature tensors on measured surface topographies with two different methods of specifying the scale. The curvature tensors are calculated as functions of scale, i.e., size, and position from a regular, orthogonal array of measured heights. Multiscale characterization of curvature is important because, like slope and area, it changes with the scale of observation, or calculation, on irregular surfaces. Curvatures can be indicative of the topographically dependent behavior of a surface and, in turn, curvatures are influenced by the processing and use of the surface. Curvatures of surface topographies have not been well- characterized yet. Curvature has been used for calculations in contact mechanics and for the evaluation of cutting edges. Manufactured surfaces are studied for further validation of the calculation method because they provide certain expectations for curvatures, which depend on scale and the degree of curvature. To study a range of curvatures on manufactured surfaces, square edges are machined and honed, then rounded progressively by mass finishing; additionally, a set of surfaces was made by turning with different feeds. Topographic measurements are made with a scanning laser confocal microscope. The calculations use vectors, normal to the measured surface, which are calculated first, then the eigenvalue problem is solved for the curvature tensor. Plots of principal curvatures as a function of position and scale are presented. Statistical analyses show expected interactions between curvature and these manufacturing processes.
TL;DR: In this paper, the complexity of a static anisotropic sphere in the context of self-interacting Brans-Dicke theory was studied and it was shown that the complexity increases with the inclusion of the scalar field and potential function.
Abstract: In this paper, we study the complexity factor of a static anisotropic sphere in the context of self-interacting Brans-Dicke theory. We split the Riemann tensor using Bel's approach to obtain structure scalars relating to comoving congruence and Tolman mass in the presence of a scalar field. We then define the complexity factor with the help of these scalars to demonstrate the complex nature of the system. We also evaluate the vanishing complexity condition to obtain solutions for two stellar models. It is concluded that the complexity of the system increases with the inclusion of the scalar field and potential function.
TL;DR: In this article, the authors investigated the complexity of static charged anisotropic cylindrical configuration and found that the complexity increases for α ≥ 1 in the presence of the electromagnetic field.
TL;DR: In this article, the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity, was established.
Abstract: In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. This extends a previous result by Q. Zhang where a $$C^1$$
bound on the whole Riemann tensor was assumed. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like function with uniformly controlled gradient and Hessian on complete non-compact manifolds with bounded Ricci curvature and strictly positive injectivity radius and a general growth estimate for the norm of the soliton vector field. This latter is based on a new Toponogov type lemma for manifolds with bounded Ricci curvature, and represents the first known growth estimate for the whole norm of the soliton field in the non-gradient case.
27 Aug 2019
TL;DR: This work considers the manifold of rank-p positive-semidefinite matrices of size n as a quotient of the set of full-rank n-by-p matrices by the orthogonal group in dimension p, and obtains expressions for the Riemannian curvature tensor and the sectional curvature of the manifold.
Abstract: We consider the manifold of rank-p positive-semidefinite matrices of size n, seen as a quotient of the set of full-rank n-by-p matrices by the orthogonal group in dimension p. The resulting distance coincides with the Wasserstein distance between centered degenerate Gaussian distributions. We obtain expressions for the Riemannian curvature tensor and the sectional curvature of the manifold. We also provide tangent vectors spanning planes associated with the extreme values of the sectional curvature.
TL;DR: In this article, a new construction of conserved charges in asymptotically anti-de Sitter spacetimes in Einstein's gravity was given, which is explicitly gauge invariant and makes direct use of the linearized curvature tensor instead of the metric perturbation.
Abstract: We give a new construction of conserved charges in asymptotically anti-de Sitter spacetimes in Einstein's gravity. The new formula is explicitly gauge-invariant and makes direct use of the linearized curvature tensor instead of the metric perturbation. As an example, we compute the mass and angular momentum of the Kerr-AdS black holes.
TL;DR: In this article, a detailed analysis of conformally flat spherically symmetric fluid distributions governed by a polytropic equation of state utilizing the metric f (R) gravity, R being the Ricci scalar is provided.
Abstract: This paper provides a detailed analysis of conformally flat spherically symmetric fluid distributions governed by a polytropic equation of state utilizing the metric f (R) gravity, R being the Ricci scalar. The Lane-Emden equation is formulated using Bianchi identity and couple of polytropic equation of states. We discussed two families of relativistic polytropes and the constraints are evaluated for their necessary physical applications. The physical practicability of polytropes is scrutinized via energy conditions. An explicit relation of the Weyl tensor with material variables is explored. The condition of vanishing Weyl tensor is imposed on the Lane-Emden equation in the background of particular f (R) model in both cases to explore physical constraints on polytropes.
11 Mar 2019
TL;DR: In this paper, an approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying the intrinsic spirit of Riemann's curvature tensor.
Abstract: The mathematical modelling in mechanics has a long-standing history as related to geometry, and significant progresses have often been achieved by the invention of new geometrical tools. Also, it happened that the elucidation of practical issues led to the invention of new scientific concepts, and possibly new paradigms, with potential impact far beyond. One such example is Riemann’s intrinsic view in geometry, that offered a radically new insight in the Physics of the early 20th century. On the other hand, the rather recent intrinsic approaches in elasticity and elasto-plasticity also share this philosophical standpoint of looking from inside, i.e., from the “manifold” point of view. Of course, this approach requires smoothness, and is thus incomplete for an analyst. Nevertheless, its first aim is to highlight the concepts of metric, curvature and torsion; these notions are addressed in the first part of this survey paper. In a second part, they are given a precise functional meaning and their properties are studied systematically. Further, a novel approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying this intrinsic spirit. The main mathematical object in this theory is the incompatibility operator, i.e., a linearized version of Riemann’s curvature tensor. So far, this route not only has led the authors to a new model with a solid functional foundation and proof of existence results, but also to a framework with a minimal amount of ad-hoc assumptions, and complying with both the basic principles of thermodynamics and invariance principles of Physics. The questions arising from this novel approach are complex and intriguing, but we believe that the model is now sufficiently well posed to be studied simultaneously as a problem of mathematics and of mechanics. Most of the research programme remains to be done, and this survey paper is written to present our model, with a particular care to put this approach into a historical perspective.
TL;DR: In this article, a divergence-free rank-4 tensor of which the trace is the cosmological Einstein tensor is constructed for asymptotically anti-de Sitter spacetimes.
Abstract: Starting from a divergence-free rank-4 tensor of which the trace is the cosmological Einstein tensor, we give a construction of conserved charges in Einstein's gravity and its higher derivative extensions for asymptotically anti-de Sitter spacetimes. The current yielding the charge is explicitly gauge invariant, and the charge expression involves the linearized Riemann tensor at the boundary. Hence, to compute the mass and angular momenta in these spacetimes, one just needs to compute the linearized Riemann tensor. We give two examples.
TL;DR: In this article, the authors developed the formulation of DFT on pre-QP-manifold and unified the consistency conditions like section condition and closure constraint by a weak master equation.
Abstract: We develop the formulation of DFT on pre-QP-manifold. The consistency conditions like section condition and closure constraint are unified by a weak master equation. The Bianchi identities are also characterized by the pre-Bianchi identity. Then, the background metric and connections are formulated by using covariantized pre-QP-manifold. An application to the analysis of the DFT on group manifold is given.
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TL;DR: In this article, the first and second moments of the Frechet mean were studied in affine connection spaces, and it was shown that the empirical mean has a bias in 1/n proportional to the gradient of the curvature tensor.
Abstract: The asymptotic concentration of the Frechet mean of IID random variables on a Rieman-nian manifold was established with a central limit theorem by Bhattacharya & Patrangenaru (BP-CLT) [6]. This asymptotic result shows that the Frechet mean behaves almost as the usual Euclidean case for sufficiently concentrated distributions. However, the asymptotic covariance matrix of the empirical mean is modified by the expected Hessian of the squared distance. This Hessian matrix was explicitly computed in [5] for constant curvature spaces in order to relate it to the sectional curvature. Although explicit, the formula remains quite difficult to interpret, and the intuitive effect of the curvature on the asymptotic convergence remains unclear. Moreover, we are most often interested in the mean of a finite sample of small size in practice. In this work, we aim at understanding the effect of the manifold curvature in this small sample regime. Last but not least, one would like computable and interpretable approximations that can be extended from the empirical Frechet mean in Rie-mannian manifolds to the empirical exponential barycenters in affine connection manifolds. For distributions that are highly concentrated around their mean, and for any finite number of samples, we establish explicit Taylor expansions on the first and second moment of the empirical mean thanks to a new Taylor expansion of the Riemannian log-map in affine connection spaces. This shows that the empirical mean has a bias in 1/n proportional to the gradient of the curvature tensor contracted twice with the covariance matrix, and a modulation of the convergence rate of the covariance matrix proportional to the covariance-curvature tensor. We show that our non-asymptotic high concentration expansion is consistent with the asymptotic expansion of the BP-CLT. Experiments on constant curvature spaces demonstrate that both expansions are very accurate in their domain of validity. Moreover, the modulation of the convergence rate of the empirical mean's covariance matrix is explicitly encoded using a scalar multiplicative factor that gives an intuitive vision of the impact of the curvature: the variance of the empirical mean decreases faster than in the Euclidean case in negatively curved space forms, with an infinite speed for an infinite negative curvature. This suggests potential links with the stickiness of the Frechet mean described in stratified spaces. On the contrary, the variance of the empirical mean decreases more slowly than in the Euclidean case in positive curvature space forms, with divergence when we approach the limits of the Karcher & Kendall concentration conditions with a uniform distribution on the equator of the sphere, for which the Frechet mean is not a single point any more.
TL;DR: In this article, the authors rigorously derive a Kirchhoff plate theory via Γ-convergence from a three-dimensional model that describes the finite elasticity of an elastically heterogeneous, thin sheet.
Abstract: We rigorously derive a Kirchhoff plate theory, via Γ-convergence, from a three-dimensional model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables x ′. At the same time, the spontaneous strain is h -close to the identity, where h is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into ℝ3 , with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from an x ′-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.
TL;DR: In this paper, the authors show that if M is a compact submanifold of euclidean space, then the volumes of small tubes about M are given by a polynomial in the radius r, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of M with respect to the induced metric.
Abstract: A famous theorem of Weyl states that if M is a compact submanifold of euclidean space, then the volumes of small tubes about M are given by a polynomial in the radius r, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of M with respect to the induced metric. It is natural to interpret this phenomenon in terms of curvature measures and smooth valuations, in the sense of Alesker, canonically associated to the Riemannian structure of M. This perspective yields a fundamental new structure in Riemannian geometry, in the form of a certain abstract module over the polynomial algebra $${\mathbb{R}[t]}$$
that reflects the behavior of Alesker multiplication. This module encodes a key piece of the array of kinematic formulas of any Riemannian manifold on which a group of isometries acts transitively on the sphere bundle. We illustrate this principle in precise terms in the case where M is a complex space form.
TL;DR: The dimensional complete non-Ricci flat gradient steady Ricci soliton with potential function for aninline-formula content-typeMath/mathml is shown.
Abstract: We show that for an $n$ dimensional complete non Ricci flat gradient steady Ricci soliton with potential function $f$ bounded above by a constant and curvature tensor $Rm$ satisfying $\overline{\lim}_{r\to \infty} r|Rm| 0$, improving a result of [36]. For any four dimensional complete non Ricci flat gradient steady Ricci soliton with scalar curvature $S\to 0$ as $r\to \infty$, we prove that $|Rm|\leq cS$ for some constant $c>0$, improving an estimate in [11]. As an application, we show that for a four dimensional complete non Ricci flat gradient steady Ricci soliton, $|Rm|$ decays exponentially provided that $\overline{\lim}_{r\to \infty} rS$ is sufficiently small and $f$ is bounded above by a constant.