Showing papers on "Riemann curvature tensor published in 2014"
TL;DR: In this paper, an extension of the Ryu-Takayanagi prescription for curvature squared theories of gravity in the bulk, and comment on a prescription for more general theories, is presented.
Abstract: We derive an extension of the Ryu-Takayanagi prescription for curvature squared theories of gravity in the bulk, and comment on a prescription for more general theories. This results in a new entangling functional, that contains a correction to Wald’s entropy. The new term is quadratic in the extrinsic curvature. The coefficient of this correction is a second derivative of the lagrangian with respect to the Riemann tensor. For Gauss-Bonnet gravity, the new functional reduces to Jacobson-Myers’.
261 citations
TL;DR: This paper employs a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau to derive lower RicCI curvature bounds on graphs in terms of such local clustering coefficients.
Abstract: In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature-dimension inequalities on graphs, building upon previous work of several authors.
185 citations
TL;DR: In this paper, the authors studied the most general covariant action of gravity up to terms that are quadratic in curvature and derived the equations of motion for such actions containing an arbitrary number of the covariant D'Alembertian operators.
Abstract: In this paper we study the most general covariant action of gravity up to terms that are quadratic in curvature. In particular this includes non-local, infinite derivative theories of gravity which are ghost-free and exhibit asymptotic freedom in the ultraviolet. We provide a detailed algorithm for deriving the equations of motion for such actions containing an arbitrary number of the covariant D'Alembertian operators, and this is our main result. We also perform a number of tests on the field equations we derive, including checking the Bianchi identities and the weak-field limit. Lastly, we consider the special subclass of ghost and asymptotically free theories of gravity by way of an example.
165 citations
TL;DR: It is shown that by inscribing a director field gradient across the sheet's thickness, one can obtain a nontrivial hyperbolic reference curvature tensor, which together with the prescription of a reference metric allows dictation of actual configurations for a thin sheet of nematic elastomer.
Abstract: A thin sheet of nematic elastomer attains 3D configurations depending on the nematic director field upon heating. In this Letter, we describe the intrinsic geometry of such a sheet and derive an expression for the metric induced by general nematic director fields. Furthermore, we investigate the reverse problem of constructing a director field that induces a specified 2D geometry. We provide an explicit recipe for how to construct any surface of revolution using this method. Finally, we show that by inscribing a director field gradient across the sheet's thickness, one can obtain a nontrivial hyperbolic reference curvature tensor, which together with the prescription of a reference metric allows dictation of actual configurations for a thin sheet of nematic elastomer.
110 citations
TL;DR: In this paper, the surface equation of motion for general four-derivative gravity theory is derived by minimizing the holographic entanglement entropy functional resulting from this proposed formula. But the two results do not match in their entirety.
Abstract: In arXiv:1310.5713
[1] and arXiv:1310.6659
[2] a formula was proposed as the entanglement entropy functional for a general higher-derivative theory of gravity, whose lagrangian consists of terms containing contractions of the Riemann tensor. In this paper, we carry out some tests of this proposal. First, we find the surface equation of motion for general four-derivative gravity theory by minimizing the holographic entanglement entropy functional resulting from this proposed formula. Then we calculate the surface equation for the same theory using the generalized gravitational entropy method of arXiv:1304.4926
[3]. We find that the two do not match in their entirety. We also construct the holographic entropy functional for quasi-topological gravity, which is a six-derivative gravity theory. We find that this functional gives the correct universal terms. However, as in the R2 case, the generalized gravitational entropy method applied to this theory does not give exactly the surface equation of motion coming from minimizing the entropy functional.
101 citations
TL;DR: In this paper, a conformal-anomaly driven inflation in the flat homogeneous and isotropic universe is studied, and it is shown that the Ricci scalar decreases during inflation and the standard evolution history of the universe is recovered at the small curvature regime.
Abstract: We explore conformal-anomaly driven inflation in $F(R)$ gravity without invoking the scalar-tensor representation. We derive the stress-energy tensor of the quantum anomaly in the flat homogeneous and isotropic universe. We investigate a suitable toy model of exponential gravity plus the quantum contribution due to the conformal anomaly, which leads to the de Sitter solution. It is shown that in $F(R)$ gravity model, the curvature perturbations with its enough amplitude consistent with the observations are generated during inflation. We also evaluate the number of $e$-folds at the inflationary stage and the spectral index $n_\mathrm{s}$ of scalar modes of the curvature perturbations by analogy with scalar tensor theories, and compare them with the observational data. As a result, it is found that the Ricci scalar decreases during inflation and the standard evolution history of the universe is recovered at the small curvature regime. Furthermore, it is demonstrated that in our model, the tensor-to-scalar ratio of the curvature perturbations can be a finite value within the $68\%\,\mathrm{CL}$ error of the very recent result found by the BICEP2 experiment.
92 citations
TL;DR: In this paper, a general study on the collapse of axially (and reflection-)symmetric sources in the context of general relativity is carried out, where all basic equations and concepts required to perform such a general analysis are deployed.
Abstract: We carry out a general study on the collapse of axially (and reflection-)symmetric sources in the context of general relativity. All basic equations and concepts required to perform such a general study are deployed. These equations are written down for a general anisotropic dissipative fluid. The proposed approach allows for analytical studies as well as for numerical applications. A causal transport equation derived from the Israel-Stewart theory is applied, to discuss some thermodynamic aspects of the problem. A set of scalar functions (the structure scalars) derived from the orthogonal splitting of the Riemann tensor are calculated and their role in the dynamics of the source is clearly exhibited. The characterization of the gravitational radiation emitted by the source is discussed.
88 citations
TL;DR: In this paper, the generalized flux formulation of Double Field Theory was extended to include all the first order bosonic contributions to the expansion of the heterotic string low energy effective theory, and the generalized tangent space and duality group were enhanced by $\alpha'$ corrections.
Abstract: We extend the generalized flux formulation of Double Field Theory to include all the first order bosonic contributions to the $\alpha '$ expansion of the heterotic string low energy effective theory. The generalized tangent space and duality group are enhanced by $\alpha'$ corrections, and the gauge symmetries are generated by the usual (gauged) generalized Lie derivative in the extended space. The generalized frame receives derivative corrections through the spin connection with torsion, which is incorporated as a new degree of freedom in the extended bein. We compute the generalized fluxes and find the Riemann curvature tensor with torsion as one of their components. All the four-derivative terms of the action, Bianchi identities and equations of motion are reproduced. Using this formalism, we obtain the first order $\alpha'$ corrections to the heterotic Buscher rules. The relation of our results to alternative formulations in the literature is discussed and future research directions are outlined.
86 citations
TL;DR: In this paper, the wave operators do not factorize in general, and identify the Weyl tensor and its derivatives as the obstruction to factorization, and give a manifestly factorized form for them on (A)dS backgrounds for arbitrary spin and on Einstein backgrounds for spin 2.
Abstract: We analyze free conformal higher spin actions and the corresponding wave operators in arbitrary even dimensions and backgrounds. We show that the wave operators do not factorize in general, and identify the Weyl tensor and its derivatives as the obstruction to factorization. We give a manifestly factorized form for them on (A)dS backgrounds for arbitrary spin and on Einstein backgrounds for spin 2. We are also able to fix the conformal wave operator in d = 4 for s = 3 up to linear order in the Riemann tensor on generic Bach-flat backgrounds.
77 citations
TL;DR: In this article, the most general N = 2 supersymmetric solutions of D = 11 supergravity consisting of a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold Y7 were analyzed.
Abstract: We analyse the most general N = 2 supersymmetric solutions of D = 11 supergravity consisting of a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold Y7. We show that the necessary and sufficient conditions for supersymmetry can be phrased interms of a local SU(2)-structure on Y7. Solutions with non-zero M2-brane charge also admit a canonical contact structure, in terms of which many physical quantities can be expressed, including the free energy and the scaling dimensions of operators dual to supersymmetric wrapped M5-branes. We show that a special class of solutions is singled out by imposing an additional symmetry, for which the problem reduces to solving a second order non-linear ODE. As well as recovering a known class of solutions, that includes the IR fixed point of a mass deformation of the ABJM theory, we also find new solutions which are dual to cubic deformations. In particular, we find a new supersymmetric warped AdS4 × S 7 solution with non-trivial four-form flux.
75 citations
TL;DR: In this article, the authors considered the problem of the scalar, electromagnetic, and linearized gravitational field equations in Minkowski spacetime, with source given by a particle moving on a null geodesic.
Abstract: We consider the retarded solution to the scalar, electromagnetic, and linearized gravitational field equations in Minkowski spacetime, with source given by a particle moving on a null geodesic. In the scalar case and in the Lorenz gauge in the electromagnetic and gravitational cases, the retarded integral over the infinite past of the source does not converge as a distribution, so we cut off the null source suitably at a finite time ${t}_{0}$ and then consider two different limits: (i) the limit as the observation point goes to null infinity at fixed ${t}_{0}$, from which the ``$1/r$'' part of the fields can be extracted and (ii) the limit ${t}_{0}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ at fixed ``observation point.'' The limit (i) gives rise to a ``velocity kick'' on distant test particles in the scalar and electromagnetic cases, and it gives rise to a ``memory effect'' (i.e., a permanent change in relative separation of two test particles) in the linearized gravitational case, in agreement with previous analyses. As already noted, the second limit does not exist in the scalar case or for the Lorenz gauge vector potential and Lorenz gauge metric perturbation in the electromagnetic and linearized gravitational cases. However, in the electromagnetic case, we obtain a well-defined distributional limit for the electromagnetic field strength, and in the linearized gravitational case, we obtain a well-defined distributional limit for the linearized Riemann tensor. In the gravitational case, this limit agrees with the Aichelberg-Sexl solution. There is no memory effect associated with this limiting solution. This strongly suggests that the memory effect---including nonlinear memory effect of Christodoulou---should not be interpreted as arising simply from the passage of (effective) null stress energy to null infinity but rather as arising from a ``burst of radiation'' associated with the creation of the null stress energy [as in case (i) above] or, more generally, with radiation present in the spacetime that was not ``produced'' by the null stress energy.
TL;DR: Yang et al. as discussed by the authors proposed a modified couple stress theory for anisotropic elasticity, in which the curvature (rotation gradient) tensor was asymmetric and the couple stress moment tensor is symmetric.
Abstract: A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.
TL;DR: In this paper, necessary and sufficient conditions for a three-dimensional contact subriemannian manifold to satisfy the Ricci curvature bound were discovered, which is one of the possible generalizations of Ricci curve bound to more general metric measure spaces.
Abstract: Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.
TL;DR: In this article, the geometric properties of the manifold of states described as (uniform) matrix product states are studied, and the main interest is in the states living in the tangent space to the base manifold, which have been shown to be interesting in relation to time dependence and elementary excitations.
Abstract: We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e., the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a Kahler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.
TL;DR: In this article, the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on a complete, simply connected Riemannian manifold with negative curvature were obtained.
Abstract: Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on M. Furthermore, if M is with strictly negative curvature, we show that the Lp Hardy inequalities can be globally refined by adding remainder terms like the Brezis–Vazquez improvement in case p ≥ 2, which is contrary to the case of Euclidean spaces.
TL;DR: In this paper, a minimal isoparametric hypersurface of a closed Riemannian manifold is shown to have positive Ricci curvature. But this is not unique in the family of isopararomorphic hypersurfaces.
Abstract: In our previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exists at least one minimal isoparametric hypersurface. In this paper, we show such a minimal isoparametric hypersurface is also unique in the family if the ambient manifold has positive Ricci curvature. Moreover, we give a proof of Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on a complete Riemannian manifold are minimal. Further, we study isoparametric hypersurfaces with constant principal curvatures in general Riemannian manifolds. It turns out that in this case the focal submanifolds have the same properties as those in the standard sphere, i.e., the shape operator with respect to any normal direction has common constant principal curvatures. Some necessary conditions involving Ricci curvature and scalar curvature are also derived.
TL;DR: A tensor is presented by combining Riemann–Christoffel curvature Tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases and is proved to have equivalency of different geometric structures.
Abstract: In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann–Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann–Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see Theorems 6.3, 6.4, 6.5, 6.6 and 6.7; Tables 1 and 2).
TL;DR: In this paper, necessary and sufficient conditions for warped product manifolds (M,g) of dimension 4, with 1-dimensional base, and in particular for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition were given.
Abstract: We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - a g) \leqslant 1, for some a \in R, or non-quasi-Einstein.
TL;DR: In this paper, a general study of the geometric properties of universal spacetimes in arbitrary dimension and a broader class of such metrics is presented, which admit a non-vanishing cosmological constant and in general do not have to possess a covariant constant or recurrent null vector field.
Abstract: Universal spacetimes are spacetimes for which all conserved symmetric rank-2 tensors, constructed as contractions of polynomials from the metric, the Riemann tensor and its covariant derivatives of arbitrary order, are multiples of the metric. Consequently, metrics of universal spacetimes solve vacuum equations of all gravitational theories, with the Lagrangian being a polynomial curvature invariant constructed from the metric, the Riemann tensor and its derivatives of arbitrary order. In the literature, universal metrics are also discussed as metrics with vanishing quantum corrections and as classical solutions to string theory. Widely known examples of universal metrics are certain Ricci-flat pp waves. In this paper, we start a general study of the geometric properties of universal metrics in arbitrary dimension and arrive at a broader class of such metrics. In contrast with pp waves, these universal metrics also admit a non-vanishing cosmological constant and in general do not have to possess a covariant constant or recurrent null vector field. First, we show that a universal spacetime is necessarily a constant curvature invariant spacetime, i.e. all curvature invariants constructed from the Riemann tensor and its derivatives are constant. Then we focus on type N spacetimes, where we arrive at a simple necessary and sufficient condition: a type N spacetime is universal if and only if it is an Einstein Kundt spacetime. A class of type III Kundt universal metrics is also found. Several explicit examples of universal metrics are presented.
TL;DR: In this paper, a covariant constraint analysis of massive gravity was performed for its entire parameter space, demonstrating that the model generically propagates 5 degrees of freedom; this was also verified by a new and streamlined Hamiltonian description.
Abstract: We perform a covariant constraint analysis of massive gravity valid for its entire parameter space, demonstrating that the model generically propagates 5 degrees of freedom; this is also verified by a new and streamlined Hamiltonian description. The constraint’s covariant expression permits computation of the model’s caustics. Although new features such as the dynamical Riemann tensor appear in the characteristic matrix, the model still exhibits the pathologies uncovered in earlier work: superluminality and likely acausalities.
TL;DR: In this article, the authors extend the Einstein-aether theory to include the Maxwell field in a nontrivial manner by taking into account its interaction with the time-like unit vector field characterizing the aether.
TL;DR: In this paper, a compact formulation of the field-strengths, Bianchi identities and gauge transformations for tensor hierarchies in gauged maximal supergravity theories is given, and a key role in the construction is played by the recently-introduced tensor hierarchy algebra.
Abstract: A compact formulation of the field-strengths, Bianchi identities and gauge transformations for tensor hierarchies in gauged maximal supergravity theories is given. A key role in the construction is played by the recently-introduced tensor hierarchy algebra.
TL;DR: In this paper, it was shown that if the critical point metric has harmonic curvature, then it is isometric to a standard sphere, and this conjecture was proved in 1984 by Besse, but has yet to be proved.
Abstract: On a compact $n$-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove that if the manifold with the critical point metric has harmonic curvature, then it is isometric to a standard sphere.
TL;DR: In this article, curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds were studied and a new curvature degree condition was obtained.
Abstract: We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain:
Geometric conditions ensuring the compactness of the underlying manifold (Bonnet–Myers type results);
Volume estimates of metric balls;
Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian;
Spectral gap estimates.
TL;DR: In this article, the authors introduce a new operator in Loop Quantum Gravity (LQG) related to the 3D scalar curvature, which is based on Regge Calculus.
Abstract: We introduce a new operator in Loop Quantum Gravity - the 3D curvature operator - related to the 3-dimensional scalar curvature. The construction is based on Regge Calculus. We define it starting from the classical expression of the Regge curvature, then we derive its properties and discuss some explicit checks of the semi-classical limit.
TL;DR: In this article, the authors introduced the notion of a Kummer tensor density of rank four, K i j k l, which is a cubic algebraic functional of a tensor densifier of rank 4 T i jk l, and decompose K irreducibly under the 4-dimensional linear group G L ( 4, R ) and subsequently under the Lorentz group S O ( 1, 3 ).
Journal Article•
TL;DR: A new objective metric for assessing the visual difference between a reference triangular mesh and its distorted version produced by lossy operations, such as noise addition, simplification, compression and watermarking is presented.
Abstract: Perceptual quality assessment of 3D triangular meshes is crucial for a variety of applications. In this paper, we present a new objective metric for assessing the visual difference between a reference triangular mesh and its distorted version produced by lossy operations such as noise addition, simplification, compression and watermarking. The proposed metric is based on the measurement of a distance between curvature tensors of the two meshes under comparison. Our algorithm uses not only tensor eigenvalues (i.e., curvature amplitudes) but also tensor eigenvectors (i.e., principal curvature directions) to derive a perceptually-oriented tensor distance. The proposed metric also accounts for the visual masking effect of the human visual system, through a roughness-based weighting of the local tensor distance. A final score that reflects the visual difference between two meshes is obtained via a Minkowski pooling of the weighted local tensor distances over the mesh surface. We validate the performance of our algorithm on four subjectively-rated mesh visual quality databases, and compare the proposed method with state-of-the-art objective metrics. Experimental results show that our approach achieves high correlation between objective scores and subjective assessments.
TL;DR: In this article, the authors investigate the AdS/CFT interpretation of the class of algebraically special solutions of Einstein gravity with a negative cosmological constant, and they introduce a formalism for studying conformal, relativistic fluids in 2 + 1 dimensions that reduces everything to the manipulation of scalar quantities.
Abstract: We investigate the AdS/CFT interpretation of the class of algebraically special solutions of Einstein gravity with a negative cosmological constant. Such solutions describe a CFT living in a 2 + 1 dimensional time-dependent geometry that, generically, has no isometries. The algebraically special condition implies that the expectation value of the CFT energy-momentum tensor is a local function of the boundary metric. When such a spacetime is slowly varying, the fluid/gravity approximation is valid and one can read off the values of certain higher order transport coefficients. To do this, we introduce a formalism for studying conformal, relativistic fluids in 2 + 1 dimensions that reduces everything to the manipulation of scalar quantities.
TL;DR: In this paper, the link between the recently found E7(7) generalised geometric structures, which are based on the SU(8) invariant reformulation of D = 11 supergravity proposed long ago, and newer results obtained in the framework of recent approaches to generalised geometry is built in and manifest from the outset.
Abstract: In this paper we establish and clarify the link between the recently found E7(7) generalised geometric structures, which are based on the SU(8) invariant reformulation of D = 11 supergravity proposed long ago, and newer results obtained in the framework of recent approaches to generalised geometry, where E7(7) duality is built in and manifest from the outset. In making this connection, the so-called generalised vielbein postulate plays a key role. We explicitly show how this postulate can be used to define an E7(7) valued affine connection and an associated covariant derivative, which yields a generalised curvature tensor for the E7(7) based exceptional geometry. The analysis of the generalised vielbein postulate also provides a natural explanation for the emergence of the embedding tensor from higher dimensions.
TL;DR: A method for computing the coarse Ricci curvature is described, to give sharper results, in the specific, but crucial case of polyhedral surfaces.
Abstract: The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu & Yau, Jost & Liu have used and extended this notion for graphs giving estimates for the curvature and hence the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific but crucial case of polyhedral surfaces.