Non-vanishing Curvature Research Articles

The Gauss map and second fundamental form of surfaces in a Lie group

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $$\mathbb {S}^3$$ and establish a correspondence between simply connected surfaces having extrinsic curvature K, K different from 0 and $$-1$$ , immersed in $$\mathbb {S}^3$$ with simply connected surfaces having non-vanishing extrinsic curvature immersed in the euclidean space $$\mathbb {R}^3$$ . Moreover, we show that a surface isometrically immersed in $$\mathbb {S}^3$$ has positive constant extrinsic curvature if, and only if, its Gauss map is a harmonic map into the Riemann sphere.

Instanton solutions from Abelian sinh-Gordon and Tzitzeica vortices

We study the Abelian Higgs vortex solutions to the sinh-Gordon equation and the elliptic Tzitzeica equation. Starting from these particular vortices, we construct solutions to the Taubes equation with higher vortex number, on surfaces with conical singularities. We then, analyse more general properties of vortices on such singular surfaces and propose a method to obtain vortices on conifolds from vortices on surfaces of revolution. We apply our method to construct explicit vortex solutions on the Poincar\'e disk with a conical singularity in the centre, to which we refer as the "hyperbolic cone". We uplift the Abelian sinh-Gordon and Tzitzeica vortex solutions to four dimensions and construct cylindrically symmetric, self-dual Yang-Mills instantons on a non-self-dual (nor anti-self-dual) $4$-dimensional K\"ahler manifold with non-vanishing scalar curvature. The instantons we construct in this way cannot be obtained via a twistorial approach.

Flat slices in Minkowski space

Minkowski space, flat spacetime, with a distance measure in natural units of , or equivalently, with spacetime metric diag(−1, +1, +1, +1), is recognized as a fundamental arena for physics. The Poincaré group, the set of all rigid spacetime rotations and translations, is the symmetry group of Minkowski space. The action of this group preserves the form of the spacetime metric. Each t = constant slice of each preferred coordinate system is flat. We show that there are also nontrivial non-singular representations of Minkowski space with complete flat slices. If the embedding of the flat slices decays appropriately at infinity, the only flat slices are the standard ones. However, if we remove the decay condition, we find non-trivial flat slices with non-vanishing extrinsic curvature. We write out explicitly the coordinate transformation to a frame with such slices.

Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature

In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner’s metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold’s approach. The corresponding Ruppeiner’s metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures.

$L^p$-nondegenerate Radon-like operators with vanishing rotational curvature

We consider the $L^p \rightarrow L^q$ mapping properties of a model family of Radon-like operators integrating functions over n-dimensional submanifolds of ${\mathbb R}^{2n}$. It is shown that nonvanishing rotational curvature is never generic when $n \geq 2$ and is, in fact, impossible for all but finitely many values of $n$. Nevertheless, operators satisfying the same $L^p \rightarrow L^q$ estimates as the nondegenerate case (modulo the endpoint) are dense in the model family for all $n$.

A characterization of Laguerre isoparametric hypersurfaces of the Euclidian space

We consider \(M^n,\,n\ge 3\), umbilic-free hypersurfaces in the Euclidean space, with nonvanishing principal curvatures. We prove that \(M\) is a Laguerre isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Laguerre curvatures.

Brane cosmological constant in two-brane warped geometry model

In the backdrop of generalized Randall–Sundrum braneworld scenario, we look for the possible origin of an effective four-dimensional cosmological constant (Ω vis ) on the visible three-brane due to the effects of bulk curvature and the modulus field that can either be a constant or a time-dependent quantity. In case of constant modulus field, the induced Ω vis leads to an exponentially expanding universe and the presence of vacuum energy densities on either of the three-branes as well as a nonvanishing bulk curvature [Formula: see text] are essential to generate an effective Ω vis . The Hubble constant turns out to be equal to the visible brane cosmological constant which agrees with the present result. In an alternative scenario, a time-dependent modulus field is found to be capable of decelerating the universe. The Hubble parameter, in this case is determined for a slowly time-varying modulus field.

Gamma-Limit of a Model for the Elastic Energy of an Inextensible Ribbon

A Γ-convergence result involving the elastic bending energy of a narrow inextensible ribbon is established. As a consequence of the result, the energy is reduced to a one-dimensional integral, over the centerline of the ribbon, in which the aspect ratio of the ribbon appears as a small parameter. That integral is observed to increase monotonically with the aspect ratio. The Γ-limit of the family of energies is taken in a Sobolev space of centerlines with nonvanishing curvature. In that space, it is shown that the Γ-limit is a functional first proposed by Sadowsky in the context of narrow ribbons that form Möbius bands. The results obtained here do not apply to such ribbons, since the centerline of a Möbius band must have at least one inflection point. As a first step toward dealing with such inflection points, a result concerning the lower semicontinuity of the Sadowsky functional with inflection points comprising a set of measure zero within the domain of an arclength parameterization is presented.

Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces

We describe natural K\ahler or para-K\ahler structures of the spaces of geodesics of pseudo-Riemannian space forms and relate the local geometry of hypersurfaces of space forms to that of their normal congruences, or Gauss maps, which are Lagrangian submanifolds. The space of geodesics L(S^{n+1}_{p,1}) of a pseudo-Riemannian space form S^{n+1}_{p,1} of non-vanishing curvature enjoys a K\ahler or para-K\ahler structure (J,G) which is in addition Einstein. Moreover, in the three-dimensional case, L(S^{n+1}_{p,1}) enjoys another K\ahler or para-K\ahler structure (J',G') which is scalar flat. The normal congruence of a hypersurface s of S^{n+1}_{p,1} is a Lagrangian submanifold \bar{s} of L(S^{n+1}_{p,1}), and we relate the local geometries of s and \bar{s}. In particular \bar{s} is totally geodesic if and only if s has parallel second fundamental form. In the three-dimensional case, we prove that \bar{s} is minimal with respect to the Einstein metric G (resp. with respect to the scalar flat metric G') if and only if it is the normal congruence of a minimal surface s (resp. of a surface s with parallel second fundamental form); moreover \bar{s} is flat if and only if s is Weingarten.

Massless particles in generalized Robertson–Walker 4-spacetimes

The model of massless relativistic particle relying in a curvature-dependent action functional is considered in the framework of generalized Robertson–Walker 4-spacetimes. The discussion is based on the number of nonvanishing Frenet curvatures, and we obtain characterizations, examples, and nonexistence results for the critical points of the action functional, which are included in the fibers.

Type II hidden symmetries for the homogeneous heat equation in some general classes of Riemannian spaces

We study the reduction of the heat equation in Riemannian spaces which admit a gradient Killing vector, a gradient homothetic vector and in Petrov Type D, N, II and Type III space–times. In each reduction we identify the source of the Type II hidden symmetries. More specifically we find that (a) if we reduce the heat equation by the symmetries generated by the gradient KV the reduced equation is a linear heat equation in the nondecomposable space. (b) If we reduce the heat equation via the symmetries generated by the gradient HV the reduced equation is a Laplace equation for an appropriate metric. In this case the Type II hidden symmetries are generated from the proper CKVs. (c) In the Petrov space–times the reduction of the heat equation by the symmetry generated from the nongradient HV gives PDEs which inherit the Lie symmetries hence no Type II hidden symmetries appear. We apply the general results to cases in which the initial metric is specified. We consider the case that the irreducible part of the decomposed space is a space of constant nonvanishing curvature and the case of the spatially flat Friedmann–Robertson–Walker space–time used in Cosmology. In each case we give explicitly the Type II hidden symmetries provided they exist.

Constraining neutrino properties with a Euclid-like galaxy cluster survey

We perform a forecast analysis on how well a Euclid-like photometric galaxy cluster survey will constrain the total neutrino mass and effective number of neutrino species. We base our analysis on the Monte Carlo Markov Chains technique by combining information from cluster number counts and cluster power spectrum. We find that combining cluster data with Cosmic Microwave Background (CMB) measurements from Planck improves by more than an order of magnitude the constraint on neutrino masses compared to each probe used independently. For the ΛCDM+mν model the 2σ upper limit on total neutrino mass shifts from ∑mν < 0.35 eV using cluster data alone to ∑mν < 0.031 eV when combined with Planck data. When a non-standard scenario with Neff≠3.046 number of neutrino species is considered, we estimate an upper limit of Neff < 3.14 (95%CL), while the bounds on neutrino mass are relaxed to ∑mν < 0.040 eV. This accuracy would be sufficient for a 2σ detection of neutrino mass even in the minimal normal hierarchy scenario (∑mν ≃ 0.05 eV). In addition to the extended ΛCDM+mν+Neff model we also consider scenarios with a constant dark energy equation of state and a non-vanishing curvature. When these models are considered the error on ∑mν is only slightly affected, while there is a larger impact of the order of ∼ 15% and ∼ 20% respectively on the 2σ error bar of Neff with respect to the standard case. To assess the effect of an uncertain knowledge of the relation between cluster mass and optical richness, we also treat the ΛCDM+mν+Neff case with free nuisance parameters, which parameterize the uncertainties on the cluster mass determination. Adopting the over-conservative assumption of no prior knowledge on the nuisance parameter the loss of information from cluster number counts leads to a large degradation of neutrino constraints. In particular, the upper bounds for ∑mν are relaxed by a factor larger than two, ∑mν < 0.083 eV (95%CL), hence compromising the possibility of detecting the total neutrino mass with good significance. We thus confirm the potential that a large optical/near-IR cluster survey, like that to be carried out by Euclid, could have in constraining neutrino properties, and we stress the importance of a robust measurement of masses, e.g. from weak lensing within the Euclid survey, in order to full exploit the cosmological information carried by such survey.

Scalar Curvature and Q-Curvature of Random Metrics

We define a family of probability measures on the set of Riemannian metrics lying in a fixed conformal class, induced by Gaussian probability measures on the (logarithms of) conformal factors. We control the smoothness of the resulting metric by adjusting the decay rate of the variance of the random Fourier coefficients of the conformal factor. On a compact surface, we evaluate the probability of the set of metrics with non-vanishing Gauss curvature, lying in a fixed conformal class. On higher-dimensional manifolds, we estimate the probability of the set of metrics with non-vanishing scalar curvature (or Q-curvature), lying in a fixed conformal class.

Second order parallel tensor in trans-Sasakian manifolds and connection with Ricci soliton

In this paper we have solved the Eisenhart problem for the symmetric case in the trans-Sasakian manifold of type (α, β) with non-vanishing ξ-sectional curvature and studied some of its consequences. Then we apply our result to obtain a Ricci soliton and studied its behavior for a particular case. Finally we studied the possible consequence for an affine Killing vector field.

Restriction of toral eigenfunctions to hypersurfaces and nodal sets

We give uniform upper and lower bounds for the L2 norm of the restriction of eigenfunctions of the Laplacian on the three-dimensional standard flat torus to surfaces with non-vanishing curvature. We also present several related results concerning the nodal sets of eigenfunctions.

On the Mattila-Sjölin theorem for distance sets

We extend a result, due to Mattila and Sjolin, which says that if the Hausdor dimension of a compact set E ‰ R d , d ‚ 2, is greater than d+1 2 , then the distance set ¢(E) = fjxi yj: x;y 2 Eg contains an interval. We prove this result for distance sets ¢B(E) = fkx i yk B : x;y 2 Eg, where k ¢ k B is the metric induced by the norm defined by a symmetric bounded convex body B with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.

Autonomous three-dimensional Newtonian systems which admit Lie and Noether point symmetries

We determine the autonomous three-dimensional Newtonian systems which admit Lie point symmetries and the three-dimensional autonomous Newtonian Hamiltonian systems which admit Noether point symmetries. We apply the results in order to determine the two-dimensional Hamiltonian dynamical systems which move in a space of constant non-vanishing curvature and are integrable via Noether point symmetries. The derivation of the results is geometric and can be extended naturally to higher dimensions.

Quaternionic Kähler and Spin(7) metrics arising from quaternionic contact Einstein structures

We construct left-invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven-dimensional manifold, equipped with a certain qc structure, has a quaternionic Kahler metric as well as a metric with holonomy contained in Spin(7). As a consequence, we determine explicit quaternionic Kahler metrics and Spin(7)-holonomy metrics, which seem to be new. Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds that are not quaternionic Kahler with either a closed fundamental four form or fundamental two forms defining a differential ideal.

Riemann–Cartan geometry of nonlinear disclination mechanics

In the continuous theory of defects in nonlinear elastic solids, it is known that a distribution of disclinations leads, in general, to a non-trivial residual stress field. To study this problem, we consider the particular case of determining the residual stress field of a cylindrically symmetric distribution of parallel wedge disclinations. We first use the tools of differential geometry to construct a Riemannian material manifold in which the body is stress-free. This manifold is metric compatible, has zero torsion, but has non-vanishing curvature. The problem then reduces to embedding this manifold in Euclidean 3-space following the procedure of a classical nonlinear elastic problem. We show that this embedding can be elegantly accomplished by using Cartan’s method of moving frames and compute explicitly the residual stress field for various distributions in the case of a neo-Hookean material.

Measuring the neutrino mass from future wide galaxy cluster catalogues

We present forecast errors on a wide range of cosmological parameters obtained from a photometric cluster catalogue of a future wide-field Euclid-like survey. We focus in particular on the total neutrino mass as constrained by a combination of the galaxy cluster number counts and correlation function. For the latter we consider only the shape information and the Baryon Acoustic Oscillations (BAO), while marginalising over the spectral amplitude and the redshift space distortions. In addition to the cosmological parameters of the standard ΛCDM+ν model we also consider a non-vanishing curvature, and two parameters describing a redshift evolution for the dark energy equation of state. For completeness, we also marginalise over a set of ``nuisance'' parameters, representing the uncertainties on the cluster mass determination. We find that combining cluster counts with power spectrum information greatly improves the constraining power of each probe taken individually, with errors on cosmological parameters being reduced by up to an order of magnitude. In particular, the best improvements are for the parameters definingthe dynamical evolution of dark energy, where cluster counts break degeneracies. Moreover, the resulting error on neutrino mass is at the level of σ(Mν) ∼ 0.9 eV, comparable with that derived from present Lyα forest measurements and Cosmic Microwave background (CMB) data in the framework of a non-flat Universe. Further adopting Planck priors and reducing the number of free parameters to a ΛCDM+ν cosmology allows to place constraints on the total neutrino mass of σ(Mν) ∼ 0.08 eV, close to the lower bound enforced by neutrino oscillation experiments. Finally, in the optimistic case where uncertainties in the calibration of the mass-observable relation were so small to be neglected, the combination of Planck priors with cluster counts and power spectrum would constrain the total neutrino mass down to σ(Mν) ∼ 0.034 eV, i.e. the minimum neutrino mass predicted by oscillation experiments would be detected in a ΛCDM framework. We thus show that galaxy clusters from future wide galaxy surveys will be an excellent tool for studying cosmology and fundamental physics.