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We study generic conformally flat hypersurfaces in the Euclidean 4-space satisfying a certain condition on the conformal class of the first fundamental form. We first classify such hypersurfaces by determining all conformal-equivalence classes of generic conformally flat hypersurfaces satisfying the condition. Next, as an application of the classification theorem, we give some examples of flat Riemannian metrics which are not conformal to the first fundamental form of any generic conformally flat hypersurface. These flat Riemannian metrics seem to provide counter-examples to Hertrich–Jeromin's claim [3, 5].

We impose perfect fluid concept along with slow expansion approximation to derive new solutions which, considering non-static spherically symmetric metrics, can be treated as Black Holes (BHs). We will refer to these solutions as Quasi BHs. Mathematical and physical features such as Killing vectors, singularities, and mass have been studied. Their horizons and thermodynamic properties have also been investigated. In addition, relationship with other related works (including McVittie's) are described.

Since the conformal transformations of metric do not change its causal structure, we use these transformations to embed the Lemaître metrics into the FRW background. In our approach, conformal transformation is in agreement with the universe expansion regimes. Indeed, we use the Lemaître metrics because the horizon singularity is eliminated in these metrics. Moreover, some physical and mathematical properties of the introduced metrics have been addressed. We show that the resultant metrics include event horizons while their physical radii are increasing as a function of the universe expansion which may provide suitable metrics for investigating the effects of the universe expansion on the black holes.

We develop a technique relating scalar fields with different masses in different conformally flat spacetimes. We apply this technique to the case of FRW spacetimes, with $k=\pm 1,0$, and discuss several examples. We also study various energy conditions and discuss how they constrain the spacetimes related by this technique. We calculate the two-point scalar correlator in the radiation-dominated universe with a hyperbolic spatial section from the one in the Milne universe using the above mapping. Finally, we consider trace anomaly and renormalized stress tensor for conformally flat spacetimes, especially Milne and radiation-dominated universe (with $k=-1$) using the transformation.

The purely affine Lagrangian for linear electrodynamics, that has the form of the Maxwell Lagrangian in which the metric tensor is replaced by the symmetrized Ricci tensor and the electromagnetic field tensor by the tensor of homothetic curvature, is dynamically equivalent to the Einstein–Maxwell equations in the metric–affine and metric formulation. We show that this equivalence is related to the invariance of the Maxwell Lagrangian under conformal transformations of the metric tensor. We also apply to a purely affine Lagrangian the Legendre transformation with respect to the tensor of homothetic curvature to show that the corresponding Legendre term and the new Hamiltonian density are related to the Maxwell–Palatini Lagrangian for the electromagnetic field. Therefore the purely affine picture, in addition to generating the gravitational Lagrangian that is linear in the curvature, justifies why the electromagnetic Lagrangian is quadratic in the electromagnetic field.

The purely affine, metric-affine and purely metric formulation of general relativity are dynamically equivalent and the relation between them is analogous to the Legendre relation between the Lagrangian and Hamiltonian dynamics. We show that one cannot construct a dynamically equivalent, purely affine Lagrangian from a metric-affine or metric F(R) Lagrangian, nonlinear in the curvature scalar. Thus the equivalence between the purely affine picture and the two other formulations does not hold for metric-affine and metric theories of gravity with a nonlinear dependence on the curvature, i.e. F(R) gravity does not have a purely affine formulation. We also show that this equivalence is restored if the metric tensor is conformally transformed from the Jordan to the Einstein frame, in which F(R) gravity turns into general relativity with a scalar field. This peculiar behavior of general relativity, among relativistic theories of gravitation, with respect to purely affine, metric-affine and purely metric variation could indicate the physicality of the Einstein frame. On the other hand, it could explain why this theory cannot interpolate among phenomenological behaviors at different scales.

It is well known that the Klein–Gordon equation in curved spacetime is conformally noninvariant, both with and without a mass term. We show that such a noninvariance provides nontrivial physical insights at different levels, first within the fully relativistic regime, then in the nonrelativistic regime leading to the Schrödinger equation, and then within the de Broglie–Bohm causal interpretation of quantum mechanics. The conformal noninvariance of the Klein–Gordon equation coupled to a vector potential is confronted with the conformal invariance of Maxwell’s equations in the presence of a charged current. The conformal invariance of the nonminimally coupled Klein–Gordon equation to gravity is then examined in light of the conformal invariance of Maxwell’s equations. Finally, the consequence of the noninvariance of the equation on the Aharonov–Bohm effect in curved space–time is discussed.

We consider a scalar–tensor action model, both simple and peculiar at the same time, with a boundary term added that it is form-invariant both under the conformal transformation of the metric as well as under the reparametrization of the scalar field. Under this assumption, the boundary term proposed matches (except factors) with the Gibbons–Hawking–York term proposed in such a way it cancels the term(s) involving $\delta ({\partial }_{\lambda }{g}_{\mu \nu })$. A matter term is added to the action also. In the Jordan frame, the pure gravitational action becomes the Einstein–Hilbert action with a cosmological constant when the scalar field takes the constant value of their vev, ${\varphi }_0=〈0|\varphi |0〉$, which is no otherwise but the only stable minimum of the classical potential chosen which is logarithmic type. This potential presents a large plateau in which it is almost fully flat and the decay of the field from its initial value (assumed to be close to $M_P$) to vacuum draws a Low Field (or New Inflation) scenario. The model fits very well with the current paradigm of the universe and, under some assumptions, gives a good account of the value of the Higgs mass, the current value of the cosmological constant, e-folding, the Hubble parameter, the spectral index of scalar perturbations, etc. The model is not studied in Einstein’s frame. In this, only the expressions of the field and the potential, for a simple case, are achieved. As a result, we get that the nature of the decay of the initial value to its ground state for the scalar field, in this frame, should be by quantum tunneling and not by slow rolling.

The concept of conformal transformation is proposed through the study of the spatial structure of $n$-dimensional hypercubes. Based on conformal transformation, a novel algorithm, called topological equivalence classification algorithm, is proposed for classifying balanced linearly separable Boolean functions. By the proposed algorithm, the topological equivalence classes of all balanced linearly separable Boolean functions and the number of Boolean functions in each of the topological equivalence classes are obtained. In addition, the properties of conformal transformation also show an application prospect for decomposing nonlinearly separable Boolean functions.

Conductance fluctuations are easily probed by passing a constant current through a pair of driver electrodes and measuring the voltage fluctuations across a pair of sensor electrodes. We calculate the geometrical factor for these fluctuations from the current distribution derived by conformal transformations. In four-electrode configurations on a flat square sample this factor depends only slightly on electrode size, whereas a strong dependence is found for two-electrode configurations. The calculations also give accurate values for the resistance, which allows a comparison with the thermal noise.

Thin films for material characterization and noise tests are preferentially manufactured with rectangular electrode shapes. In the companying paper Vandamme presents expressions for resistance and geometrical noise factor f_Q for the case of circular electrodes. The present paper employs numerical conformal mapping to obtain accurate values for rectangular and circular electrodes. The sample size is either assumed very large, or is taken comparable to the electrodes.

In this paper, we study the projective invariant quantities in Finsler geometry which remain invariant under the conformal change of metrics. In particular, we obtain the necessary and sufficient conditions of a given Douglas and Weyl and generalized Douglas–Weyl (GDW) metric to be invariant under the conformal transformations. Finally, we introduce some explicit examples of these metrics. Also, some of these $r$-conformal transformations of Einstein metrics are considered.

In this work, first, we define quasi-Einstein Weyl manifold which is one of the generalization of Einstein–Weyl manifold. Then, we prove its existence and construct an example. Moreover, we consider quasi-Einstein Weyl manifolds with semi-symmetric and Ricci-quarter symmetric connections. Finally, we examine conformal and generalized concircular mappings of quasi-Einstein Weyl manifolds and prove that quasi-Einstein Weyl manifolds are invariant under the generalized concircular mappings.

This paper presents the study of rectifying curves on smooth surfaces using the Darboux frame in Euclidean-3 space. We obtain sufficient conditions for the image of an asymptotic and non-asymptotic rectifying curve to remain invariant under conformal transformation. Furthermore, we investigate the variations of these asymptotic and non-asymptotic rectifying curves along any tangent vector $\mathcal{𝒯}=(a{\sigma }_{\psi }+b{\sigma }_{\omega })$, along the surface normal $\mathcal{𝒰}$, and along $\mathcal{𝒫}(=\mathcal{𝒰}\times \mathcal{𝒯})$.