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Miguel Brozos-Vázquez

Publications and source records attributed to Miguel Brozos-Vázquez.

14 recordsLinked to original sources
arXiv2025

Conformally weighted Einstein manifolds: the uniqueness problem

We discuss smooth metric measure spaces admitting two weighted Einstein representatives of the same weighted conformal class. First, we describe the local geometries of such manifolds in terms of certain Einstein and quasi-Einstein warped products. Secondly, a global classification result is obtained when one of the underlying metrics is complete, showing that either it is a weighted space form, a special Einstein warped product, or a specific family of quasi-Einstein warped products. As a consequence, it must be a weighted sphere in the compact case.

math.DG↗
arXiv2024

The vacuum weighted Einstein field equations on pr-waves

We classify solutions of the vacuum weighted Einstein field equations on smooth metric measure spacetimes $(M,g,h\, dvol_g)$ of dimension 4, where the underlying manifold $(M,g)$ is a $pr$-wave. We use this result to provide examples of solutions with some special geometric properties. The gradient $\nabla h$ is lightlike or spacelike. In the first case, the underlying manifold is a $pp$-wave. In the second case, the Ricci operator is nilpotent. Moreover, $2$-step nilpotent solutions are also realized on $pp$-waves.

math.DG↗
arXiv2023

Rigidity of weighted Einstein smooth metric measure spaces

We study the geometric structure of weighted Einstein smooth metric measure spaces with weighted harmonic Weyl tensor. A complete local classification is provided, showing that either the underlying manifold is Einstein, or decomposes as a warped products in a specific way. Moreover, if the manifold is complete, then it either is a weighted analogue of a space form, or it belongs to a particular family of Einstein warped products.

math.DG↗
arXiv2022

Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case

On a smooth metric measure spacetime $(M,g,e^{-f} dvol_g)$, we define a weighted Einstein tensor. It is given in terms of the Bakry-Émery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is $2$-step nilpotent and a Kundt spacetime if it is $3$-step nilpotent. More specific results are obtained in dimension $3$, where all isotropic solutions are given in local coordinates as plane waves or Kundt spacetimes.

math.DG↗
arXiv2021

Contact detection between an ellipsoid and a combination of quadrics

We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of discriminants of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial.

cs.CG↗
arXiv2019

Isotropic quasi-Einstein manifolds

We investigate the local structure of four-dimensional Lorentzian quasi-Einstein manifolds under conditions on the Weyl tensor. We show that if the Weyl tensor is harmonic and the potential function preserves this harmonicity then, in the isotropic case, the manifold is necessarily a $pp$-wave. Using the quasi-Einstein equation, further conclusions are obtained for $pp$-waves.

math.DG↗
arXiv2019

Spaces of locally homogeneous affine surfaces

We examine the topology of various spaces of locally homogeneous affine manifolds which arise from the classification result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] as orbits of the action of $GL(2,\mathbb{R})$ (Type $\mathcal{A}$) and the $ax+b$ group (Type $\mathcal{B}$). We determine the topology of the spaces of Type $\mathcal{A}$ models in relation to the rank of the Ricci tensor. We determine the topology of the spaces of Type $\mathcal{B}$ models which either are flat or where the Ricci tensor is alternating.

math.DG↗
arXiv2018

Classification of the relative positions between an ellipsoid and an elliptic paraboloid

We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid ({\it small} meaning that the ellipsoid cannot be tangent to the paraboloid at two points simultaneously). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial.

cs.CG↗
arXiv2017

The affine quasi-Einstein Equation for homogeneous surfaces

We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature $(2,2)$ manifolds, to conformally Einstein manifolds and also to new Einstein manifolds through a warped product construction.

math.DG↗
arXiv2017

Half conformally flat generalized quasi-Einstein manifolds

We provide classification results for and examples of half conformally flat generalized quasi Einstein manifolds of signature $(2,2)$. This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation that we explore in further detail.

math.DG↗
arXiv2016

Homogeneous affine surfaces: Moduli spaces

We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $\mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $\mathcal{B}$ surfaces which are not Type $\mathcal{A}$ we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to $1$.

math.DG↗
arXiv2014

Four-dimensional neutral signature self-dual gradient Ricci solitons

We describe the local structure of self-dual gradient Ricci solitons in neutral signature. If the Ricci soliton is non-isotropic then it is locally conformally flat and locally isometric to a warped product of the form $I\times_φN(c)$, where $N(c)$ is a space of constant curvature. If the Ricci soliton is isotropic, then it is locally isometric to the cotangent bundle of an affine surface equipped with the Riemannian extension of the connection, and the Ricci soliton is described by the underlying affine structure. This provides examples of self-dual gradient Ricci solitons which are not locally conformally flat.

math.DG↗