arXiv subjects

Francisco J. Palomo

Publications and source records attributed to Francisco J. Palomo.

16 recordsLinked to original sources

arXiv2026

A Cartan geometry from a correspondence space in Riemannian geometry

Cartan geometries are curved analogues of homogeneous spaces, and the correspondence space construction produces, from a Cartan geometry of one type, another of a different type over a larger base manifold. The unit tangent bundle of a Riemannian manifold is, in this language, a Cartan geometry of type $(\operatorname{Euc}(n),O(n-1))$ obtained in this way; we study Cartan geometries of this type in general, whether or not they arise as correspondence spaces. We show that every such geometry induces on its base manifold an almost contact metric structure, together with an orthogonal splitting of the tangent bundle and a compatible linear connection; conversely, these data determine the geometry, so that the correspondence is a bijection. When $n\ge3$, we single out a distinguished Cartan connection, which we call normal, by a condition on its torsion formulated in terms of the Spencer differential. This normalization is dictated by the correspondence space construction: the Cartan connection induced on the unit tangent bundle of a Riemannian manifold is normal.

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arXiv2025

Cartan geometries with model the future lightlike cone of Lorentz-Minkowski spacetime

This paper develops the theory of Cartan geometries modeled on the future lightlike cone of Lorentz Minkowski spacetime, which we refer to as lightlike Cartan geometries. We show that such geometries naturally induce on the base manifold a lightlike metric, a globally defined radical vector field, and two additional compatible structures. Within this framework, we construct the standard tractor bundle associated with a lightlike Cartan geometry, showing that it extends the tangent bundle of the base manifold and carries a canonical metric linear connection. This construction provides an alternative characterization of lightlike Cartan geometries purely in terms of vector bundle data. Using this alternative description, we analyze the additional geometric information encoded in the Cartan connection beyond the metric and radical data, and we show how it can be extracted via natural decompositions of the standard tractor bundle. Our results underscore the intrinsic significance of Cartan connection methods in the study of lightlike geometry and open new avenues for the analysis of geometric structures that are neither parabolic nor reductive.

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arXiv2025

Spacelike Submanifolds of Codimension Two with Parallel Mean Curvature Vector Field in Lorentz-Minkowski Spacetime Contained in the Light Cone

A general integral inequality is established for compact spacelike submanifolds of codimension two in the Lorentz-Minkowski spacetime under the assumption that the mean curvature vector field is parallel. This inequality is then used to derive a rigidity result. Specifically, we obtain a complete characterization of all compact spacelike submanifolds with parallel mean curvature vector field that lie in the light cone of the Lorentz-Minkowski spacetime: they must be totally umbilical spheres contained in a spacelike hyperplane in Lorentz-Minkowski spacetime.

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arXiv2024

Exploring new extrinsic upper bounds on the first eigenvalue of the Laplace operator for compact submanifolds in Euclidean spaces

Upper bounds of the first non-trivial eigenvalue $λ_1$ of the Laplace operator of a compact submanifold $M^n$ of Euclidean space $\R^{m+1}$, by means of a new technique, are obtained. Each of the upper bounds of $λ_1$ depends on the length of mean curvature vector field, the dimension $n$, the volume of $M^n$, and of a vector of $\R^{m+1}$. When $M^n$ does not lie minimally in a hypersphere of $\R^{m+1}$, classical Reilly's inequality \cite{Re} is improved and new upper bounds are explicitly computed. For instance, considering a torus of revolution whose generating circle has a radius of $1$ and is centered at distance $\sqrt{2}$ from the axis of revolution, we find $λ_1 < \frac{4}{3}(\sqrt{2}-1)\approx 0.552284$, whereas Reilly's upper bound gives $λ_1 < 1/\sqrt{2}\approx 0.707106$.

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arXiv2023

Spacelike immersions in certain Lorentzian manifolds with lightlike foliations

The generalized Schwarzschild spacetimes are introduced as warped manifolds where the base is an open subset of $\mathbb{R}^2$ equipped with a Lorentzian metric and the fiber is a Riemannian manifold. This family includes physically relevant spacetimes closely related to models of black holes. The generalized Schwarzschild spacetimes are endowed with involutive distributions which provide foliations by lightlike hypersurfaces. In this paper, we study spacelike submanifolds immersed in the generalized Schwarzschild spacetimes, mainly, under the assumption that such submanifolds lie in a leaf of the above foliations. In this scenario, we provide an explicit formula for the mean curvature vector field and establish relationships between the extrinsic and intrinsic geometry of the submanifolds. We have derived several characterizations of the slices, and we delve into the specific case where the warping function is the radial coordinate in detail. This subfamily includes the Schwarzschild and Reissner-Nordström spacetimes.

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arXiv2023

Codimension two spacelike submanifolds in Lorentzian manifolds and conformal structures

Starting from a Riemannian conformal structure on a manifold M, we provide a method to construct a family of Lorentzian manifolds. The construction relies on the choice of a metric in the conformal class and a smooth 1-parameter family of self-adjoint tensor fields. Then, every metric in the conformal class corresponds to the induced metric on M seen as a codimension two spacelike submanifold into these Lorentzian manifolds. Under suitable choices of the 1-parameter family of tensor fields, there exists a lightlike normal vector field along such spacelike submanifolds whose Weingarten endomorphism provide a Mobius structure on the Riemannian conformal structure. Conversely, every Mobius structure on a Riemannian conformal structure arises in this way. Flat Mobius structures are characterized in terms of the extrinsic geometry of the corresponding spacelike surfaces.

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arXiv2022

Reductive homogeneous spaces of the compact Lie group $G_2$

The first author defended her doctoral thesis Espacios homogéneos reductivos y álgebras no asociativas in 2001, supervised by P. Benito and A. Elduque. This thesis contained the classification of the Lie-Yamaguti algebras with standard enveloping algebra $\mathfrak{g}_2$ over fields of characteristic zero, which in particular gives the classification of the homogeneous reductive spaces of the compact Lie group $G_2$. In this work we revisit this classification from a more geometrical approach. We provide too geometric models of the corresponding homogeneous spaces and make explicit some relations among them.

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arXiv2022

Normal tractor conformal bundles and codimension two spacelike submanifolds in Lorentzian manifolds

For every codimension two spacelike submanifold of a Lorentz manifold and each choice of a normal lightlike vector field, we introduce a canonical way to construct a tractor conformal bundle. We characterize when the induced connection of a such submanifold defines a tractor connection and then, in this case, when this tractor conformal bundle with the induced connection is standard for the induced metric. Finally, the normality condition for this tractor conformal bundle endowed with the induced connection is characterized in terms of a strong relationship between the intrinsic and the extrinsic geometry of the starting spacelike submanifold.

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arXiv2022

Rigidity results for complete spacelike submanifolds in plane fronted waves

New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.

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arXiv2020

Lightlike manifolds and Cartan geometries

Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold $N$ provides a lightlike metric $h$ with radical distribution globally spanned by a vector field $Z$. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold $(N,h)$, we show a method to construct a family of ambient Lorentzian manifolds that realize $(N,h)$ as a hypersurface. This method is inspired on the Feffermann-Graham ambient metric construction in conformal geometry and provides a lightlike Cartan geometry on the original manifold when $(N,h)$ is generic.

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arXiv2019

On the First eigenvalue of the Laplace operator for Compact Spacelike submanifolds in Lorentz-Minkowski Spacetime $\mathbb{L}^{m}$

By means of a family of counter-examples, it is shown that the Reilly upper bound for the first eigenvalue of the Laplace operator for a compact submanifold in Euclidean space does not work for $n$-dimensional compact spacelike submanifolds of Lorentz-Minkowski spacetime $\mathbb{L}^m$, $m\geq n+2$. We develop a new suitable technique, based on an integral formula on compact spacelike sections of the light cone in $\mathbb{L}^m$. Then, a family of extrinsic upper bounds for the first eigenvalue of the Laplace operator for a compact spacelike submanifold in $\mathbb{L}^m$ is proved. For each one of these inequalities, becoming an equality can be characterized in geometric terms. In particular, the eigenvalue achieves one of these upper bounds if and only if the submanifold lies minimally in certain hypersphere of a spacelike hyperplane.

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arXiv2019

Affine Connections on 3-Sasakian Homogeneous Manifolds

The space of invariant affine connections on every $3$-Sasakian homogeneous manifold of dimension at least $7$ is described. In particular, the remarkable subspaces of invariant affine metric connections, and the subclass with skew-torsion, are also determined. To this aim, an explicit construction of all $3$-Sasakian homogeneous manifolds is exhibited. The unique $3$-Sasakian homogeneous manifolds which admit nontrivial Einstein with skew-torsion invariant affine connections are those of dimension $7$, that is, $\mathbb{S}^7=\mathrm{Sp} (2)/ \mathrm{Sp(1)}$, $\mathbb{R} P^7=\mathrm{Sp}(2)/ \mathrm{Sp(1)}\times \mathbb{Z}_{2}$ and the Aloff-Wallach space $\mathfrak{W}^{7}_{1,1}= \mathrm{SU}(3)/ \mathrm{U}(1)$. For $\mathbb{S}^7$ and $\mathbb{R} P^7$, the set of such connections is in one to one correspondence with two copies of the conformal linear transformation group of the Euclidean space, while it is strictly bigger for $\mathfrak{W}^{7}_{1,1}$. In addition, the set of invariant connections with totally skew-symmetric torsion whose Ricci tensor is multiple of the metric, with different factors, on the canonical vertical and horizontal distributions, is fully described on every $3$-Sasakian homogeneous manifold. An affine connection satisfying these conditions is distinguished, characterized by parallelizing all the characteristic vector fields associated to the $3$-Sasakian structure. This connection is Einstein with skew-torsion for the $7$-dimensional examples. Several results have also been adapted to the nonnecessarily homogeneous setting. In this case, the above mentioned sets of affine connections are, in general, only proper subsets satisfying the properties.

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arXiv2017

Einstein connections with skew-torsion on Berger spheres

The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstein with skew-torsion up to $\mathbb{S}^3$. For Riemannian signature, the existence of such connections strongly depends on the dimension of the sphere and on the scale of the deformation used for the Berger metric. In particular, there are Riemaniann Berger spheres, not Einstein, which admit invariant metric affine connections Einstein with skew-torsion.

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arXiv2016

Compact spacelike surfaces in four-dimensional Lorentz-Minkowski spacetime with a non-degenerate lightlike normal direction

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $η$. Several sufficient assumptions on such a surface with non-degenerate $η$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With this aim, a new formula which relates the Gauss curvatures of the induced metric and of the $η$-second fundamental form is developed. Then, totally umbilical round spheres are characterized as the only compact spacelike surfaces through the lightcone such that its $η$-second fundamental form is non-degenerate and has constant Gauss curvature two. Another characterizations of totally umbilical round spheres in terms of the Gauss-Kronecker curvature of $η$ and the area of the $η$-second fundamental form are also given.

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arXiv2015

Invariant affine connections on odd-dimensional spheres

A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natural way. For the case of the odd dimensional spheres $\mathbb{S}^{2n+1}$ viewed as homogeneous spaces of the special unitary groups, the classical Nomizu's Theorem on invariant connections has permitted to obtain an algebraical description of all the connections which turn the spheres $\mathbb{S}^{2n+1}$ into homogeneous Riemann-Cartan spaces. The expressions of such connections as covariant derivatives are given by means of several invariant tensors: the ones of the usual Sasakian structure of the sphere; an invariant 3-differential form coming from a $3$-Sasakian structure on $\mathbb{S}^7$; and the involved ones in the almost contact metric structure of $\mathbb{S}^5$ provided by its natural embedding into the nearly Kähler manifold $\mathbb{S}^6$. Furthermore, the invariant connections sharing geodesics with the Levi-Civita one have also been completely described. Finally, $\mathbb{S}^3$ and $\mathbb{S}^7$ are characterized as the unique odd-dimensional spheres which admit nontrivial invariant connections satisfying an Einstein-type equation.

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arXiv2012

On spacelike surfaces in 4-dimensional Lorentz-Minkowski spacetime through a lightcone

On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann type result. Two remarkable families of examples of spacelike surfaces in a lightcone are explicitly constructed. Finally, several results which involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a lightcone are obtained.

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