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- Miguel Angel Javaloyes

# Miguel Angel Javaloyes

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Research Items (51)

- Apr 2019

We show how to compute tensor derivatives and curvature tensors using affine connections. This allows us to make all the computations without using coordinate systems in a way that parallels computations in classical Riemannian Geometry. In particular, we obtain Bianchi identities for the curvature tensor of any anisotropic connection, we compare the curvature tensors of any two anisotropic connections and we find a family of anisotropic connections which are good to study the geometry of Finsler metrics.

We develop a systematic study of (smooth, strong) cone structures $\mathcal{C}$ and (smooth) Lorentz-Finsler metrics $L$ which includes: (a) to associate each $L$ (defined on some conic domain) with a $\mathcal{C}$, (b) to describe any $\mathcal{C}$ by means of a cone triple $(\Omega,T, F)$, where $\Omega$ (resp. $T$) is a 1-form (resp. vector field) with $\Omega(T)\equiv 1$ and $F$, a Finler metric on $\ker (\Omega)$, (c) to associate each $(\Omega,T, F)$ with a continuous Lorentz-Finsler metric $G$, which is smooth everywhere up to Span$(T)$, (d) to develop a procedure in order to smoothen continuous Lorentz-Finsler metrics maintaining its cone, in particular, applicable to previous $G$, (e) to associate bijectively each $\mathcal{C}$ with a class of anisotropically conformal metrics $L$, developing then natural notions such as cone geodesics, (f) to provide systematic procedures in order to construct mathematically and physically interesting smooth Lorentz-Finsler metrics, including a general characterization of any $L$ in terms of Finslerian and Riemannian metrics. As a non-relativistic application, the time-dependent Zermelo navigation problem is stated and solved.

- Apr 2018

Using the relativistic Fermat's principle, we establish a bridge between stationary-complete manifolds which satisfy the observer-manifold condition and pre-Randers metrics, namely, Randers metrics without any restriction on the one-form. As a consequence, we give a description of the causal ladder of such spacetimes in terms of the elements associated with the pre-Randers metric: its geodesics and the associated distance. We obtain, as applications of this interplay, the description of conformal maps of Killing submersions, and existence and multiplicity results for geodesics of pre-Randers metrics and magnetic geodesics.

- Oct 2017

In this note, we prove that given a submanifold $P$ in a Finsler manifold $(M,F)$, (i) the orthogonal geodesics to $P$ minimize the distance from $P$ at least in some interval, (ii) there exist tubular neighbourhoods around each point of $P$, (iii) the distance from $P$ is smooth in some open neighbourhood of $P$ (but not necessarily in $P$).

- Aug 2017

In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if $\mathcal{F}$ is a singular Finsler foliation on a Randers manifold $(M,Z)$ with Zermelo data $(\mathtt{h},W),$ then $\mathcal{F}$ is a singular Riemannian foliation on the Riemannian manifold $(M,\mathtt{h} )$. We also present a slice theorem that relates local singular Finsler foliations on Finsler manifolds with singular Finsler foliations on Minkowski spaces.

Recently, a link between Lorentzian and Finslerian Geometries has been carried out, leading to the notion of wind Riemannian structure (WRS), a generalization of Finslerian Randers metrics. Here, we further develop this notion and its applications to spacetimes, by introducing some characterizations and criteria for the completeness of WRS's. As an application, we consider a general class of spacetimes admitting a time function $t$ generated by the flow of a complete Killing vector field (SSTK spacetimes) and derive simple criteria ensuring that its slices $t=$ constant are Cauchy. Moreover, a brief summary on the Finsler/Lorentz link for readers with some acquaintance in Lorentzian Geometry plus some simple examples of applications in Mathematical Relativity are provided.

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to obtain the local classification. For the global one, a suitable result on completeness for WRS yields the complete simply connected models. In particular, any of the local models in the Randers classification does admit an extension to a unique model of wind Riemannian structure, even if it cannot be extended as a complete Finslerian manifold. Thus, WRS's emerge as the natural framework for the analysis of Randers spaceforms and, prospectively, wind Finslerian structures would become important for other global problems too. For the sake of completeness, a brief overview about WRS (including a useful link with the conformal geometry of a class of relativistic spacetimes) is also provided.

- Jan 2017
- Lorentzian Geometry and Related Topics

We introduce the anisotropic tensor calculus, which is a way of handling with tensors that depend on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application we show how to define derivations using anisotropic linear connections in a manifold. In particular, we show that the Chern connection of a Finsler metric can be interpreted as the Levi-Civita connection. Moreover, we also introduce the concept of anisotropic Lie derivative.

We prove a monodromy theorem for local vector fields belonging to a sheaf
satisfying the unique continuation property. In particular, in the case of
admissible regular sheaves of local fields defined on a simply connected
manifold, we obtain a global extension result for every local field of the
sheaf. This generalizes previous works of Nomizu for semi-Riemannian Killing
fields, of Ledger--Obata for conformal fields, and of Amores for fields
preserving a $G$-structure of finite type. The result applies to Finsler or
pseudo-Finsler Killing fields and, more generally, to affine fields of a spray.
Some applications are discussed.

We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler
metrics possibly defined in conic subsets. The translation of a pseudo-Finsler
metric $F$ is a new pseudo-Finsler metric whose indicatrix is the translation
of the indicatrix of $F$ by a vector field $W$ at each point, where $W$ is an
arbitrary vector field. Then we show that the Matsumoto tensor of a
pseudo-Finsler metric is equal to zero if and only if it is the translation of
a semi-Riemannian metric, and when $W$ is homothetic, the flag curvature of the
translation coincides with the one of the original one up to the addition of a
non-positive constant. In this case, we also give a description of the geodesic
flow of the translation.

We show that Penrose's singularity theorem translates readily to the setting
of a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are
extended to the Finsler spacetime setting, including definitions and properties
of focal points, Cauchy hypersurfaces, and trapped surfaces. We work out in
detail all the modifications that are required in doing so, with careful
attention to the differences that arise in the Finsler spacetime setting.

We introduce the notion of wind Finslerian structure; this is a
generalization of Finsler metrics where the indicatrices at the tangent spaces
may not contain the zero vector. In the particular case that these indicatrices
are ellipsoids, called here wind Riemannian structures, they admit a double
interpretation which provides: (a) a model for classical Zermelo's navigation
problem even when the trajectories of the moving object are influenced by
strong winds or streams, and (b) a natural description of the causal structure
of relativistic spacetimes endowed with a non-vanishing Killing vector field
(SSTK splittings), in terms of Finslerian elements. These elements can be
regarded as conformally invariant Killing initial data on a partial Cauchy
hypersurface. The links between both interpretations as well as the possibility
to improve the results on one of them using the other viewpoint are stressed.
The wind Finslerian structure is described in terms of two (conic, pseudo)
Finsler metrics having convex and concave indicatrices and notions such as
balls and geodesics are extended. However, the spacetime viewpoint for the wind
Riemannian case gives a useful unified viewpoint. A thorough study of the
causal properties of such a spacetime is carried out in Finslerian terms.
Randers-Kropina metrics appear as the Finslerian counterpart to the case of an
SSTK splitting when the Killing vector field is either timelike or lightlike.

We consider the Chern connection of a (conic) pseudo-Finsler manifold (M, L) as a linear connection del(V) on any open subset Omega subset of M associated to any vector field V on Omega which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor g. Then we show some properties of the curvature tensor R-V associated to del(V) and in particular we prove that the Jacobi operator of R-V along a geodesic coincides with the one given by the Chern curvature.

In this paper, we obtain the first and the second variation of the energy
functional of a pseudo-Finsler metric using the family of affine connections
associated to the Chern connection. This allows us to accomplish the
computations with the free-coordinate methods of Modern Differential Geometry.
We also introduce the index form using the formula for the second variation and
give some properties of Jacobi fields. Finally we prove that lightlike
geodesics and its focal points are preserved up to reparametrization by
conformal changes.

For a standard Finsler metric F on a manifold M, its domain is the whole
tangent bundle TM and its fundamental tensor g is positive-definite. However,
in many cases (for example, the well-known Kropina and Matsumoto metrics),
these two conditions are relaxed, obtaining then either a pseudo-Finsler metric
(with arbitrary g) or a conic Finsler metric (with domain a "conic" open domain
of TM).
Our aim is twofold. First, to give an account of quite a few subtleties which
appear under such generalizations, say, for conic pseudo-Finsler metrics
(including, as a previous step, the case of Minkowski conic pseudo-norms on an
affine space).
Second, to provide some criteria which determine when a pseudo-Finsler metric
F obtained as a general homogeneous combination of Finsler metrics and
one-forms is again a Finsler metric ---or, with more accuracy, the conic domain
where g remains positive definite. Such a combination generalizes the known
(alpha,beta)-metrics in different directions. Remarkably, classical examples of
Finsler metrics are reobtained and extended, with explicit computations of
their fundamental tensors.

Recent links between Finsler Geometry and the geometry of spacetimes are
briefly revisited, and prospective ideas and results are explained. Special
attention is paid to geometric problems with a direct motivation in Relativity
and other parts of Physics.

- Sep 2013

We give the details of the proof of the equality between the critical
groups, with respect the H^1 and C^1 topology, at a non-degenerate
critical point of the energy functional of a non-reversible Finsler
manifold (M,F), defined on the Hilbert manifold of the H^1 curves
connecting two given points on M.

In this review, we collect several results for conformally standard
stationary spacetimes (SxR,g) obtained in terms of a Finsler metric of Randers
type on the orbit manifold S that we call Fermat metric. This metric is
obtained by applying the relativistic Fermat principle and it turns out that it
encodes all the causal aspects of the spacetime.

- May 2012

We develop the basics of a theory of almost isometries for spaces endowed
with a quasi-metric. The case of non-reversible Finsler (more specifically,
Randers) metrics is of particular interest, and it is studied in more detail.
The main motivation arises from General Relativity, and more specifically in
spacetimes endowed with a timelike conformal field K, in which case
\emph{conformal diffeomorphisms} correspond to almost isometries of the Fermat
metrics defined in the spatial part. A series of results on the topology and
the Lie group structure of conformal maps are discussed.

In this paper we prove the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.fs. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

We show that any two non-conjugate points on a forward or backward complete
connected Finsler manifold can be joined by infinitely many geodesics which are
not covered by finitely many closed ones, provided that the Betti numbers of
the based loop space grow unbounded.

- May 2010

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime (M0×R,g) is equal to the index of its spatial projection as a geodesic of a Finsler metric F on M0 associated to (M0×R,g). Moreover we obtain the Morse relations of lightlike geodesics connecting a point p to a curve γ(s)=(q0,s) by using Morse theory on the Finsler manifold (M0,F). To this end, we prove a splitting lemma for the energy functional of a Finsler metric. Finally, we show that the reduction to Morse theory of a Finsler manifold can be done also for timelike geodesics.RésuméOn démontre que l'indice d'un rayon de lumière dans un espace-temps stationnaire (M0×R,g) conformément standard est égal à l'indice de sa projection spatiale vue comme une géodésique d'une métrique de Finsler F sur M0 associée à (M0×R,g). De plus, on obtient les relations de Morse de géodésiques isotropes reliant un point p à une courbe γ(s)=(q0,s) en utilisant la théorie de Morse sur la variété de Finsler (M0,F). À cette fin, on démontre un lemme de séparation de la fonctionnelle de l'énergie d'une métrique de Finsler. Enfin, on montre que la réduction à la théorie de Morse d'une variété de Finsler peut être faite aussi pour les géodésiques temporelles.

- Apr 2010

We improve to the C
∞–category the genericity result of light rays nondegeneracy in the stationary case given in [2].

- Feb 2010

Given a Lorentzian manifold (M, g), a geodesic υ in M and a timelike Jacobi field Υ along υ, we introduce a special class of instants along υ that we call ΥY- pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Υ-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Υ is obtained as the restriction of a globally defined timelike Killing vector field.

We obtain a result about the existence of only a finite number of geodesics between two fixed non-conjugate points in a Finsler manifold endowed with a convex function. We apply it to Randers and Zermelo metrics. As a by-product, we also get a result about the finiteness of the number of lightlike and timelike geodesics connecting an event to a line in a standard stationary spacetime. Comment: 16 pages, AMSLaTex. v2 is a minor revision: title changed, references updated, typos fixed; it matches the published version. This preprint and arXiv:math/0702323v3 [math.DG] substitute arXiv:math/0702323v2 [math.DG]

We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a
given compact manifold M, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the Ck-topology, k=2, …, ∞, in the set of metrics of a given index on M. A higher-order genericity Riemannian result of Klingenberg and Takens is extended to semi-Riemannian geometry.

We study focal points and Maslov index of a horizontal geodesic $\gamma:I\to
M$ in the total space of a semi-Riemannian submersion $\pi:M\to B$ by
determining an explicit relation with the corresponding objects along the
projected geodesic $\pi\circ\gamma:I\to B$ in the base space. We use this
result to calculate the focal Maslov index of a (spacelike) geodesic in a
stationary space-time which is orthogonal to a timelike Killing vector field.

We obtain some results in both Lorentz and Finsler geometries, by using a
correspondence between the conformal structure (Causality) of standard
stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In
particular, for stationary spacetimes, we give a simple characterization of
when they are causally continuous or globally hyperbolic (including in the
latter case, when $S$ is a Cauchy hypersurface), in terms of an associated
Randers metric. Consequences for the computability of Cauchy developments are
also derived. Causality suggests that the role of completeness in many results
of Riemannian Geometry (geodesic connectedness by minimizing geodesics,
Bonnet-Myers, Synge theorems) is played, in Finslerian Geometry, by the
compactness of symmetrized closed balls. Moreover, under this condition we show
that for any Randers metric there exists another Randers metric with the same
pregeodesics and geodesically complete. Even more, results on the
differentiability of Cauchy horizons in spacetimes yield consequences for the
differentiability of the Randers distance to a subset, and vice versa.

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is never vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold $M$ admits a Lorentzian metric with a never vanishing Killing vector field which is timelike somewhere if and only if $M$ admits a smooth circle action without fixed points.

- Jan 2009

In this paper we prove the existence of closed geodesics in certain types of compact Riemannian good orbifolds. This gives us an elementary alternative proof of a result due to Guruprasad and Haefliger. In addition, we prove some results about horizontal periodic geodesics of Riemannian foli- ations and stress the relation between them and closed geodesics in Riemannian orbifolds. In particular we note that each singular Riemannian foliation with flat sections and compact leaves on a compact simply connected space has horizontal periodic geodesics in each section.

- Sep 2008

We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.

Let $(M,g)$ be a spacetime which admits a complete timelike conformal Killing
vector field $K$. We prove that $(M,g)$ splits globally as a standard
conformastationary spacetime with respect to $K$ if and only if $(M,g)$ is
distinguishing (and, thus causally continuous). Causal but non-distinguishing
spacetimes with complete stationary vector fields are also exhibited. For the
proof, the recently solved "folk problems" on smoothability of time functions
(moreover, the existence of a {\em temporal} function) are used.

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Based on an idea of B. White, we prove an abstract genericity result that employs the infinite dimensional Sard--Smale theorem. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.

In this paper we prove several multiplicity results of $t$-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of $t$-periodic light rays and compute a lower bound for the period of the light rays when the flag curvature of the Fermat metric is $\eta$-pinched.

We study the Maslov index of continuous paths in the Grassmannian Lagrangian using an isotropic reduction of the symplectic space, and we discuss a few applications.

Given a Lorentzian manifold $(M,g)$, a geodesic $\gamma$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $\gamma$, we introduce a special class of instants along $\gamma$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field $\mathcal Y$ is obtained as the restriction of a globally defined timelike Killing vector field.

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.

- Oct 2007

This paper has been withdrawn by the authors because it has been merged with paper arXiv:0903.3501v1 [math.DG]

- Oct 2007

We give the notion of a conjugate instant along a solution of the relativistic Lorentz force equation (LFE). Electromagnetic conjugate instants are defined as zeros of solutions of the linearized LFE with fixed value of the charge-to-mass ratio; equivalently, we show that electromagnetic conjugate points are the critical values of the corresponding electromagnetic exponential map. We prove a second-order variational principle relating every solution of the LFE to a canonical lightlike geodesic in a Kaluza–Klein manifold, whose metric is defined using the value of the charge-to-mass ratio. Electromagnetic conjugate instants correspond to conjugate points along the lightlike geodesic, and therefore they are isolated; based on such correspondence and on a recent result of bifurcation for light rays, we prove a bifurcation result for solutions of the LFE in the exact case.

Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λγ on the unit circle with the property that the Morse index of the iterated γN
is equal, up to a correction term εγ∈{0,1}, to the sum of the values of Λγ at the N-th roots of unity. The discontinuities of Λγ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.

In this paper we first study some global properties of the energy functional
on a non-reversible Finsler manifold. In particular we present a fully detailed
proof of the Palais--Smale condition under the completeness of the Finsler
metric. Moreover we define a Finsler metric of Randers type, which we call
Fermat metric, associated to a conformally standard stationary spacetime. We
shall study the influence of the Fermat metric on the causal properties of the
spacetime, mainly the global hyperbolicity. Moreover we study the relations
between the energy functional of the Fermat metric and the Fermat principle for
the light rays in the spacetime. This allows us to obtain existence and
multiplicity results for light rays, using the Finsler theory. Finally the case
of timelike geodesics with fixed energy is considered.

- Jan 2007

Given a Lorentzian manifold (M,g), a geodesic in M and a time- like Jacobi field Y along , we introduce a special class of conjugate (or focal) instants along , called Y-parallel. We prove that the Y-parallel instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

- Jun 2005

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are given for the lack of local injectivity of the exponential map around its singularities.

Models describing relativistic particles, where Lagrangian densities depend linearly on both the curvature and the torsion of the trajectories, are revisited in D = 3 Lorentzian spacetimes with constant curvature. The moduli spaces of trajectories are completely and explicitly determined. Trajectories are Lancret curves including ordinary helices. To get the geometric integration of the solutions, we design algorithms that essentially involve the Lancret program as well as the notions of scrolls and Hopf tubes. The most interesting and consistent models appear in anti-de Sitter spaces, where the Hopf mappings, both the standard and the Lorentzian ones, play an important role. The moduli subspaces of closed solitons in anti-de Sitter settings are also obtained. Our main tool is the isoperimetric inequality in the hyperbolic plane. The mass spectra of these models are also obtained. The main characteristic of the anti-de Sitter space comes from the presence of real gravity, which becomes essential to find some system with only massive states. This fact, on one hand, has no equivalent in flat spaces, where spectra necessarily present tachyonic sectors and, on the other hand, solves an early stated problem.

We consider the motion of relativistic particles described by an action which is a function of the curvature and torsion of the particle path. The Euler–Lagrange equations and the dynamical constants of the motion are expressed in a simple way in terms of a suitable coordinate system. The moduli spaces of solutions in a three-dimensional pseudo-Riemannian space form are completely exhibited.

We consider the motion of relativistic particles described by an action that is linear in the torsion (second curvature) of the particle path. The Euler-Lagrange equations and the dy-namical constants of the motion associated with the Poincaré group, the mass and the spin of the particle, are expressed in a simple way in terms of the curvatures of the embedded world-line. The moduli spaces of solutions are completely exhibited in 4-dimensional background spaces and in the 5-dimensional case we explicitly obtain the curvatures of the worldline. PACS numbers(s): 04.20.-q, 02.40.-k Keywords: torsion, spinning massless and massive particles, moduli spaces of solutions.

We study the total lifts of curves by means of a submersion π:M<sup>3</sup><sub>s</sub> → B<sup>2</sup><sub>r</sub> that satisfy the condition ΔH = λH analyzing, in particular, the cases in which the submersion has totally geodesic fibres or integrable horizontal distribution. We also consider in detail the case λ = 0 (biharmonic lifts).Moreover, we obtain a biharmonic lift in R<sup>3</sup> by means of a Riemannian submersion that has non-constant mean curvature, getting so a counterexample to the Chen conjecture for R<sup>3</sup> with a non-flat Riemannian metric.

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