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Rui Albuquerque

Publications and source records attributed to Rui Albuquerque.

At least 19 recordsLinked to original sources
arXiv2022

On the volume of a unit vector field in 3 dimensions via calibrations

We give a new proof of the well-known result that the minimal volume vector fields on $\mathbb{S}^3(r)$ are the Hopf vector fields. Such proof relies again on calibration theory, arising here from a systematic point of view given by a natural source of differential forms. Our results serve in particular for all $r$. A classification of relevant calibrations on $T^1M$ for every oriented 3-manifold $M$ of constant sectional curvature is given, continuing the study of the \textit{usual} fundamental differential system of Riemannian geometry. Showing applications of this differential system is also one of the purposes of this article. We deduce new properties of the geodesic flow vector field of space forms, which interacts with the solutions of the minimal volume problem both in elliptic and hyperbolic geometry, in any dimension. The solution -- unknown -- for the hyperbolic case in 3-dimensions being most dependent on the homology class of the domain and boundary values of the vector fields. This is illustrated with a noteworthy example which ironically works just for curvature $-1$.

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arXiv2021

Vector fields with big and small volume on the 2-sphere

We consider the problem of minimal volume vector fields on a given Riemann surface, specialising on the case of $M^\star$, that is, the arbitrary radius 2-sphere with two antipodal points removed. We discuss the homology theory of the unit tangent bundle $(T^1M^\star,\partial T^1M^\star)$ in relation with calibrations and a certain minimal volume equation. A particular family $X_{\mathrm{m},k},\:k\in\mathbb{N}$, of minimal vector fields on $M^\star$ is found in an original fashion. The family has unbounded volume, $\lim_k\mathrm{vol}({X_{\mathrm{m},k}}_{|\Omega})=+\infty$, on any given open subset $\Omega$ of $M^\star$ and indeed satisfies the necessary differential equation for minimality. Another vector field $X_\ell$ is discovered on a region $\Omega_1\subset\mathbb{S}^2$, with volume smaller than any other known \textit{optimal} vector field restricted to $\Omega_1$.

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arXiv2020

An invariant Kähler metric on the tangent disk bundle of a space-form

We find a family of Kähler metrics invariantly defined on the radius $r_0>0$ tangent disk bundle ${{\cal T}_{M,r_0}}$ of any given real space-form $M$ or any of its quotients by discrete groups of isometries. Such metrics are complete in the non-negative curvature case and non-complete in the negative curvature case. If $\dim M=2$ and $M$ has constant sectional curvature $K\neq0$, then the Kähler manifolds ${{\cal T}_{M,r_0}}$ have holonomy $\mathrm{SU}(2)$; hence they are Ricci-flat. For $M=S^2$, just this dimension, the metric coincides with the Stenzel metric on the tangent manifold ${{\cal T}_{S^2}}$, giving us a new most natural description of this well-know metric.

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arXiv2019

Self-duality and associated parallel or cocalibrated ${\mathrm{G}}_2$ structures

We find a remarkable family of $\mathrm{G}_2$ structures defined on certain principal $\mathrm{SO}(3)$-bundles $P_\pm\longrightarrow M$ associated with any given oriented Riemannian 4-manifold $M$. Such structures are always cocalibrated. The study starts with a recast of the Singer-Thorpe equations of 4-dimensional geometry. These are applied to the Bryant-Salamon cons\-truction of complete $\mathrm{G}_2$-holonomy metrics on the vector bundle of self- or anti-self-dual 2-forms on $M$. We then discover new examples of that special holonomy on disk bundles over ${\cal H}^4$ and ${\cal H}^2_{\mathbb{C}}$, respectively, the real and complex hyperbolic space. Only in the end we present the new $\mathrm{G}_2$ structures on principal bundles.

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arXiv2019

A fundamental differential system of Riemannian geometry

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree $n$ associated to any given oriented Riemannian manifold $M$ of dimension $n+1$. The framework is that of the tangent sphere bundle of $M$. We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler-Lagrange differential systems.

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arXiv2018

Notes on the Sasaki metric

We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago. Following the results of Sasaki, we try to write and deduce them by different means. Questions of vector fields, mainly those arising from the base, are related as invariants of the classical metric, contact and Hermitian structures. Attention is given to the natural notion of extension or complete lift of a vector field, from the base to the tangent manifold. Few results are original, but finally new equations of the mirror map are considered.

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arXiv2017

On vector bundle manifolds with spherically symmetric metrics

We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle $E\rightarrow M$, over a Riemannian manifold $M$, when $E$ is endowed with a metric connection. The tangent bundle of $E$ admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of $E$; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant-Salamon type $\mathrm{G}_2$ manifolds.

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arXiv2016

A fundamental differential system of 3-dimensional Riemannian geometry

We briefly recall a fundamental exterior differential system introduced by the author and then apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemaniann 3-manifolds. The system leads to a remarkable Weingarten type equation for surfaces on hyperbolic 3-space. An independent proof for low dimensions of the structural equations gives new insight on the intrinsic exterior differential system.

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arXiv2014

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles $S_rM$ of a Riemannian manifold $M,g$ of $\dim\geq 3$, assuming different variable radius functions $r$ and weighted Sasaki metrics induced by the conformal class of $g$. New examples are shown of manifolds with constant positive or with constant negative scalar curvature, which are not Einstein. Recalling results on the associated almost complex structure $I^G$ and symplectic structure $ω^G$ on the manifold $TM$, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel-Whitney characteristic classes of the manifolds $TM$ and $S_rM$.

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arXiv2012

On the characteristic connection of gwistor space

We give a brief presentation of gwistor space, which is a new concept from G_2 geometry. Then we compute the characteristic torsion T^c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T^c is \nabla^c-parallel; this allows for the classification of the G_2 structure with torsion and the characteristic holonomy according to known references. The case with the Einstein base manifold is envisaged.

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arXiv2012

Variations of gwistor space

We study natural variations of the G2 structure σ_0 \in Λ^3_+ existing on the unit tangent sphere bundle SM of any oriented Riemannian 4-manifold M. We find a circle of structures for which the induced metric is the usual one, the so-called Sasaki metric, and prove how the original structure has a preferred role in the theory. We deduce the equations of calibration and cocalibration, as well as those of W3 pure type and nearly-parallel type.

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arXiv2011

Curvatures of weighted metrics on tangent sphere bundles

We determine the curvature equations of natural metrics on tangent bundles and radius r tangent sphere bundles S_rM of a Riemannian manifold M. A family of positive scalar curvature metrics on S_rM is found, for any M with bounded sectional curvature and any chosen constant r.

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arXiv2011

On invariants of almost symplectic connections

We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.

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arXiv2010

Weighted metrics on tangent sphere bundles

Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of $TM$ to the almost Hermitian category. Our purpose is the study of the natural contact structure on $S_rM$ and the $G_2$-twistor space of any oriented Riemannian 4-manifold.

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arXiv2010

Isometries and curvatures of tangent sphere bundles

Natural metric structures on tangent bundles and tangent sphere bundles enclose many important problems, from the topology of the base to the determination of their holonomy. We make here a brief study of the topic. We find the characteristic classes of some of those structures. We solve the question of when two given tangent sphere bundles S_rM of a Riemannian manifold M,g are homothetic, assuming different variable radius functions r and weighted metrics induced only by the conformal class of g. We determine their Riemannian, Ricci, scalar and mean curvatures in some cases. We find a family of positive scalar curvature metrics on S_rM when M has positive scalar curvature or when it has bounded sectional curvature and index of nullity 0. Our objective is the study of contact structures and gwistor spaces, a recently found natural G_2-structure on S_1M.

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arXiv2009

On the G_2 bundle of a Riemannian 4-manifold

We study the natural G_2 structure on the unit tangent sphere bundle SM of any given orientable Riemannian 4-manifold M, as it was discovered in \cite{AlbSal}. A name is proposed for the space. We work in the context of metric connections, or so called geometry with torsion, and describe the components of the torsion of the connection which imply certain equations of the G_2 structure. This article is devoted to finding the G_2-torsion tensors which classify our structure according to the theory in \cite{FerGray}.

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