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 ${\omega}^G$ 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 $S_rM$.Comment: 15 pages, to appear in Journal of Geometr

Natural SU(2)-structures on tangent sphere bundles

We define and study natural $\mathrm{SU}(2)$-structures, in the sense of Conti-Salamon, on the total space $\cal S$ of the tangent sphere bundle of any given oriented Riemannian 3-manifold $M$. We recur to a fundamental exterior differential system of Riemannian geometry. Essentially, two types of structures arise: the contact-hypo and the non-contact and, for each, we study the conditions for being hypo, nearly-hypo or double-hypo. We discover new double-hypo structures on $S^3\times S^2$, of which the well-known Sasaki-Einstein are a particular case. Hyperbolic geometry examples also appear. In the search of the associated metrics, we find a theorem, useful for explicitly determining the metric, which applies to all $\mathrm{SU}(2)$-structures in general. Within our application to tangent sphere bundles, we discover a whole new class of metrics specific to 3d-geometry. The evolution equations of Conti-Salamon are considered; leading to a new integrable $\mathrm{SU}(3)$-structure on ${\cal S}\times\mathbb{R}_+$ associated to any flat $M$.Comment: 27 pages, 1 figure, final version, accepted in The Asian Journal of Mathematic

Self-duality and associated parallel or cocalibrated G2{\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.Comment: 20 pages; final version, to appear in Annales Academi{\ae} Scientiarum Fennic{\ae

“Flower of Aristolochia gigas var. sturtevantii used as a hat by a native of British Guiana” – a photograph from Everard im Thurn at the Royal Botanic Gardens, Kew

The Royal Botanic Gardens, Kew, holds a small but valuable and unstudied collection of photographs by Everard im Thurn (1852–1932) who contributed to botany, with specimens and publications, and to anthropology, publishing works that highlight his interest in photography.FCT (SFRH/BD/45965/2008, Portugal) e AHRC (Arts and Humanities Research Council, U

Variations of gwistor space

We study natural variations of the G2 structure {\sigma}_0 \in {\Lambda}^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.Comment: 16 page

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.Comment: Accepted and about to appear in Quarterly Journal of Mathematics. 16 pages. This was the final draft submitted, before print proof

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.Comment: Final version, very close to the one published; 32 p

Educational Technologies. Analysis of Master dissertations carried out in Portugal

Acknowledging the importance scientific research may have in terms of the foundation, orientation and evaluation of the use of technologies in an educational context, it is only natural that there should also be greater knowledge on the research carried out in this particular field. This is what we set out to achieve in a recent conference held in Portugal on the theme Research in Education (1). Using the theme of the conference itself – Research in Education between 1960 and 2005 – as our basis, we decided to focus on what has actually been the object of research in our country within the framework of Educational Technology. We realised, at a very early stage, that there were hardly any studies in this field before Portuguese universities took on a more active role, nor in the field of educational technologies, particularly after the appearance of the first Masters courses at the University of Minho in 1987. Even though we are not aware of any in-depth study to characterise scientific research developed in Portugal in this area, several Portuguese authors have referred to this issue in some way or another (Abrantes, 1981, 1998; Blanco & Silva, 1993; Caldas, 2001; Fernandes, 1969;Ponte, 1994; Silva, 2000). On the other hand, since we have stated that a considerable part of research, in this particular field, focuses precisely on this academic qualification, we have decided to construct our analysis around this aspect. Therefore, in this article we present the result of the studies on Masters dissertations carried out in Portugal, with a view to furthering understanding of the studied themes, their theoretical and methodological frameworks, and to finding out where they are carried out, who does the research, what the collection techniques are and the type of data analysis used, just to mention some of the aspects around which our analysis is centred. This is an exploratory analysis within a restricted context, however, we hope that it may contribute to the acquisition of more profound knowledge regarding research practices in this specific field of Educational Sciences in Portugal