Dias de Geometria,
Topologia e Aplicações

Ana Cristina Ferreira (Univ. Minho)
Special Riemannian geometries and characteristic connections of traceless cyclic type

The study of affine metric connections dates back to Cartan who proved that the torsion tensor of a metric connection splits under the action of the orthogonal group into three modules: vectorial torsion, skew torsion and traceless cyclic torsion. Vectorial and skew torsion have been intensively studied in the literature but the traceless cyclic module, which is the one of largest dimension and less symmetries, is not so well understood. In dimension 4, this module splits further into two submodules, due to the phenomenon of self-duality. In this talk, we intend to explore this type of connections and their relation to G-structures via intrinsic torsion. We will then focus on dimension 4 and special almost hermitian structures. This is joint work with I. Agricola and H. Winther.

Miguel Brozos Vázquez (Univ. A Corunha)
The Einstein equation in spacetimes with density

Ricardo Brasil (Univ. Minho)
Some maps between Grassmannians

For positive integers k ≤ n, the Grassmannian Gr(k, n) is the manifold of linear subspaces of dimension k in euclidean space ℝⁿ. In particular, Gr(2,n) is the manifold of linear planes in ℝⁿ and Gr(1,n) is the (n-1)- dimensional projective space ℝP^n-1. In this talk, we will use the quaternions and octonions to construct examples of maps from Gr(2,n) to ℝP^m, where m is less than the dimension of Gr(2,n), that is, less than 2(n-2). Some of these maps yield new results regarding the Lusternik-Schnirelmann category of Gr(2,n).

David Mosquera Lois (Univ. Santiago de Compostela)
Integration with respect to topological measures and its applications

In this talk, our main focus is discussing the integration with respect to topological measures and exploring its applications. Specifically, we delve into the integration with respect to the Euler-Poincaré characteristic and its implications in the counting and computation of the Euler-Poincaré characteristic for fiber bundles and fibrations. Additionally, in order to address certain limitations associated with integration with respect to the Euler-Poincaré characteristic, we introduce integration with respect to the Lefschetz number.

Catarina Faustino (Univ. Minho)
The homology digraph of a preordered space

Directed algebraic topology studies spaces equipped with some structure representing the flow of time or a direction of traversal. An important line of research in directed algebraic topology is directed homology. A natural approach to directed homology is to consider ordinary homology with a supplementary directional structure. Examples of such concepts of directed homology are the one defined by M. Grandis for cubical sets and the homology graph of a precubical set introduced by T. Kahl. In this work, we consider preordered topological spaces and define a modified version of the homology graph, the homology digraph, which has more satisfactory properties. We show that the homology digraph is a directed homotopy invariant and examine its compatibility with the main ingredients of ordinary singular homology theory. In particular, we show that, over a field, the homology digraph of the product of two preordered spaces is determined by the homology digraphs of the components. No analogous result holds for directed homology in the sense of Grandis or the original homology graph.

João Nuno Mestre (Univ. Coimbra)
Deformations of holomorphic groupoids

Lie groupoids can encode geometric objects such as smooth actions, and foliations; deformations of Lie groupoids also relate to deformations of such objects. After a brief introduction to Lie groupoids, we'll see the deformation cohomology of a (real) Lie groupoid. We'll also mention a few relations to the mentioned examples, and how an interpretation in terms of vector bundles over a Lie groupoid can be used to study deformations of holomorphic groupoids.

The deformation complex obtained for a holomorphic groupoid arises as the total complex of a double complex. It combines the deformation complex of the groupoid structure and the Kodaira-Spencer complex controlling deformations of the underlying complex manifold. The talk is based on ongoing work with Luca Vitagliano.

Henrik Winther (Univ. Tromsø)
Large automorphism groups of parabolic geometries

A parabolic geometry is a Cartan geometry modelled on the homogeneous space G/P, where G is semi-simple, and P is a parabolic subgroup. This is a class which contains conformal geometry (for any signature), projective geometry, quaternionic geometry and many other well-known examples. We will discuss maximal and submaximal global symmetry dimensions, i.e. dimensions of automorphism groups. It is known that the maximal dimension is achieved for the flat model G/P itself, with automorphism group G. This is true both when considering local infinitesimal and global symmetries. The question of which geometry has the most symmetries, by dimension, after the flat model, is called the submaximal problem. The infinitesimal version has been intensely studied, is related to the so-called gap-phenomenon and symmetry breaking, and this comes from the presence of (harmonic) curvature. In contrast we are going to discuss the submaximal problem for global symmetries. Here it turns out that submaximal models typically admit zero harmonic curvature, and the reductions in symmetry are due to algebraic and topological reasons. We will also be interested in the global submaximal problem restricted to the class of compact manifolds. This turns out to be the more complicated case. We will show two constructions giving rise to submaximal models: via compact fiber bundles over generalizations of Hopf manifolds, and via finite group actions on the flat model. Joint work with Boris Kruglikov

Said Hamoun (Univ. Meknès)
An upper bound for the rational topological complexity of a family of elliptic spaces

In this work, we show that, for any simply-connected elliptic space S admitting a pure minimal Sullivan model with a differential of constant length, we have TC₀(S)≤ 2 cat₀(S)+ χ_π(S) where TC₀ and cat₀ are respectively the rational topological complexity and Lusternik-Schnirelmann category and χ_π(S) is the homotopy characteristic. This is a consequence of a structure theorem for this type of models, which is actually our main result.

Miguel Domínguez Vázquez (Univ. Santiago de Compostela)
Isoparametric families of hypersurfaces in symmetric spaces of higher rank

An isoparametric family of hypersurfaces is a decomposition of a Riemannian manifold into equidistant hypersurfaces of constant mean curvature and possibly up to two focal submanifolds with codimension greater than one. Fundamental examples are given by the so-called homogeneous isoparametric families, that is, the orbit foliations of Lie group isometric actions with codimension one orbits. The focal submanifolds of isoparametric families are known to be minimal, and all the examples known so far are indeed austere (that is, the multiset of their principal curvatures is invariant under change of sign).

In this talk I will present a simple method to extend submanifolds from certain Euclidean spaces embedded in a symmetric space of noncompact type to the whole ambient symmetric space, in such a way that the codimension, the mean curvature, and other geometric properties are preserved. As a direct application, we will construct the first examples of inhomogeneous isoparametric hypersurfaces in every symmetric space of noncompact type and rank higher than two. Surprisingly, these examples have nonaustere focal set. This is based on a joint work with Víctor Sanmartín-López.

Salah Chaib (Univ. Minho)
Left-invariant metrics on Lie groups: Completeness, Isometries, and Dynamics

A striking difference between Riemannian and semi-Riemannian geometry is that, for a smooth Cauchy-complete manifold, indefinite metrics are not necessarily geodesically complete, not even in the compact case. In the literature, geodesic completeness on 3-dimensional Lie groups has only been investigated and characterized for unimodular Lie groups. Our techniques provided some new results in the non-unimodular case in terms of providing a full classification. In this talk, we shall set up the problem, give some new results, and finally present further questions about our project (joint with A.C. Ferreira and A. Zeghib).

Ahmed Elsafei (Univ. Minho)
Lie groups with all left-invariant semi-Riemannian metrics complete

In this joint work with A.C. Ferreira, M. Sànchez and A. Zeghib, we introduce for every left-invariant semi-Riemannian metric g on a Lie group G, a class of bi-Lipschitz equivalent Riemannian metrics, called Clairaut metrics, whose completeness implies the completeness of g. We present a sufficient condition for the completeness of all Clairaut metrics associated to any metric g, as an at most linear growth bound on the adjoint representation of G. In particular, we prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by any semidirect product K ⋉_ρ ℝⁿ , where K is the direct product of a compact and an abelian Lie group and ρ(K) is pre-compact. This presented list includes all the known examples of complete Lie groups, i.e. all their left-invariant metrics complete. Finally, a detailed study of the case of the affine group of the real line is considered, to illustrate how these techniques work even in the the absence of linear growth and suggest new questions.

Sandro Caeiro Oliveira (Univ. Vigo)
Curvature homogeneous metrics and quadratic curvature functionals

Maria Piedade Ramos (Univ. Minho)
Relativistic polyatomic gases in a Robertson-Walker spacetime

We study rarefied relativistic polyatomic gases in both Minkowski spacetime and Robertson-Walker spacetime. A generalized relativistic BGK-type model for polyatomic gases is applied to both Minkowski spacetime and flat and non flat Robertson-Walker spacetimes. The equation of state for the non-equilibrium dynamical pressure is obtained from the Chapman-Enskog method applied to a variant of the Anderson and Witting model for polyatomic gases.

Irene Brito (Univ. Minho)
On the propagation velocity of sound waves in spherically symmetric elastic spacetimes in general relativity

An expression for the propagation speed of sound waves in the radial direction in spherically symmetric spacetimes with elastic matter is obtained in terms of density, radial pressure and elasticity tensor components. Static and non-static shear free spherically symmetric elastic solutions of the Einstein field equations are considered and the local causality condition for the propagation speed of elastic wave fronts relative to the flow is analysed.