Short survey on the existence of slices for the space of Riemannian metrics

We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.Comment: 21 pages; added references [DR19] and [Kan05]; corrected typos; added propositions 2.2 and 4.6, remarks 2.6, 2.18 and 2.22; to appear in the Proceedings of the IV Meeting of Mexican Mathematicians Abroad 201

A-Foliations of codimension two on compact simply-connected manifolds

We show that a singular Riemannian foliation of codimension $2$ on a compact simply-connected Riemannian $(n + 2)$-manifold, with regular leaves homeomorphic to the $n$-torus, is given by a smooth effective $n$-torus action.Comment: 26 pages, 4 figure

Manifolds with aspherical singular Riemannian foliations

In the present work we study $A$-foliations, i.e. singular Riemannian foliations with regular leaf aspherical. The main result is that, for a simply-connected closed $(n+2)$-manifold $M$, an $A$-foliation with regular leaves of codimension $2$ in $M$ is homogeneous. In other words it is given by a smooth effective action of the torus $\mathbb{T }^n$ on $M$ by isometries. We will give some conditions to compare two simply-connected, closed manifolds with $A$-foliations, up to foliated homeomorphism, via their leaf spaces

Core reduction for singular Riemannian foliations in positive curvature

We show that for a smooth manifold equipped with a singular Riemannian foliation, if the foliated metric has positive sectional curvature, and there exists a pre-section, that is a proper submanifold retaining all the transverse geometric information of the foliation, then the leaf space has boundary. In particular, we see that polar foliations of positively curved manifolds have leaf spaces with nonempty boundary.Comment: 12 page

Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions

We show that for each n 1, there exist infinitely many spin and non-spin diffeomorphism types of closed, smooth, simply-connected (n + 4)- manifolds with a smooth, effective action of a torus T n+2 and a metric of positive Ricci curvature invariant under a T n-subgroup of T n+2. As an application, we show that every closed, smooth, simply-connected 5- and 6-manifold admitting a smooth, effective torus action of cohomogeneity two supports metrics with positive Ricci curvature invariant under a circle or T2-action, respectively

Procesos de enseñanza en escuelas rurales multigrado de México mediante Comunidades de Aprendizaje

El artículo analiza los procesos de enseñanza desarrollados en dos escuelas primarias rurales multigrado en México, a partir de la implementación de un nuevo modelo pedagógico inspirado en el concepto de Comunidades de Aprendizaje. De manera inicial, se describen las características generales del modelo y del organismo estatal que lo desarrolla, para después destacar las fortalezas y limitaciones observadas durante la implementación del modelo en las escuelas seleccionadas. Entre las fortalezas se identifican el fomento de procesos de autonomía y el desarrollo de competencias investigativas y de expresión oral y escrita en los alumnos. Como limitantes, se señala la convivencia de prácticas innovadoras y tradicionales en las aulas, además de la pobre dotación de recursos por parte del Estado hacia la educación en las poblaciones ruralesThis article analyzes the teaching methods developed in two primary rural multigrade schools in Mexico, based on the implementation of a new pedagogical model inspired by the concept of Learning Communities. The text examines the general characteristics of the model, and of the state agency that develops it, and also highlights both the strengths and limitations observed during the implementation of the model in the selected schools. Among the strengths is the promotion of autonomy processes and the development of investigative skills and also oral and written communication in students. The limitations are noted as the coexistence of innovative practices along with traditional classroom practices, as well as the poor provision of resources by the state towards education in rural communitie

Torus actions on Alexandrov 4-spaces

We obtain an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed four-dimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivariantly homeomorphic to $4$-dimensional Riemannian orbifolds with isometric $T^2$-actions. We also obtain a partial homeomorphism classification.Comment: 30 pages, 6 figures, 2 tables. We added subsections 2.4 and 2.5 for convenience of the reader, and modified sections 3, 4, 5, 6, and 7, to improve the clarity of the proof of the main result

Torus Actions on Alexandrov 4-Spaces

We obtain an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed four-dimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivariantly homeomorphic to 4-dimensional Riemannian orbifolds with isometric $T^2$-actions. We also obtain a partial homeomorphism classification

Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics

Let $E$ be a smooth bundle with fiber an $n$-dimensional real projective space $\mathbb{R}P^n$. We show that, if every fiber carries a positively curved pointwise strongly $1/4$-pinched Riemannian metric that varies continuously with respect to its base point, then the structure group of the bundle reduces to the isometry group of the standard round metric on $\mathbb{R}P^n$.Comment: 11 pages, to appear in Proceedings of the American Mathematical Societ

Singular Riemannian Foliations, variational problems and Principles of Symmetric Criticalities

A singular foliation $\mathcal{F}$ on a complete Riemannian manifold $M$ is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of $M$ into orbits of an isometric action. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRF with special properties, e.g. isoparametric foliations, SRF on fibers bundles with Sasaki metric, and orbit-like foliations. More precisely, we prove two results analogous to Palais' Principle of Symmetric Criticality, one is a general principle for $\mathcal{F}$ symmetric operators on the Hilbert space $W^{1,2}(M)$, the other one is for $\mathcal{F}$ symmetric integral operators on the Banach spaces $W^{1,p}(M)$. These results together with a $\mathcal{F}$ version of Rellich Kondrachov Hebey Vaugon Embedding Theorem allow us to circumvent difficulties with Sobolev's critical exponents when considering applications of Calculus of Variations to find solutions to PDEs. To exemplify this we prove the existence of weak solutions to a class of variational problems which includes $p$-Kirschoff problems.Comment: 54 page