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Marcos M. Alexandrino

Publications and source records attributed to Marcos M. Alexandrino.

At least 19 recordsLinked to original sources
arXiv2026

On singular Finsler foliations of $(α,β)$-spaces

We investigate singular Finsler foliations (SFFs) on a manifold equipped with an $(α,β)$-metric. To be precise, we verify that any SFF of an $(α,β)$-space is, under some hypotheses on the metric, a singular Riemannian foliation (SRF). This gives a partial answer to the general question "under which conditions a SFF is a SRF with respect to some Riemannian metric". Moreover, we extend the proof of Molino's conjecture to SFFs whenever they are also a SRFs. Finally, we prove equifocality of the regular leaves for a SFF under the same condition.

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arXiv2026

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 the orbits of a Lie group action by isometries. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRFs with special properties, which we name as AVP. Examples of such SRFs being considered include isoparametric foliations, SRFs on Euclidean fiber bundles, and the partition of $M$ into the orbits of a Lie group acting by isometries. More precisely, we prove an analog to Palais' Principle of Symmetric Criticality for $\mathcal{F}$-symmetric integral operators on the Banach spaces $W^{1,p}(M)$. This result together with a version of the Rellich--Kondrachov--Hebey--Vaugon Embedding Theorem for $\mathcal{F}$-basic Sobolev functions allows us to circumvent difficulties with Sobolev's critical exponents when considering applications of techniques from Calculus of Variations to find solutions to PDEs. To exemplify this, we prove the existence of countably infinite many weak solutions to a class of variational problems, which includes $p$-Kirchhoff problems for manifolds equipped with AVP.

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arXiv2022

Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino's structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations and discuss singular Killing foliations.

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arXiv2022

Traveling along horizontal broken geodesics of a homogenous Finsler submersion

In this paper, we discuss how to travel along horizontal broken geodesics of a homogenous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets $\mathcal{A}_{q}(\mathcal{C})$ of the set of analytic vector fields $\mathcal{C}$ determined by the family of horizontal unit geodesic vector fields $\mathcal{C}$ to the fibers $\mathcal{F}=\{ρ^{-1}(c)\}$ of a homogenous analytic Finsler submersion $ρ: M\to B$. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examples on non compact Finsler manifolds $M$ where the attainable sets (the dual leaves) are no longer orbits or even submanifolds. Nevertheless we prove that, when $M$ is compact and the orbits of $\mathcal{C}$ are embedded, then the attainable sets coincide with the orbits. Furthermore, if the flag curvature is positive then $M$ coincides with the attainable set of each point. In other words, fixed two points of $M$, one can travel from one point to the other along horizontal broken geodesics. In addition, we show that each orbit $\mathcal{O}(q)$ of $\mathcal{C}$ associated to a singular Finsler foliation coincides with $M$, when the flag curvature is positive, i.e, we prove Wilking's result in Finsler context. In particular we review Wilking's transversal Jacobi fields in Finsler case.

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arXiv2021

Singular Riemannian Foliations and the prescribing scalar curvature problem

An orbit-like foliation is a singular foliation on a complete Riemannian manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular foliation contains not only the classe of partion of the space into orbits of isometric actions, but also infinite many non homogenous examples and in particular the partition of $M$ into orbits of a proper groupoid. In this paper we prove a version of Kondrakov Embedding Theorem and an analogous Principle of Symmetric Criticality of Palais for basic funcions of orbit-like foliations. As proof of concepts, we study not only the corresponding Yamabe problem in the setting, but also to the case of fiber bundles with homogeneous fibers, seeking for the existence of metrics with constant scalar curvature that respect the respective Riemannian Foliation decomposition. An application to the existence of positive constant scalar curvature on exotic spheres is presented. In an upcoming version we shall extend the results to the corresponding Kazdan--Warner problem.

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arXiv2021

Lie groupoids and semi-local models of Singular Riemannian foliations

We describe a local model for any Singular Riemannian Foliation in a neighbourhood of a closed saturated submanifold of a regular stratum. Moreover we construct a Lie groupoid which controls the transverse geometry of the linear approximation of the Singular Riemannian Foliation around these submanifolds. We also discuss the closure of this Lie groupoid and its Lie algebroid.

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arXiv2021

On equifocal Finsler submanifolds and analytic maps

A relevant property of equifocal submanifolds is that their parallel sets are still immersed submanifolds, which makes them a natural generalization of the so-called isoparametric submanifolds. In this paper, we prove that the regular fibers of an analytic map $π:M^{m+k}\to B^{k}$ are equifocal whenever $M^{m+k}$ is endowed with a complete Finsler metric and there is a restriction of $π$ which is a Finsler submersion for a certain Finsler metric on the image. In addition, we prove that when the fibers provide a singular foliation on $M^{m+k}$, then this foliation is Finsler.

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arXiv2019

On Mean curvature flow of Singular Riemannian foliations: Non compact cases

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. We also discuss the existence of basin of attractions, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and make a few remarks on MCF of non closed leaves of generalized isoparametric foliation.

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arXiv2018

On Finsler transnormal functions

In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isoparametric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of $M$ into level sets is a Finsler partition, when the function is defined on a compact analytic manifold $M$.

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arXiv2018

On singular Finsler foliation

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 $(M,\mathtt{h} )$. As a direct consequence we infer that the regular leaves are equifocal submanifolds (a generalization of isoparametric submanifolds) when the wind $W$ is an infinitesimal homothety of $\mathtt{h}$ (e.,g when $W$ is killing vector field or $M$ has constant Finsler curvature). We also present a slice theorem that relates local singular Finsler foliations on Finsler manifolds with singular Finsler foliations on Minkowski spaces.

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arXiv2014

Smoothness of isometric flows on orbit spaces and applications to the theory of foliations

We prove here that given a proper isometric action $K\times M\to M$ on a complete Riemannian manifold $M$ then every continuous isometric flow on the orbit space $M/K$ is smooth, i.e., it is the projection of an $K$-equivariant smooth flow on the manifold $M$. As a direct corollary we infer the smoothness of isometric actions on orbit spaces. Another relevant application of our result concerns Molino's conjecture, which states that the partition of a Riemannian manifold into the closures of the leaves of a singular Riemannian foliation is still a singular Riemannian foliation. We prove Molino's conjecture for the main class of foliations considered in his book, namely orbit-like foliations.

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arXiv2013

Isometries between leaf spaces

In this paper we prove that an isometry between orbit spaces of two proper isometric actions is smooth if it preserves the codimension of the orbits or if the orbit spaces have no boundary. In other words, we generalize Myers-Steenrod's theorem for orbit spaces. These results are proved in the more general context of singular Riemannian foliations.

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arXiv2012

Progress in the Theory of Singular Riemannian Foliations

A singular foliation is called a singular Riemannian foliation (SRF) if every geodesic that is perpendicular to one leaf is perpendicular to every leaf it meets. A typical example is the partition of a complete Riemannian manifold into orbits of an isometric action. In this survey, we provide an introduction to the theory of SRFs, leading from the foundations to recent developments in research on this subject. Sketches of proofs are included and useful techniques are emphasized. We study the local structure of SRFs in general and under curvature conditions. We review the solution of the Palais-Terng problem on integrability of the horizontal distribution. Important special classes of SRFs, like polar and variationally complete foliations and their connections, are treated. A characterisation of SRFs whose leaf space is an orbifold is given. Moreover, desingularizations of SRFs are studied and applications, e.g., to Molino's conjecture, are presented.

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arXiv2011

Polar foliations and isoparametric maps

A singular Riemannian foliation $F$ on a complete Riemannian manifold $M$ is called a polar foliation if, for each regular point $p$, there is an immersed submanifold $Σ$, called section, that passes through $p$ and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of $M$ into the orbits of a polar action, i.e., an isometric action with sections. In this work we prove that the leaves of $F$ coincide with the level sets of a smooth map $H: M\to Σ$ if $M$ is simply connected. In particular, we have that the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.

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arXiv2011

On polar foliations and fundamental group

In this work we investigate the relation between the fundamental group of a complete Riemannian manifold $M$ and the quotient between the Weyl group and reflection group of a polar action on $M$, as well as the relation between the fundamental group of $M$ and the quotient between the lifted Weyl group and lifted reflection group. As applications we give short alternative proofs of two results. The first one, due to the author and Töben, states that there is non-exceptional orbit, if $M$ is simply connected. The second result, due to Lytchak, states that the orbits are closed and embedded if $M$ is simply connected. All results are proved in the more general case of polar foliations.

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arXiv2010

Introduction to Lie groups, isometric and adjoint actions and some generalizations

The main purpose of these lecture notes is to provide a concise introduction to Lie groups, Lie algebras, and isometric and adjoint actions, aiming mostly at advanced undergraduate and graduate students. In addition, the connection between such classic theories and the research area of the first author is explored. Namely, generalizations to isoparametric submanifolds, polar actions and singular Riemannian foliations with sections (s.r.f.s.) are mentioned. The first chapters cover basic concepts, giving results on adjoint representation, closed subgroups, bi-invariant metrics, Killing forms and splitting in simple ideals. In the following chapters, proper and isometric actions are recalled together with adjoint action and foliations, mostly concerning the Weyl group, normal slices and Dynkin diagrams. A special focus is given to maximal tori and roots of compact Lie groups, exploring its connection with isoparametric submanifolds and polar actions. Furthermore, in the last chapter, a survey on recent research results on s.r.f.s. is given. In this revised version, more details about fiber bundles, proper and isometric actions are explored, and further exercises and examples were added. It also features new sections with examples of singular Riemannian foliations constructed with surgery and suspension of homomorphisms. This is still a preliminary version and we expect to improve it in the future. We would be grateful for any kind of suggestions.

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arXiv2010

On closed geodesics in the leaf spaces of singular Riemannian foliations

In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

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