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Ana Cristina Ferreira

Publications and source records attributed to Ana Cristina Ferreira.

At least 19 recordsLinked to original sources
arXiv2026

Fourier-Helgason transform as infinite geodesic time limit in geometric quantization

The Fourier-Helgason (FH) transform for a noncompact symmetric space $G/K$ establishes the direct integral decomposition of the unitary representation of $G$ on $L^2(G/K)$ into irreducible principal series representations. By applying techniques of geometric quantization to the symplectic manifold $T^*(G/K),$ Lisiecki in 1987 gave a geometric interpretation of the FH transform in the case when $G$ is complex. He defined for general $G$ a ''horizontal'' polarization on $T^*(G/K)$ and showed that, for complex $G$, the Blattner-Kostant-Sternberg (BKS) pairing between the Schrödinger vertical polarization Hilbert space, $L^2(G/K)$, and the Hilbert space of horizontally polarized functions coincides with the FH transform. However, in the same paper, Lisiecki showed that for noncomplex Lie groups the BKS pairing is nonequivalent to the FH transform and nonunitary in general. In the present paper, we resolve this discrepancy between the FH transform and geometric quantization in the case when $G$ is not complex. First, we show that the horizontal polarization is the infinite-time limit of the push-forward of the vertical polarization with respect to the geodesic flow for a $G$-invariant Riemannian metric. Then we lift the geodesic flow to an intertwining unitary parallel transport on the quantum bundle that we call quantum geodesic transform (QGT). Finally we show that the QGT has a well-defined limit, as the geodesic time goes to infinity, and that it is equal, up to the phase of the Harish-Chandra $c$-function and an irrelevant multiplicative constant, to the FH transform.

math.SG↗
arXiv2025

The completeness problem on 3-dimensional non-unimodular Lie groups

We consider the completeness problem for left-invariant Lorentzian metrics on 3-dimensional non-unimodular Lie groups, all of which have Lie algebra of the form $\mathbb{R} \ltimes_A \mathbb{R}^2$, where $A$ is a real $2 \times 2$ matrix with nonzero trace. The case where $A$ is not diagonalizable over $\mathbb{C}$ was addressed in previous work by the authors, and the limiting case where $A$ is a scalar multiple of the identity is also known from the literature. In this paper, we determine all geodesically (in)complete left-invariant Lorentzian metrics for all other cases where $A$ is diagonalizable over $\mathbb{R}$. Additionally, we show that, when $A$ is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$, there exists at least one incomplete metric. As a consequence of prior work and our results, we obtain that every 3-dimensional non-unimodular Lie group admits an incomplete left-invariant Lorentzian metric.

math.DG↗
arXiv2025

Isometries of 3-dimensional semi-Riemannian Lie groups

Let $G$ be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) metric $g$. By definition, the isometry group $\mathrm{Isom}(G, g)$ contains $G$ itself, acting by left translations. It turns out that, generically, $\mathrm{Isom}(G, g)$ is actually equal to $G$, and the natural question then becomes to classify those special metrics for which this is not the case. Using Lie-theoretical methods, we present a unified approach to obtain all pairs $(G, g)$ whose full isometry group $\mathrm{Isom}(G, g)$ has dimension greater than or equal to four. As a consequence, we determine, for every pair $(G, g)$, up to automorphism and scaling, the dimension of $\mathrm{Isom}(G, g)$, which can be three, four, or six.

math.DG↗
arXiv2025

The completeness problem on the pseudo-homothetic Lie group

Let us call pseudo-homothetic group the non-unimodular 3-dimensional Lie group that is the semi-direct product of $\mathbb{R}$ acting non-semisimply on $\mathbb{R}^2$. In this article, we solve the geodesic completeness problem on this Lie group. In particular, we exhibit a family of complete metrics such that all geodesics have bounded velocity. As an application, we show that the set of complete metrics is not closed.

math.DG↗
arXiv2025

Left-invariant ${\rm G}_2^*$-structures of type III

We investigate left-invariant ${\rm G}_2^*$-structures on 7-dimensional Lie groups, focusing on those whose holonomy algebras are indecomposable and of type III, the latter meaning that the socle of the holonomy representation is maximal. Building on the classification of indecomposable holonomy algebras contained in $\mathfrak g_2^*$, we determine which ones arise as infinitesimal holonomy algebras of type III for left-invariant ${\rm G}_2^*$-structures. Our main result shows that only abelian subalgebras occur, and these are necessarily of dimension two or three. Moreover, we provide explicit Lie groups with left-invariant ${\rm G}_2^*$-structures realizing these abelian holonomies.

math.DG↗
arXiv2025

Maps from Grassmannians of 2-planes to projective spaces

Using quaternions and octonions, we construct some maps from the Grassmannian of 2-dimensional planes of $\mathbb{R}^n$, $\mathrm{Gr}_2(\mathbb{R}^n)$, to the projective space $\mathbb{R}\mathrm{P}^k$, for certain values of $n$ and $k$. All of our maps induce an isomorphism at the level of fundamental groups, and two of them are shown to be submersions. As an application, we obtain new estimates of the Lusternik-Schnirelmann category of $\mathrm{Gr}_2(\mathbb{R}^n)$ for specific values of $n$.

math.AT↗
arXiv2024

Lie groups with all left-invariant semi-Riemannian metrics complete

For each left-invariant semi-Riemannian metric $g$ on a Lie group $G$, we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the completeness of $g$. When the adjoint representation of $G$ satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any $g$. We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products $K \ltimes_ρ\mathbb{R}^n$ , where $K$ is the direct product of a compact and an abelian Lie group and $ρ(K)$ is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the the absence of linear growth and suggest new questions.

math.DG↗
arXiv2024

Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

In this paper, we describe holomorphic quantizations of the cotangent bundle of a symmetric space of compact type $T^*(U/K)\cong U_\mathbb{C}/K_\mathbb{C}$, along Mabuchi rays of $U$-invariant Kähler structures. At infinite geodesic time, the Kähler polarizations converge to a mixed polarization $\mathcal{P}_\infty$. We show how a generalized coherent state transform relates the quantizations along the Mabuchi geodesics such that holomorphic sections converge, as geodesic time goes to infinity, to distributional $\mathcal{P}_\infty$-polarized sections. Unlike in the case of $T^*U$, the gCST mapping from the Hilbert space of vertically polarized sections are not asymptotically unitary due to the appearance of representation dependent factors associated to the isotypical decomposition for the $U$-action. In agreement with the general program outlined in [Bai+23], we also describe how the quantization in the limit polarization $\mathcal{P}_\infty$ is given by the direct sum of the quantizations for all the symplectic reductions relative to the invariant torus action associated to the Hamiltonian action of $U$.

math.SG↗
arXiv2022

Locally conformal SKT structures

A Hermitian metric on a complex manifold is called SKT (strong Kähler with torsion) if the Bismut torsion $3$-form $H$ is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called \emph{locally conformal SKT} (or shortly LCSKT). More precisely, a Hermitian structure $(J,g)$ is said to be LCSKT if there exists a closed non-zero $1$-form $α$ such that $d H = α\wedge H$. In the paper we consider non-trivial LCSKT structures, i.e. we assume that $d H \neq 0$ and we study their existence on Lie groups and their compact quotients by lattices. In particular, we classify 6-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a $6$-dimensional $3$-step nilpotent Lie algebra admitting a non-trivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a non-trivial LCSKT structure, which allows us to construct explicit examples of $6$-dimensional unimodular almost abelian Lie algebras admitting a non-trivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a 6-dimensional nilpotent or a $2n$-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is Kähler.

math.DG↗
arXiv2022

Einstein manifolds with skew torsion

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar curvature, it is necessarily compact and it has finite first fundamental group. The longest part of the paper is devoted to the systematic construction of large families of examples. We discuss when a Riemannian Einstein manifold can be Einstein with skew torsion. We give examples of almost Hermitian, almost metric contact, and G2 manifolds that are Einstein with skew torsion. For example, we prove that any Einstein-Sasaki manifold and any 7-dimensional 3-Sasakian manifolds admit deformations into an Einstein metric with parallel skew torsion.

math.DG↗
arXiv2022

Geodesic completeness of pseudo and holomorphic Riemannian metrics on Lie groups

This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler-Arnold formalism in the holomorphic setting. We study the real Lie group $\mathrm{SL}(2, \mathbb{R})$ and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group $\mathrm{SL}(2, \mathbb{C})$.

math.DG↗
arXiv2019

Palais leaf-space manifolds and surfaces carrying holomorphic flows

Starting from some remarkable singularities of holomorphic vector fields, we construct (open) complex surfaces over which the singularities in question are realized by complete vector fields. Our constructions lead to manifolds and vector fields beyond the algebraic setting and provide examples of complete vector fields with some new dynamical phenomena.

math.CA↗
arXiv2017

Dirichlet principal eigenvalue comparison theorems in geometry with torsion

We describe min-max formulas for the principal eigenvalue of a $V$-drift Laplacian defined by a vector field $V$ on a geodesic ball of a Riemannian manifold $N$. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under pointwise comparison of the corresponding radial sectional and Ricci curvatures, and of the radial component of the vector fields. These results generalize the known case $V=0$.

math.DG↗
arXiv2016

Tangent Lie groups are Riemannian naturally reductive spaces

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such that $(TG,g,\nabla)$ is naturally reductive. An alternative spinorial description of the same connection on the direct product $G\times \mathfrak{g}$ generalizes in a rather subtle way to $TS^7$, which is in many senses almost a tangent Lie group.

math.DG↗
arXiv2015

Quaternionic Heisenberg groups as naturally reductive homogeneous spaces

In this note, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost $3$-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the $3$-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact structure of the quaternionic Heisenberg groups. We focus on the $7$-dimensional case and prove that the canonical connection can also be obtained by means of a cocalibrated $G_2$ structure. We then study the spinorial properties of this group and present the noteworthy fact that it is the only known example of a manifold which carries generalized Killing spinors with three different eigenvalues.

math.DG↗
arXiv2014

The classification of naturally reductive homogeneous spaces in dimensions $n \leq 6$

We present a new method for classifying naturally reductive homogeneous spaces -- i.\,e.~homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion \emph{and} curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such a connection with the geometric structure on the given Riemannian manifold. It allows to reproduce by easier arguments the known classifications in dimensions $3,4$, and $5$, and yields as a new result the classification in dimension $6$. In each dimension, one obtains a `hierarchy' of degeneracy for the torsion form, which we then treat case by case. For the completely degenerate cases, we obtain results that are independent of the dimension. In some situations, we are able to prove that any Riemannian manifold with parallel skew torsion has to be naturally reductive. We show that a `generic' parallel torsion form defines a quasi-Sasakian structure in dimension $5$ and an almost complex structure in dimension $6$.

math.DG↗