Full Isometry Group Research Articles

A Class of Locally Inhomogeneous Complete Quaternionic Kähler Manifolds

AbstractWe prove that the one-loop deformation of any quaternionic Kähler manifold in the class of c-map spaces is locally inhomogeneous. As a corollary, we obtain that the full isometry group of the one-loop deformation of any homogeneous c-map space has precisely cohomogeneity one.

Isometric actions on the four dimensional Minkowski spacetime

In this paper we give representations of all connected Lie groups acting isometrically on the four dimensional Minkowski spacetime, up to conjugacy within the full isometry group of the space. For each obtained group, we study its induced orbits. Then we classify the Lie groups up to orbit equivalence.

Consistent truncations to 3-dimensional supergravity

We show how to construct consistent truncations of 10-/11-dimensional supergravity to 3-dimensional gauged supergravity, preserving various amounts of supersymmetry. We show, that as in higher dimensions, consistent truncations can be defined in terms of generalised G-structures in Exceptional Field Theory, with G ⊂ E8(8) for the 3-dimensional case. Differently from higher dimensions, the generalised Lie derivative of E8(8) Exceptional Field Theory requires a set of “covariantly constrained” fields to be well-defined, and we show how these can be constructed from the G-structure itself. We prove several general features of consistent truncations, allowing us to rule out a higher-dimensional origin of many 3-dimensional gauged supergravities. In particular, we show that the compact part of the gauge group can be at most SO(9) and that there are no consistent truncations on a 7-or 8-dimensional product of spheres such that the full isometry group of the spheres is gauged. Moreover, we classify which matter-coupled mathcal{N} ≥ 4 gauged supergravities can arise from consistent truncations. Finally, we give several examples of consistent truncations to three dimensions. These include the truncations of IIA and IIB supergravity on S7 leading to two different mathcal{N} = 16 gauged supergravites, as well as more general IIA/IIB truncations on Hp,7−p. We also show how to construct consistent truncations of IIB supergravity on S5 fibred over a Riemann surface. These result in 3-dimensional mathcal{N} = 4 gauged supergravities with scalar manifold mathcal{M}=frac{mathrm{S}mathrm{O}left(6,4right)}{mathrm{S}mathrm{O}(6)times mathrm{SO}(4)}times frac{mathrm{S}mathrm{U}left(2,1right)}{mathrm{S}left(mathrm{U}(1)times mathrm{U}(2)right)} with a ISO(3) × U(1)4 gauging and for hyperboloidal Riemann surfaces contain mathcal{N} = (2, 2) AdS3 vacua.

Isometry groups of three-dimensional Lie groups

We compute the full isometry group of any left-invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of the moduli space of left-invariant metrics, up to isometric automorphism are contained in the subspace of classes of metrics with maximal index of symmetry.

From vortices to instantons on the Euclidean Schwarzschild manifold

The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a complete description of a connected component of the moduli space of unit energy $\mathrm{SU} (2)$ instantons; new examples of instantons with non-integer energy (and non-trivial holonomy at infinity); a complete classification of finite energy, spherically symmetric, $\mathrm{SU} (2)$ instantons. As opposed to the previously known solutions, the generic instanton coming from our construction is not invariant under the full isometry group, in particular not static. Hence disproving a conjecture of Tekin.

Максимальная группа изометрий лоренцевой группы Ли 𝐴+(1) × 𝐴+(1)

We consider a connected simply connected four-dimensional Lie group𝐺𝐼𝑉 , whose Lie algebra belongs to the IV Bianchi type. It is proved that there is only one way of defining a left-invariant Lorentzian metric on it, for which the full isometry group of the resulting homogeneous manifold is five-dimensional. In the coordinates associated with the matrix representation of the Lie group, the matrix of the metric tensor is found and formulas are written according to which the one-parameter group of isometries acts, which leaves the unit element of the Lie group fixed. Also, formulas are written out according to which the full group of isometries acts.

Symmetries of quaternionic Kähler manifolds with S1‐symmetry

We study symmetry properties of quaternionic K\"ahler manifolds obtained by the HK/QK correspondence. To any Lie algebra $\mathfrak{g}$ of infinitesimal automorphisms of the initial hyper-K\"ahler data we associate a central extension of $\mathfrak{g}$, acting by infinitesimal automorphisms of the resulting quaternionic K\"ahler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the $c$-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special K\"ahler manifold gives rise to a Lie algebra of Killing fields of the corresponding one-loop deformed $c$-map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous then the one-loop deformed metric is of cohomogeneity at most one. As an example, we consider the one-loop deformation of the symmetric quaternionic K\"ahler metric on $SU(n,2)/S(U(n)\times U(2))$, which we prove is of cohomogeneity exactly one. This family generalizes the so-called universal hypermultiplet ($n=1$), for which we determine the full isometry group.

On the Description of Masses and Charges in the 6D Theory of Kaluza-Klein Type

We analyze two branches of five-dimensional theories from a methodological point of view: Kaluza’s theory and the Klein-Fock-Rumer (KFR) 5-optics, aimed at a geometric description of electromagnetism and masses of elementary particles, respectively. These two theories have a number of problematic points such as the Planckian masses in Kaluza’s theory, or appearing of a configuration space in the KFR scheme. We propose a simple six-dimensional toy model of Kaluza-Klein type with compactification on a two-dimensional torus $$\mathbb{T}^2$$, which demonstrates a possible way to overcome these difficulties in quite a simple manner. The key features of our approach are: merging of the above two 5D theories into a unique 6D theory; using the signature (+-) of extra space allowing one to renormalize the mass spectrum; and a special kind of truncation of the full isometry group to the electromagnetic gauge group U(1). The latter feature allows a geometrical interpretation in terms of an effective 5D theory on a hypersurface $$\Sigma\subset\mathbb{T}^2$$ being a leaf of a linear foliation of the 2-torus. Possible consequences and open questions arising in such a scheme are briefly discussed.

Discrete Scale Invariance in Holography Revisited

AbstractIn 2013, Balasubramanian presented a 5 + 1 dimensional holographic toy model that allows for an exact solution to Einstein's equations in the bulk in which the isometries of appear to be broken to an isometry group describing a discretely scale invariant and Poincaré invariant setup. In this paper, we investigate this solution in more detail. By analytically solving the Killing equations, we prove that the full isometry group is still present, although in a somewhat hidden way. We will also comment on the prospects of finding other holographic bottom up toy models which allow for solutions with discrete scale invariance or scale invariance without conformal invariance in the future.

Homogeneity for a class of Riemannian quotient manifolds

We study Riemannian coverings φ:M˜→Γ\\M˜ where M˜ is a normal homogeneous space G/K1 fibered over another normal homogeneous space M=G/K and K is locally isomorphic to a nontrivial product K1×K2. The most familiar such fibrations π:M˜→M are the natural fibrations of Stiefel manifolds SO(n1+n2)/SO(n1) over Grassmann manifolds SO(n1+n2)/[SO(n1)×SO(n2)] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings φ:M˜→Γ\\M˜ are the universal Riemannian coverings of spherical space forms. When M=G/K is reasonably well understood, in particular when G/K is a Riemannian symmetric space or when K is a connected subgroup of maximal rank in G, we show that the Homogeneity Conjecture holds for M˜. In other words we show that Γ\\M˜ is homogeneous if and only if every γ∈Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M˜ we work out the full isometry group of M˜, extending Élie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.

Factoring Euclidean isometries

Every isometry of a finite-dimensional Euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this paper we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimum length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.

An indiscrete Bieberbach theorem: from amenable CAT$(0)$ groups to Tits buildings

Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geodesically complete, it must be a product of flats, symmetric spaces, biregular trees and Bruhat–Tits buildings. We provide moreover a sufficient condition for a spherical building arising as the visual boundary of a proper CAT(0) space to be Moufang, and deduce that an irreducible locally finite Euclidean building of dimension ≥ 2 is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.

Generic elements in isometry groups of Polish ultrametric spaces

This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) has a neighborhood basis at the identity consisting of open subgroups with ample generics. It also gives a characterization of the existence of an open subgroup in Iso(X) with a comeager conjugacy class. We also study the transfinite sequence defined by the projection of a Polish ultrametric space X on the ultrametric space of orbits of X under the action of Iso(X).

Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds

AbstractLet Γ1 and Γ2 be Bieberbach groups contained in the full isometry group G of ℝn. We prove that if the compact flat manifolds Γ1\ℝn and Γ2\ℝn are strongly isospectral, then the Bieberbach groups Γ1 and Γ2 are representation equivalent; that is, the right regular representations L2(Γ1\G) and L2(Γ2\G) are unitarily equivalent.

Isometry groups of proper metric spaces

Given a locally compact Polish space X, a necessary and sufficient condition for a group G of homeomorphisms of X to be the full isometry group of (X,d) for some proper metric d on X is given. It is shown that every locally compact Polish group G acts freely on GxY as the full isometry group of GxY with respect to a certain proper metric on GxY, where Y is an arbitrary locally compact Polish space with (card(G),card(Y)) different from (1,2). Locally compact Polish groups which act effectively and almost transitively on complete metric spaces as full isometry groups are characterized. Locally compact Polish non-Abelian groups on which every left invariant metric is automatically right invariant are characterized and fully classified. It is demonstrated that for every locally compact Polish space X having more than two points the set of proper metrics d such that Iso(X,d) = {id} is dense in the space of all proper metrics on X.

A note on the uniqueness of the canonical connection of a naturally reductive space

We extend the result in Olmos and Reggiani (J. Reine Angew. Math. 664:29–53, 2012) to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In particular, the canonical connection is unique for the hyperbolic space when the dimension is different from three. We also prove that the canonical connection on the sphere is unique for the symmetric presentation. Finally, we compute the full isometry group (connected component) of a compact and locally irreducible naturally reductive space.

Representation Equivalence and p-Spectrum of Constant Curvature Space Forms

We study the p-spectrum of a locally symmetric space of constant curvature Γ∖X, in connection with the right regular representation of the full isometry group G of X on \(L^{2}(\varGamma \backslash G)_{\tau_{p}}\), where τ p is the complexified p-exterior representation of \(\operatorname{O}(n)\) on \(\bigwedge^{p}(\mathbb {R}^{n})_{\mathbb{C}}\). We give an expression of the multiplicity d λ (p,Γ) of the eigenvalues of the p-Hodge–Laplace operator in terms of multiplicities n Γ (π) of specific irreducible unitary representations of G.As a consequence, we extend results of Pesce for the spectrum on functions to the p-spectrum of the Hodge–Laplace operator on p-forms of Γ∖X, and we compare p-isospectrality with τ p -equivalence for 0≤p≤n. For spherical space forms, we show that τ-isospectrality implies τ-equivalence for a class of τ’s that includes the case τ=τ p . Furthermore, we prove that p−1 and p+1-isospectral implies p-isospectral.For nonpositive curvature space forms, we give examples showing that p-isospectrality is far from implying τ p -equivalence, but a variant of Pesce’s result remains true. Namely, for each fixed p, q-isospectrality for every 0≤q≤p implies τ q -equivalence for every 0≤q≤p. As a byproduct of the methods we obtain several results relating p-isospectrality with τ p -equivalence.

Geodesic orbit manifolds and Killing fields of constant length

The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold $(M,g)$ and the structure of its full isometry group. The Lie algebra of the full isometry group of $(M,g)$ is identified with the Lie algebra of Killing fields $\mathfrak{g}$ on $(M,g)$. We prove the following result: If $\mathfrak{a}$ is an abelian ideal of $\mathfrak{g}$, then every Killing field $X\in \mathfrak{a}$ has constant length. On the ground of this assertion we give a new proof of one result of C. Gordon: Every Riemannian geodesic orbit manifold of nonpositive Ricci curvature is a symmetric space.

Classification of compact lorentzian 2-orbifolds with noncompact full isometry groups

Among closed Lorentzian surfaces, only flat tori can admit noncompact full isometry groups. Moreover, for every n ≥ 3 the standard n-dimensional flat torus equipped with canonical metric has a noncompact full isometry Lie group. We show that this fails for n = 2 and classify the flat Lorentzian metrics on the torus with a noncompact full isometry Lie group. We also prove that every two-dimensional Lorentzian orbifold is very good. This implies the existence of a unique smooth compact 2-orbifold, called the pillow, admitting Lorentzian metrics with a noncompact full isometry group. We classify the metrics of this type and make some examples.

Automorphisms of the graph of free splittings

We prove that every simplicial automorphism of the free splitting graph of a free group Fn is induced by an outer automorphism of Fn for n ≥ 3. In this note we consider the graph Gn of free splittings of the free group Fn of rank n ≥ 3. Loosely speaking, Gn is the graph whose vertices are non-trivial free splittings of Fn up to conjugacy, and where two vertices are adjacent if they are represented by free splittings admitting a common refinement. The group Out(Fn) of outer automorphisms of Fn acts simplicially on Gn. Denoting by Aut(Gn) the group of simplicial automorphisms of the free splitting graph, we prove: Theorem 1. The natural map Out(Fn)→ Aut(Gn) is an isomorphism for n ≥ 3. We briefly sketch the proof of Theorem 1. We identify Gn with the 1skeleton of the sphere complex Sn and observe that every automorphism of Gn extends uniquely to an automorphism of Sn. It is due to Hatcher [4] that the sphere complex contains an embedded copy of the spine Kn of Culler-Vogtmann space. We prove that the latter is invariant under Aut(Sn), and that the restriction homomorphism Aut(Sn) → Aut(Kn) is injective. The claim of Theorem 1 then follows from a result of Bridson-Vogtmann [1] which asserts that Out(Fn) is the full automorphism group of Kn. Before concluding this introduction we would like to point out that recently Martino and Francaviglia [9] have proved that Out(Fn) is also the full isometry group of Culler-Vogtmann space when the latter is endowed with the Lipschitz metric. While one could claim that Theorem 1 is the analogue of Ivanov’s theorem on the isometries of the curve complex [6], the result of Martino and Francaviglia is the analogue of Royden’s theorem on the isometries of Teichmuller space [10]. The first author was partially supported by M.E.C. grant MTM2006/14688 and NUI Galway’s Millennium Research Fund; the second author was partially supported by NSF grant DMS-0706878 and Alfred P. Sloan Foundation.