Spectral properties of flat manifolds

Miatello, Roberto J.; Rossetti, Juan Pablo

doi:10.1090/conm/491/09610

Cited by 8 publications

(6 citation statements)

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“…Recently, the research on flat Riemannian manifolds has shown many developments [22,23,2,17,18,19,20]. For instance, a quantization scheme for euclidean space forms based on path integrals is developed in [1].…”

Section: Discussionmentioning

confidence: 99%

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Holomorphic Path Integrals in Tangent Space for Flat Manifolds

Capobianco¹,

Reartes²

2017

Preprint

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In this paper we study the quantum evolution in a flat Riemannian manifold. The holomorphic functions are defined on the cotangent bundle of this manifold. We construct Hilbert spaces of holomorphic functions in which the scalar product is defined using the exponential map. The quantum evolution is proposed by means of an infinitesimal propagator and the holomorphic Feynman integral is developed via the exponential map. The integration corresponding to each step of the Feynman integral is performed in the tangent space. Moreover, in the case of S 1 , the method proposed in this paper naturally takes into account paths that must be included in the development of the corresponding Feynman integral.

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Section: Discussionmentioning

confidence: 99%

“…Recently, the research on flat Riemannian manifolds (known as euclidean space forms [27,16]) has shown many developments. For instance, its spectral properties are studied in detail in [22,23]. In cosmology, the euclidean space forms are used to model the spatial part of flat universe models [2,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Holomorphic Path Integrals in Tangent Space for Flat Manifolds

Capobianco¹,

Reartes²

2017

Preprint

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“…• representation theoretic methods such as Sunada's Theorem [53] and its generalizations (see the survey [23]); • the method of torus actions (see, for example, [22,27,29,47]); and • methods specific to special Riemannian manifolds such as flat closed manifolds (e.g., [13,42]) or Lens spaces (e.g., [36,39,49]) in which the spectrum can be "explicitly" computed, e.g., through the use of a generating function. In [25], P. Herbrich, D. Webb and the third author showed that both the original Sunada technique and the torus action method, when applied to compact Riemannian manifolds with boundary, result in manifolds that are Steklov isospectral as well as Laplace isospectral.…”

Section: Examples Of Steklov Isospectral Orbifoldsmentioning

confidence: 99%

Spectral Geometry of the Steklov Problem on Orbifolds

Arias-Marco

Dryden

Gordon

et al. 2017

International Mathematics Research Notices

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We consider how the geometry and topology of a compact n-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions.In dimension two, we show that a flat disk is not only Steklov isospectral to a cone but, in fact, a disk and cone of appropriate size have identical Dirichlet-to-Neumann operators. This provides a counterexample to the inverse tomography problem in the orbifold setting and contrasts with results of Lassas and Uhlmann in the manifold setting.In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components.

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“…On the other hand, compact flat manifolds are well understood due to the classical Bieberbach's theorems and they have been used to study different phenomena in geometry. For instance, questions about isospectrality (see [18] and the references therein), Kähler flat metrics with holonomy in SU(n) ( [7]), among others.…”

Section: Introductionmentioning

confidence: 99%

Classification of 6-dimensional splittable flat solvmanifolds

Tolcachier¹

2021

Preprint

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A flat solvmanifold is a compact quotient Γ\G where G is a simply-connected solvable Lie group endowed with a flat left invariant metric and Γ is a lattice of G. Any such Lie group can be written as G = R k ⋉ φ R m with R m the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting G by a lattice Γ that can be decomposed as Γ = Γ 1 ⋉ φ Γ 2 , where Γ 1 and Γ 2 are lattices of R k and R m , respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5.

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Spectral properties of flat manifolds

Cited by 8 publications

References 0 publications

Holomorphic Path Integrals in Tangent Space for Flat Manifolds

Holomorphic Path Integrals in Tangent Space for Flat Manifolds

Spectral Geometry of the Steklov Problem on Orbifolds

Classification of 6-dimensional splittable flat solvmanifolds

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