scholarly journals Symmetries of holomorphic geometric structures on tori

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Sorin Dumitrescu ◽  
Benjamin McKay
Keyword(s):  
Homogeneous Space ◽  
Geometric Structure ◽  
Connected Components ◽  
Deformation Space ◽  
Higher Dimension ◽  
Geometric Structures ◽  
Translation Invariant ◽  
Homogeneous Surface ◽  
Complex Torus ◽  
Locally Homogeneous

AbstractWe prove that any holomorphic locally homogeneous geometric structure on a complex torus of dimension two, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true in any dimension. In higher dimension, we prove it for G nilpotent. We also prove that for any given complex algebraic homogeneous space (X, G), the translation invariant (X, G)-structures on tori form a union of connected components in the deformation space of (X, G)-structures.

Fujiki class 𝒞 and holomorphic geometric structures

2020 ◽  
Vol 31 (05) ◽  
pp. 2050039
Author(s):  
Indranil Biswas ◽  
Sorin Dumitrescu
Keyword(s):  
Chern Class ◽  
Geometric Structure ◽  
Geometric Structures ◽  
Cartan Geometry ◽  
Compact Torus ◽  
First Chern Class ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Affine Type ◽  
Compact Complex Manifolds

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class [Formula: see text], whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class [Formula: see text] and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.

scholarly journals Holomorphic affine connections on non-Kähler manifolds

2016 ◽  
Vol 27 (11) ◽  
pp. 1650094 ◽  
Author(s):  
Indranil Biswas ◽  
Sorin Dumitrescu
Keyword(s):  
Fundamental Group ◽  
Geometric Structure ◽  
Riemannian Metric ◽  
Complex Manifolds ◽  
Geometric Structures ◽  
Holomorphic Affine ◽  
Locally Homogeneous ◽  
Dimension One ◽  
Affine Type ◽  
Compact Complex Manifolds

Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi–Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In particular, if the geometric structure is rigid in Gromov’s sense, then the fundamental group of the manifold must be infinite. We also prove that compact complex manifolds of algebraic dimension one bearing a holomorphic Riemannian metric must have infinite fundamental group.

scholarly journals A Bochner principle and its applications to Fujiki class 𝒞 manifolds with vanishing first Chern class

2019 ◽  
Vol 22 (06) ◽  
pp. 1950051 ◽  
Author(s):  
Indranil Biswas ◽  
Sorin Dumitrescu ◽  
Henri Guenancia
Keyword(s):  
Chern Class ◽  
Geometric Structure ◽  
Riemannian Metric ◽  
Vanishing Theorem ◽  
Translation Invariant ◽  
Zariski Open Subset ◽  
First Chern Class ◽  
Locally Homogeneous ◽  
Affine Type ◽  
Compact Complex Manifolds

We prove a Bochner-type vanishing theorem for compact complex manifolds [Formula: see text] in Fujiki class [Formula: see text], with vanishing first Chern class, that admit a cohomology class [Formula: see text] which is numerically effective (nef) and has positive self-intersection (meaning [Formula: see text], where [Formula: see text]). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold [Formula: see text] are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of [Formula: see text] must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold [Formula: see text] admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant.

Invariant rigid geometric structures and expanding maps

2011 ◽  
Vol 32 (3) ◽  
pp. 941-959 ◽  
Author(s):  
YONG FANG
Keyword(s):  
Dynamical Systems ◽  
Geometric Structure ◽  
Closed Manifold ◽  
Expanding Maps ◽  
Geometric Structures ◽  
Global Rigidity ◽  
Chaotic Dynamical Systems ◽  
Expanding Map ◽  
Locally Homogeneous ◽  
Rigid Geometric Structures

AbstractIn the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let φ be a C∞ expanding map of a closed manifold. If φ preserves a topologically complete C∞ rigid geometric structure, then φ is C∞ conjugate to an expanding infra-nilendomorphism.

scholarly journals Directed Rooted Forests in Higher Dimension

10.37236/5819 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Olivier Bernardi ◽  
Caroline J. Klivans
Keyword(s):  
Generating Function ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Graph Laplacian ◽  
Connected Components ◽  
Higher Dimension ◽  
Cell Complexes ◽  
Open Questions ◽  
Higher Dimensional ◽  
Rooted Forest

For a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.

Riemannian and Euclidean material structures in anelasticity

2020 ◽  
Vol 25 (6) ◽  
pp. 1267-1293 ◽  
Author(s):  
Fabio Sozio ◽  
Arash Yavari
Keyword(s):  
Geometric Structure ◽  
Deformation Gradient ◽  
Linear Momentum ◽  
The Other ◽  
Governing Equations ◽  
Multiplicative Decomposition ◽  
Material Forces ◽  
Geometric Structures ◽  
Non Linear

In this paper, we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to non-linear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold. This allows one to define, in addition to the two geometric structures, a Weitzenböck connection on the material manifold. We use this connection to express natural uniformity in a geometrically meaningful way. The concept of uniformity is then extended to the Riemannian and Euclidean structures. Finally, we discuss the role of non-uniformity in the form of material forces that appear in the configurational form of the balance of linear momentum with respect to the two structures.

scholarly journals Riemannian manifolds with local symmetry

2014 ◽  
Vol 06 (02) ◽  
pp. 211-236 ◽  
Author(s):  
Wouter van Limbeek
Keyword(s):  
Riemannian Manifolds ◽  
Local Symmetry ◽  
Universal Cover ◽  
Connected Components ◽  
Curved Manifolds ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Positively Curved ◽  
Aspherical Manifolds

We give a classification of many closed Riemannian manifolds M whose universal cover [Formula: see text] possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that [Formula: see text] has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.

scholarly journals Low dimensional supersymmetries in SUSY Chern–Simons systems and geometrical implications

2016 ◽  
Vol 13 (06) ◽  
pp. 1650083 ◽  
Author(s):  
V. K. Oikonomou
Keyword(s):  
Quantum Theory ◽  
Geometric Structure ◽  
Fiber Bundles ◽  
Simons Theory ◽  
Dimensional Manifold ◽  
Geometric Structures ◽  
One Dimensional ◽  
Low Dimensional ◽  
Chern Simons ◽  
Chern Simons Theory

We study in detail the underlying graded geometric structure of abelian [Formula: see text] supersymmetric Chern–Simons theory in (2 + 1)-dimensions. This structure is an attribute of the hidden unbroken one-dimensional [Formula: see text] supersymmetries that the system also possesses. We establish the result that the geometric structures corresponding to the bosonic and to the fermionic sectors are equivalent fiber bundles over the (2 + 1)-dimensional manifold. Moreover, we find a geometrical answer to the question why some and not all of the fermionic sections are related to a [Formula: see text] supersymmetric algebra. Our findings are useful for the quantum theory of Chern–Simons vortices.

Geometric structures and electronic properties of the Bi2X2Y (X, Y = O, S, Se, and Te) ternary compound family: a systematic DFT study

2018 ◽  
Vol 6 (48) ◽  
pp. 13241-13249 ◽  
Author(s):  
Xiaoyu Ma ◽  
Dahu Chang ◽  
Chunxiang Zhao ◽  
Rui Li ◽  
Xiaoyu Huang ◽  
...  
Keyword(s):  
Density Functional Theory ◽  
Electronic Properties ◽  
Ternary Compound ◽  
Geometric Structure ◽  
First Principles ◽  
Density Functional ◽  
Dft Study ◽  
Geometric Structures ◽  
Functional Theory

The geometric structure and electronic properties of Bi2X2Y (X, Y = O, S, Se, and Te) ternary compound have been studied by means of first-principles density functional theory.

scholarly journals Labelled seeds and the mutation group

2016 ◽  
Vol 163 (2) ◽  
pp. 193-217
Author(s):  
ALASTAIR KING ◽  
MATTHEW PRESSLAND
Keyword(s):  
Automorphism Group ◽  
Homogeneous Space ◽  
Equivalence Relation ◽  
Cluster Algebra ◽  
Equivalence Relations ◽  
Connected Components ◽  
Model Field ◽  
Regular Equivalence ◽  
Mutation Group

AbstractWe study the set${\mathcal{S}}$of labelled seeds of a cluster algebra of rankninside a field${\mathcal{F}}$as a homogeneous space for the groupMnof (globally defined) mutations and relabellings. Regular equivalence relations on${\mathcal{S}}$are associated to subgroupsWof AutMn(${\mathcal{S}}$), and we thus obtain groupoidsW\${\mathcal{S}}$. We show that for two natural choices of equivalence relation, the corresponding groupsWcandW+act on${\mathcal{F}}$, and the groupoidsWc\${\mathcal{S}}$andW+\${\mathcal{S}}$act on the model field${\mathcal{K}}$=ℚ(x1,. . .,xn). The groupoidW+\${\mathcal{S}}$is equivalent to Fock–Goncharov's cluster modular groupoid. Moreover,Wcis isomorphic to the group of cluster automorphisms, andW+to the subgroup of direct cluster automorphisms, in the sense of Assem–Schiffler–Shramchenko.We also prove that, for mutation classes whose seeds have mutation finite quivers, the stabiliser of a labelled seed underMndetermines the quiver of the seed up to ‘similarity’, meaning up to taking opposites of some of the connected components. Consequently, the subgroupWcis the entire automorphism group of${\mathcal{S}}$in these cases.

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