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## An algorithm for automatic checking of exercises in a dynamic geometry system: iGeom

ISOTANI, Seiji; BRANDAO, Leonidas de Oliveira
Fonte: PERGAMON-ELSEVIER SCIENCE LTD Publicador: PERGAMON-ELSEVIER SCIENCE LTD
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
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One of the key issues in e-learning environments is the possibility of creating and evaluating exercises. However, the lack of tools supporting the authoring and automatic checking of exercises for specifics topics (e.g., geometry) drastically reduces advantages in the use of e-learning environments on a larger scale, as usually happens in Brazil. This paper describes an algorithm, and a tool based on it, designed for the authoring and automatic checking of geometry exercises. The algorithm dynamically compares the distances between the geometric objects of the students solution and the templates solution, provided by the author of the exercise. Each solution is a geometric construction which is considered a function receiving geometric objects (input) and returning other geometric objects (output). Thus, for a given problem, if we know one function (construction) that solves the problem, we can compare it to any other function to check whether they are equivalent or not. Two functions are equivalent if, and only if, they have the same output when the same input is applied. If the students solution is equivalent to the templates solution, then we consider the students solution as a correct solution. Our software utility provides both authoring and checking tools to work directly on the Internet...

## "Desenvolvimento de ferramentas no iGeom: utilizando a geometria dinâmica no ensino presencial e a distância" ; "Developing tools in iGeom: Using the dynamic geometry in the classroom and distance learning"

Isotani, Seiji
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 01/04/2005 PT
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Neste trabalho, apresentamos o desenvolvimento de ferramentas no programa iGeom - Geometria Interativa na Internet, para ensino-aprendizagem de Geometria, dando destaque aos recursos que facilitam a integração e uso deste programa, principalmente em ambientes de educação a distância via Internet. Atualmente, este tipo de programa é bastante conhecido e a Geometria que ele possibilita é usualmente denominada Geometria Dinâmica. Em poucas palavras, um programa de Geometria Dinâmica é a implementação computacional da régua e do compasso, permitindo que os objetos construídos sejam movidos mantendo-se às propriedades da construção. Dentre os principais recursos desenvolvidos, destacamos a autoria e a validação automática de exercícios e a comunicação com servidores, que podem ser utilizados para integrarem o iGeom em sistemas gerenciadores de cursos pela Web. Deste modo, se integrado a um sistema gerenciador, estes recursos podem ser utilizados para facilitar a tarefa do professor, que poderá criar exercícios diretamente pela Web e não precisará avaliar pessoalmente as respostas de cada aluno, e também para que o aluno saiba de imediato se sua solução está dentro do esperado pelo professor.¶; In this work...

## Geometria diferencial em grupos de Lie; Differential geometry on Lie groups

Eder de Moraes Correa
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 02/05/2013 PT
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Neste trabalho estudamos os aspectos geométricos dos grupos de Lie do ponto de vista da geometria Riemanniana, geometria Hermitiana e geometria Kähler, através das estruturas geométricas invariantes associadas. Exploramos resultados relacionados às curvaturas da variedade Riemanniana subjacente a um grupo de Lie através do estudo de sua álgebra de Lie correspondente. No contexto da geometria Hermitiana e geometria Kähler, para um caso concreto de grupo de Lie complexo, investigaram suas curvaturas seccionais holomorfas e verificamos a existência de uma estrutura pseudo-Kähler invariante por sua forma real compacta.; In this dissertation, we study the geometric aspects of Lie groups from the viewpoint of Riemannian geometry, Hermitian geometry, and Kähler geometry through its associated invariant geometric structures. We explore results related to curvatures of Riemannian manifold underlying a Lie group by studying its corresponding Lie algebra. In the context of Hermitian geometry and Kähler geometry, for a complex Lie group case, we investigate its holomorphic sectional curvatures and verify the existence of pseudo-Kähler structure invariant for its compact real form.

## Generalized geometry

Baraglia, David
Fonte: Universidade de Adelaide Publicador: Universidade de Adelaide
Tipo: Tese de Doutorado Formato: 730723 bytes; 54917 bytes; application/pdf; application/pdf
Publicado em //2007 EN
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Generalized geometry is a recently discovered branch of differential geometry that has received a reasonable amount of interest due to the emergence of several connections with areas of Mathematical Physics. The theory is also of interest because the different geometrical structures are often generalizations of more familiar geometries. We provide an introduction to the theory which explores a number of these generalized geometries. After introducing the basic underlying structures of generalized geometry we look at integrability which offers some geometrical insight into the theory and this leads to Dirac structures. Following this we look at generalized metrics which provide a generalization of Riemannian metrics. We then look at generalized complex geometry which is a generalization of both complex and symplectic geometry and is able to unify a number of features of these two structures. Beyond generalized complex geometry we also look at generalized Calabi-Yau and generalized Kähler structures which are also generalizations of the more familiar structures.; Thesis (M.Sc.(M&CS))--University of Adelaide, School of Mathematical Sciences, Discipline of Pure Mathematics, 2007.

## A Geometria da escola e a utilização de história em quadrinhos nos anos finais do Ensino Fundamental; The Geometry of the school and the using cartoons in the final years of primary school

Santos, Lupi Scheer dos
Fonte: Universidade Federal de Pelotas; Faculdade de Educação; Programa de Pós-Graduação em Ensino de Ciências e Matemática; UFPel; Brasil Publicador: Universidade Federal de Pelotas; Faculdade de Educação; Programa de Pós-Graduação em Ensino de Ciências e Matemática; UFPel; Brasil
Tipo: Dissertação de Mestrado
POR
Relevância na Pesquisa
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This study deals with a qualitative investigation with quantitative aspects carried out during 2012 and 2013. It aims to learn about the reality of Geometry teaching in the final years of primary school in the municipal schools in the city of Pelotas in order to provide the History of Mathematics as a mediating instrument for teaching and learning through cartoons. The theoretical foundation is made up of three stages. Firstly, the search of elements about the history of mathematics and cartoons as pedagogical resources to be used in the classroom. The Scientific Electronic Library Online (SciElO) dataset is used for free searches on the internet. It also uses the approaches provided by Miguel (1997, 2009) as well as the ones provided by Vergueiro (2010). In the second stage a bibliographic study, regarding the mediation concepts, zone of proximal development, spontaneous and scientific knowledge about Vygotsky theory (2011) and researchers in the area, was carried out. This study was used in the elaboration of the product of this research. The third stage resorts to Pavanello (1989) and to event publication datasets such as the National Meeting of Mathematics Education in order to capture elements to understand the teaching of Geometry in the schools. As far as the empirical field is concerned...

## Lorentzian geometry and physics in Kasparov's theory

van den Dungen, Koenraad Lambertus
Fonte: Universidade Nacional da Austrália Publicador: Universidade Nacional da Austrália
Tipo: Thesis (PhD)
EN
Relevância na Pesquisa
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We study two geometric themes, Lorentzian geometry and gauge theory, from the perspective of Connes’ noncommutative geometry and (the unbounded version of) Kasparov’s KK-theory. Lorentzian geometry is the mathematical framework underlying Einstein’s description of gravity. The geometric formulation of a gauge theory (in terms of principal bundles) offers a classical description for the interactions between particles. The underlying motivation is the hope that this noncommutative approach may lead to a unified description of gauge theories coupled with gravity on a Lorentzian manifold. The main objects in noncommutative geometry are spectral triples, which encompass and generalise Riemannian spin manifolds. A spectral triple defines a class in K-homology, via which one can access the topology of the (noncommutative) manifold. In this thesis we present two possible definitions for ‘Lorentian spectral triples’, which offer noncommutative generalisations of Lorentzian manifolds as well. We will prove that both definitions preserve the link with analytic K-homology. We will describe under which conditions Lorentzian (or pseudo- Riemannian) manifolds satisfy these definitions. Another main example is the harmonic oscillator...

## Absolute Parallelism Geometry: Developments, Applications and Problems

Wanas, M. I.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/09/2002
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Absolute parallelism geometry is frequently used for physical applications. It has two main defects, from the point of view of applications. The first is the identical vanishing of its curvature tensor. The second is that its autoparallel paths do not represent physical trajectories. The present work shows how these defects were treated in the course of development of the geometry. The new version of this geometry contains simultaneous non-vanishing torsion and curvatures. Also, the new paths discovered in this geometry do represent physical trajectories. Advantages and disadvantages of this geometry are given for each stage of its development. Physical applications are just mentioned without giving any details.; Comment: 12 Pages Tex file. A version of this paper has been published in the Proceeding of the 11th conference on "Finsler, Lagrange and Hamilton Geometry", held in Romania (Bacau, Feb. 2000)

## Generalized complex geometry

Gualtieri, Marco
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define and study generalized complex branes, which interpolate between flat bundles on Lagrangian submanifolds and holomorphic bundles on complex submanifolds.; Comment: 49 pages, refined from math.DG/0401221 and updated with new constructions, added references.

## Azumaya noncommutative geometry and D-branes - an origin of the master nature of D-branes

Liu, Chien-Hao
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/12/2011
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In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of Douglas and Moore in 1996. A conjectural relation of our setting with D-geometry' in the sense of Douglas is also given. The lecture is based on a series of works on D-branes with Shing-Tung Yau, and in part with Si Li and Ruifang Song.; Comment: 23 pages, 5 figures; parts delivered in the workshop `Noncommutative algebraic geometry and D-branes', December 12 -- 16, 2011, organized by Charlie Beil, Michael Douglas, and Peng Gao, at Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY

## The Hitchin Model, Poisson-quasi-Nijenhuis Geometry and Symmetry Reduction

Zucchini, Roberto
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We revisit our earlier work on the AKSZ formulation of topological sigma model on generalized complex manifolds, or Hitchin model. We show that the target space geometry geometry implied by the BV master equations is Poisson--quasi--Nijenhuis geometry recently introduced and studied by Sti\'enon and Xu (in the untwisted case). Poisson--quasi--Nijenhuis geometry is more general than generalized complex geometry and comprises it as a particular case. Next, we show how gauging and reduction can be implemented in the Hitchin model. We find that the geometry resulting form the BV master equation is closely related to but more general than that recently described by Lin and Tolman, suggesting a natural framework for the study of reduction of Poisson--quasi--Nijenhuis manifolds.; Comment: 38 pages, no figures, LaTex. One paragraph in sect. 6 and 3 references added

## Gromov compactness in non-archimedean analytic geometry

Yu, Tony Yue
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Gromov's compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov's compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of K\"ahler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin's representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.; Comment: Improved presentation

## Split Special Lagrangian Geometry

Harvey, F. Reese; Lawson Jr, H. Blaine
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 02/07/2010
Relevância na Pesquisa
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One purpose of this article is to draw attention to the seminal work of J. Mealy in 1989 on calibrations in semi-riemannian geometry where split SLAG geometry was first introduced. The natural setting is provided by doing geometry with the complex numbers C replaced by the double numbers D, where i with i^2 = -1 is replaced by tau with tau^2 = 1. A rather surprising amount of complex geometry carries over, almost untouched, and this has been the subject of many papers. We briefly review this material and, in particular, we discuss Hermitian D-manifolds with trivial canonical bundle, which provide the background space for the geometry of split SLAG submanifolds. A removable singularities result is proved for split SLAG subvarieties. It implies, in particular, that there exist no split SLAG cones, smooth outside the origin, other than planes. This is in sharp contrast to the complex case. Parallel to the complex case, space-like Lagrangian submanifolds are stationary if and only if they are theta-split SLAG for some phase angle theta, and infinitesimal deformations of split SLAG submanifolds are characterized by harmonic 1-forms on the submanifold. We also briefly review the recent work of Kim, McCann and Warren who have shown that split Special Lagrangian geometry is directly related to the Monge-Kantorovich mass transport problem.

## Strongly Homotopy Lie Algebras from Multisymplectic Geometry

Shahbazi, C. S.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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This Master Thesis is devoted to the study of $n$-plectic manifolds and the Strongly Homotopy Lie algebras, also called $L_{\infty}$-algebras, that can be associated to them. Since multisymplectic geometry and $L_{\infty}$-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce the relevant background material in order to make the exposition accessible to non-experts, perhaps interested physicists. The background material includes graded and homological algebra theory, fibre bundles, basics of group actions on manifolds and symplectic geometry. We give an introduction to $L_{\infty}$-algebras and define $L_{\infty}$-morphisms in an independent way, not yet related to multisymplectic geometry, giving explicit formulae relating $L_{\infty}[1]$-algebras and $L_{\infty}$-algebras. We give also an account of multisymplectic geometry and $n$-plectic manifolds, connecting them to $L_{\infty}$-algebras. We then introduce, closely following the work {1304.2051} of Yael Fregier, Christopher L. Rogers and Marco Zambon, the concept of homotopy moment map. The new results presented here are the following: we obtain specific conditions under which two $n$-plectic manifolds with strictly isomorphic Lie-$n$ algebras are symplectomorphic...

## Counting Algebraic Curves with Tropical Geometry

Block, Florian
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/06/2012
Relevância na Pesquisa
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Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to significant progress. In this survey, we give an introduction to tropical geometry techniques for algebraic curve counting problems. We also survey some recent developments, with a particular emphasis on the computation of the degree of the Severi varieties of the complex projective plane and other toric surfaces as well as Hurwitz numbers and applications to real enumerative geometry. This paper is based on the author's lecture at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 2011.; Comment: 14 pages, 6 figures. To appear in Contemporary Mathematics (Proceedings), "Tropical Geometry and Integrable Systems", Glasgow, July 2011

## Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry

Vacaru, Sergiu I.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/08/2004
Relevância na Pesquisa
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We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metric structure (with an additional affine connection possessing nontrivial torsion) is equivalent to a generalized Lagrange, or Finsler, geometry modeled on nonholonomic manifolds. This results in the problem of constructing noncommutative geometries with local anisotropy, in particular, related to geometrization of classical and quantum mechanical and field theories, even if we restrict our considerations only to commutative and noncommutative Riemannian spaces. We elaborate a geometric approach to the Clifford modules adapted to nonlinear connections, to the theory of spinors and the Dirac operators on nonholonomic spaces and consider possible generalizations to noncommutative geometry. We argue that any commutative Riemann--Finsler geometry and generalizations my be derived from noncommutative geometry by applying certain methods elaborated for Riemannian spaces but extended to nonholonomic frame transforms and manifolds provided with nonlinear connection structure.; Comment: 55 pages...

## Generalized complex geometry

Gualtieri, Marco
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/01/2004
Relevância na Pesquisa
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Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.; Comment: Oxford University DPhil thesis, 107 pages

## Mori geometry meets Cartan geometry: Varieties of minimal rational tangents

Hwang, Jun-Muk
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/01/2015
Relevância na Pesquisa
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We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.; Comment: to appear in Proceedings of ICM2014

## tt* geometry, Frobenius manifolds, their connections, and the construction for singularities

Hertling, Claus
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The base space of a semiuniversal unfolding of a hypersurface singularity carries a rich geometry. By work of K. Saito and M. Saito is can be equipped with the structure of a Frobenius manifold. By work of Cecotti and Vafa it can be equipped with tt* geometry if the singularity is quasihomogeneous. tt* geometry generalizes the notion of variation of Hodge structures. In the second part of this paper (chapters 6-8) Frobenius manifolds and tt* geometry are constructed for any hypersurface singularity, using essentially oscillating integrals; and the intimate relationship between polarized mixed Hodge structures and this tt* geometry is worked out. In the first part (chapters 2-5) tt* geometry and Frobenius manifolds and their relations are studied in general. To both of them flat connections with poles are associated, with distinctive common and different properties. A frame for a simultaneous construction is given.; Comment: 82 pages, amslatex, additional references, misprints corrected

## D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifolds

Liu, Chien-Hao; Yau, Shing-Tung
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/06/2014
Relevância na Pesquisa
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We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, $k$-times differentiable maps, and smooth maps from an Azumaya/matrix manifold with a fundamental module to a (commutative) real manifold $Y$ is developed. Such maps are meant to describe D-branes or matrix branes in string theory when these branes are light and soft with only small enough or even zero brane-tension. When $Y$ is a symplectic manifold (resp. a Calabi-Yau manifold; a $7$-manifold with $G_2$-holonomy; a manifold with an almost complex structure $J$), the corresponding notion of Lagrangian maps (resp. special Lagrangian maps; associative maps, coassociative maps; $J$-holomorphic maps) are introduced. Indicative examples linking to symplectic geometry and string theory are given. This provides us with a language and part of the foundation required to study themes, new or old, in symplectic geometry and string theory, including (1) $J$-holomorphic D-curves (with or without boundary), (2) quantization and dynamics of D-branes in string theory, (3) a definition of Fukaya category guided by Lagrangian maps from Azumaya manifolds with a fundamental module with a connection...

## Influence of the subducting plate velocity on the geometry of the slab and migration of the subduction hinge

Schellart, Wouter
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Geological observations indicate that along two active continental margins (East Asia and Mediterranean) major phases of overriding plate extension, resulting from subduction hinge-retreat, occurred synchronously with a reduction in subducting plate velocity. In this paper, results of fluid dynamical experiments are presented to test the influence of the velocity of the subducting plate on the hinge-migration velocity and on the geometry of the slab. Results show that hinge-retreat decreases with increasing subducting plate velocity. In addition, phases of hinge-retreat alternate with phases of hinge-advance for relatively high subducting plate velocities due to interaction of the slab with the bottom of the box, simulating the upper-lower mantle discontinuity. Such slab kinematics could explain the episodic behaviour of back-arc opening observed in convergent settings. The geometry of the slab and the kinematics of subduction are significantly affected by the velocity of the subducting plate. Three subduction modes with accompanying slab geometry can be recognized. A relatively low subducting plate velocity is accompanied by relatively fast hinge-retreat with backward sinking of the slab and a backward draping slab geometry. With increasing subducting plate velocity hinge-migration is relatively small...