IRIS Uni Torinohttps://iris.unito.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Tue, 18 Jan 2022 07:20:27 GMT2022-01-18T07:20:27Z10201Esplorazione dei solidi e oltre: fare geometria con gli Zometool. A cura di Ornella Robutti.http://hdl.handle.net/2318/1554367Titolo: Esplorazione dei solidi e oltre: fare geometria con gli Zometool. A cura di Ornella Robutti.
Abstract: Nonostante sia presente nelle Indicazioni Nazionali, l’insegnamento della geometria solida è spesso messo in secondo piano nella comune pratica didattica per le sue difficoltà concettuali e gli ampi prerequisiti teorici. Questo volume vuole offrire una proposta di didattica costruttiva della geometria solida fondata sull’uso degli ZOMETOOL, un innovativo strumento didattico adatto a costruire figure sia bidimensionali che tridimensionali.
La metodologia adottata è quella del laboratorio di matematica, guidato da schede, in modo da stimolare la discussione tra gli studenti nei singoli gruppi, puntando sia sulla leva motivazionale data dalla scoperta, sia sulla costruzione autonoma dei significati matematici.
Nel volume sono presentate le attività proposte negli anni scolastici 2013/14 e 2014/15 in diverse scuole secondarie di secondo grado della provincia di Torino e sono analizzati i risultati degli studenti e le criticità emerse in corso d’opera. Tale lavoro si configura dunque come strumento per guidare l’insegnante nella realizzazione in classe delle attività e per fornire suggerimenti utili alla progettazione di nuovi percorsi di apprendimento.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/2318/15543672016-01-01T00:00:00ZNon-integrable special geometric structures in dimensions six and sevenhttp://hdl.handle.net/2318/1557510Titolo: Non-integrable special geometric structures in dimensions six and seven
Abstract: Six-dimensional manifolds admitting an SU(3)-structure and seven-dimensional manifolds endowed with a G2-structure are the main object of study in this thesis. In the six-dimensional case, we consider SU(3)-structures (ω,ψ) satisfying the condition dω = c ψ, c ∈ R − {0}, known in literature as coupled. They are half-flat and generalize the class of nearly Kähler SU(3)-structures. We study their properties in the general case and in relation with the rôle they play in supersymmetric string theory, the conditions under which the associated metric is Einstein, their behaviour with respect to the Hitchin flow equations and various classes of examples. In the seven-dimensional case, we focus on G2-structures defined by a stable 3-form φ which is locally conformal equivalent to a closed one. We study the restrictions arising when the underlying metric is Einstein, we use warped products and the mapping torus construction to provide noncompact and compact examples of 7-manifolds endowed with such a structure starting from 6-manifolds with a coupled SU(3)-structure and, finally, we prove a structure result for compact 7-manifolds. We conclude studying a generalization of the Hitchin flow equations and a geometric flow of spinors on 6-manifolds. The latter gives rise to a flow of SU(3)-structures.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/2318/15575102016-01-01T00:00:00ZClosed warped G2-structures evolving under the Laplacian flowhttp://hdl.handle.net/2318/1655103Titolo: Closed warped G2-structures evolving under the Laplacian flow
Abstract: We study the behaviour of the Laplacian flow evolving closed G2-structures on warped products of the form M^6×S^1, where the base M^6 is a compact 6-manifold endowed with an SU(3)-structure. In the general case, we reinterpret the flow as a set of evolution equations on M^6 for the differential forms defining the SU(3)-structure and the warping function. When the latter is constant, we find sufficient conditions for the existence of solutions of the corresponding coupled flow. This provides a method to construct immortal solutions of the Laplacian flow on the product manifolds M^6×S^1. The application of our results to explicit cases allows us to obtain new examples of expanding Laplacian solitons.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/2318/16551032020-01-01T00:00:00ZSpecial Types of Locally Conformal Closed G2-Structureshttp://hdl.handle.net/2318/1684878Titolo: Special Types of Locally Conformal Closed G2-Structures
Abstract: Motivated by known results in locally conformal symplectic geometry, we study different classes of G2-structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G2-structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G2-structures.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/2318/16848782018-01-01T00:00:00ZHalf-flat structures inducing Einstein metrics on homogeneous spaceshttp://hdl.handle.net/2318/150154Titolo: Half-flat structures inducing Einstein metrics on homogeneous spaces
Abstract: In this paper, we consider half-flat SU(3)-structures and the subclasses of coupled and double structures. In the general case, we show that the intrinsic torsion form w1^- is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneous spaces. We give an example of a left invariant half-flat (non-coupled and non-double) structure inducing an Einstein metric on S^3×S^3 and we show there does not exist any left invariant coupled structure inducing an Ad(S^1)-invariant Einstein metric on it. Finally, we show that there are no coupled structures inducing the Einstein metric on Einstein solvmanifolds and on homogeneous Einstein manifolds of nonpositive sectional curvature.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2318/1501542015-01-01T00:00:00ZOn the automorphism group of a closed G2-structurehttp://hdl.handle.net/2318/1676269Titolo: On the automorphism group of a closed G2-structure
Abstract: We study the automorphism group of a compact 7-manifold M endowed with a closed non-parallel G2-structure, showing that its identity component is abelian with dimension bounded by min{6,b2(M)}. This implies the non-existence of compact homogeneous manifolds endowed with an invariant closed non-parallel G2-structure. We also discuss some relevant examples.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/2318/16762692019-01-01T00:00:00ZOn G2-Structures, Special Metrics and Related Flowshttp://hdl.handle.net/2318/1742543Titolo: On G2-Structures, Special Metrics and Related Flows
Abstract: We review results about G2-structures in relation to the existence of special metrics, such as Einstein metrics and Ricci solitons, and the evolution under the Laplacian flow on non-compact homogeneous spaces. We also discuss some examples in detail.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/2318/17425432020-01-01T00:00:00ZExact G2-structures on unimodular Lie algebrashttp://hdl.handle.net/2318/1739993Titolo: Exact G2-structures on unimodular Lie algebras
Abstract: We consider seven-dimensional unimodular Lie algebras $mathfrak{g}$ admitting exact G$_2$-structures, focusing our attention on those with vanishing third Betti number $b_3(mathfrak{g})$.
We discuss some examples, both in the case when $b_2(rg)
eq0$ and in the case when the Lie algebra $rg$ is (2,3)-trivial, i.e., when both $b_2(mathfrak{g})$ and $b_3(mathfrak{g})$ vanish.
These examples are solvable, as $b_3(mathfrak{g})=0$, but they are not strongly unimodular,
a necessary condition for the existence of lattices on the simply connected Lie group corresponding to $mathfrak{g}$.
More generally, we prove that any seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit any exact G$_2$-structure.
From this, it follows that there are no compact examples of the form $(GammaackslashG,arphi)$, where $G$ is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra,
$GammasubsetG$ is a co-compact discrete subgroup, and $arphi$ is an exact $G_2$-structure on $GammaackslashG$ induced by a left-invariant one on $G$.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/2318/17399932020-01-01T00:00:00ZHomogeneous symplectic half-flat 6-manifoldshttp://hdl.handle.net/2318/1676268Titolo: Homogeneous symplectic half-flat 6-manifolds
Abstract: We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result of Wolf and Gray allowing one to show the nonexistence of compact non-flat examples. In the noncompact setting, we classify such manifolds under the assumption that G is semisimple. Moreover, in each case, we describe all invariant symplectic half-flat SU(3)-structures up to isomorphism, showing that the Ricci tensor is always Hermitian with respect to the induced almost complex structure. This property of the Ricci tensor is characterized in the general case.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/2318/16762682019-01-01T00:00:00ZEinstein locally conformal calibrated G_2-structureshttp://hdl.handle.net/2318/135182Titolo: Einstein locally conformal calibrated G_2-structures
Abstract: We study locally conformal calibrated $G_2$-structures whose underlying Riemannian metric is Einstein, showing an integral formula for compact manifolds. We show that a compact homogeneous 7-manifold cannot admit an invariant Einstein locally conformal calibrated $G_2$-structure unless the underlying metric is flat. In contrast to the compact case we provide a non-compact example of homogeneous manifold endowed with a locally conformal calibrated $G_2$-structure whose associated Riemannian metric is Einstein and non Ricci-flat. The homogeneous Einstein metric is a rank-one extension of a Ricci soliton on the 3-dimensional complex Heisenberg group endowed with a left-invariant coupled half-flat SU(3)-structure $(\omega, \psi)$ such that $d \omega = - Re(\psi)$.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2318/1351822015-01-01T00:00:00Z