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Sergio Console

Bio: Sergio Console is an academic researcher from University of Turin. The author has contributed to research in topics: Submanifold & Holonomy. The author has an hindex of 12, co-authored 35 publications receiving 732 citations.

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Book
08 Feb 2016
TL;DR: In this article, the Berger-Simons holonomy theorem and the Skew-Torsion Holonomy Theorem were proved for complex submanifolds of Cn with nontransitive normal holonomy.
Abstract: Basics of Submanifold Theory in Space Forms The fundamental equations for submanifolds of space forms Models of space forms Principal curvatures Totally geodesic submanifolds of space forms Reduction of the codimension Totally umbilical submanifolds of space forms Reducibility of submanifolds Submanifold Geometry of Orbits Isometric actions of Lie groups Existence of slices and principal orbits for isometric actions Polar actions and s-representations Equivariant maps Homogeneous submanifolds of Euclidean spaces Homogeneous submanifolds of hyperbolic spaces Second fundamental form of orbits Symmetric submanifolds Isoparametric hypersurfaces in space forms Algebraically constant second fundamental form The Normal Holonomy Theorem Normal holonomy The normal holonomy theorem Proof of the normal holonomy theorem Some geometric applications of the normal holonomy theorem Further remarks Isoparametric Submanifolds and Their Focal Manifolds Submersions and isoparametric maps Isoparametric submanifolds and Coxeter groups Geometric properties of submanifolds with constant principal curvatures Homogeneous isoparametric submanifolds Isoparametric rank Rank Rigidity of Submanifolds and Normal Holonomy of Orbits Submanifolds with curvature normals of constant length and rank of homogeneous submanifolds Normal holonomy of orbits Homogeneous Structures on Submanifolds Homogeneous structures and homogeneity Examples of homogeneous structures Isoparametric submanifolds of higher rank Normal Holonomy of Complex Submanifolds Polar-like properties of the foliation by holonomy tubes Shape operators with some constant eigenvalues in parallel manifolds The canonical foliation of a full holonomy tube Applications to complex submanifolds of Cn with nontransitive normal holonomy Applications to complex submanifolds of CPn with nontransitive normal holonomy The Berger-Simons Holonomy Theorem Holonomy systems The Simons holonomy theorem The Berger holonomy theorem The Skew-Torsion Holonomy Theorem Fixed point sets of isometries and homogeneous submanifolds Naturally reductive spaces Totally skew one-forms with values in a Lie algebra The derived 2-form with values in a Lie algebra The skew-torsion holonomy theorem Applications to naturally reductive spaces Submanifolds of Riemannian Manifolds Submanifolds and the fundamental equations Focal points and Jacobi fields Totally geodesic submanifolds Totally umbilical submanifolds and extrinsic spheres Symmetric submanifolds Submanifolds of Symmetric Spaces Totally geodesic submanifolds Totally umbilical submanifolds and extrinsic spheres Symmetric submanifolds Submanifolds with parallel second fundamental form Polar Actions on Symmetric Spaces of Compact Type Polar actions - rank one Polar actions - higher rank Hyperpolar actions - higher rank Cohomogeneity one actions - higher rank Hypersurfaces with constant principal curvatures Polar Actions on Symmetric Spaces of Noncompact Type Dynkin diagrams of symmetric spaces of noncompact type Parabolic subalgebras Polar actions without singular orbits Hyperpolar actions without singular orbits Polar actions on hyperbolic spaces Cohomogeneity one actions - higher rank Hypersurfaces with constant principal curvatures Appendix: Basic Material Exercises appear at the end of each chapter.

250 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the Dolbeault cohomology of a compact nilmanifold endowed with an invariant complex structure is isomorphic to the cohomeology of the differential bigraded algebra associated to the complexification of complex structures.
Abstract: LetM=G/Γ be a compact nilmanifold endowed with an invariant complex structure. We prove that on an open set of any connected component of the moduli space $$\mathcal{C}\left( \mathfrak{g} \right)$$ of invariant complex structures onM, the Dolbeault cohomology ofM is isomorphic to the cohomology of the differential bigraded algebra associated to the complexification $$\mathfrak{g}^\mathbb{C} $$ of the Lie algebra ofG. to obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.

136 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds.
Abstract: Let M =Γ \G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

65 citations

Posted Content
TL;DR: In this article, it was shown that the Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result of C. Maclaughlin, H. Pedersen, Y. Poon and S.S. Salamon for 2-step nilmanifolds.
Abstract: Let $M = \Gamma \backslash G$ be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result of C. Maclaughlin, H. Pedersen, Y.S. Poon and S. Salamon for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

39 citations


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Journal ArticleDOI
TL;DR: In this paper, a ten-dimensional theory, named \beta-supergravity, was presented, which contains non-geometric fluxes and could uplift some four-dimensional gauged supergravities.
Abstract: We present a ten-dimensional theory, named \beta-supergravity, that contains non-geometric fluxes and could uplift some four-dimensional gauged supergravities. Building on earlier work, we study here its NSNS sector, where Q- and R-fluxes are precisely identified. Interestingly, the Q-flux is captured in an analogue of the Levi-Civita spin connection, giving rise to a second curvature scalar. We reproduce the ten-dimensional Lagrangian using the Generalized Geometry formalism; this provides us with enlightening interpretations of the new structures. Then, we derive the equations of motion of our theory, and finally discuss further aspects: the dimensional reduction to four dimensions and comparison to gauged supergravities, the obtention of ten-dimensional purely NSNS solutions, the extensions to other sectors and new objects, the supergravity limit, and eventually the symmetries, in particular the \beta gauge transformation. We also introduce the related notion of a generalized cotangent bundle.

176 citations

Book ChapterDOI
01 Jan 2000
TL;DR: A survey of isoparametric hypersurfaces and their generalizations can be found in this paper, where the authors present a survey of the generalizations of the Dupin hypersurface.
Abstract: This chapter discusses isoparametric hypersurfaces and their generalizations The chapter presents a survey of isoparametric hypersurfaces, discusses Dupin hypersurfaces, and describes isoparametric submanifolds in ambient spaces that are finite dimensional Euclidean spaces or infinite dimensional Hilbert spaces The taut submanifolds in Riemannian manifolds are described The classification of isoparametric hypersurfaces in spheres with three principal curvatures is related to the various characterizations of the standard embeddings of the projective planes The chapter discusses isoparametric hypersurfaces in spheres with four different principal curvatures, all of which are assumed to have the same multiplicity The Clifford examples of Ferus, Karcher, and Munzner together with the homogeneous hypersurfaces are all known examples of isoparametric hypersurfaces in spheres A difference between isoparametric and Dupin hypersurfaces is that although the parallel hypersurfaces of the Dupin ones are also Dupin, they do not foliate the ambient space as the isoparametric ones do The hypersurface of a sphere is isoparametric if and only if it is equifocal

172 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the strong Kahler with torsion structures (J,g) on M are parametrized by the points in a subset of the Euclidean space, in particular, the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold.
Abstract: Let (J,g) be a Hermitian structure on a six-dimensional compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We show that, up to equivalence of the complex structure, the strong Kahler with torsion structures (J,g) on M are parametrized by the points in a subset of the Euclidean space, in particular, the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold and the region outside to strong Kahler with torsion structures with nonabelian J on the nilmanifold \(\Gamma\backslash (H^3\times H^3),\) where H3 is the Heisenberg group. A classification of six-dimensional nilmanifolds admitting balanced Hermitian structures (J,g) is given, and as an application we classify the nilmanifolds having invariant complex structures which do not admit Hermitian structure with restricted holonomy of the Bismut connection contained in SU(3). It is also shown that on the nilmanifold \(\Gamma\backslash (H^3\times H^3)\) the balanced condition is not stable under small deformations. Finally, we prove that a compact quotient of \(H(2,1)\times \mathbb{R},\) where H(2,1) is the five-dimensional generalized Heisenberg group, is the only six-dimensional nilmanifold having locally conformal Kahler metrics, and the complex structures underlying such metrics are all equivalent. Moreover, this nilmanifold is a Vaisman manifold for any invariant locally conformal Kahler metric.

148 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered IIA compactifications on solvmanifolds with O6/D6 branes and studied the conditions for obtaining de Sitter vacua in ten dimensions.
Abstract: We consider IIA compactifications on solvmanifolds with O6/D6 branes and study the conditions for obtaining de Sitter vacua in ten dimensions. While this is a popular set-up for searching de Sitter vacua, we propose a new method to include supersymmetry breaking sources. For space-time filling branes preserving bulk supersymmetry, the energy density can easily be extremized with respect to all fields, thanks to the replacement of the DBI action by a pullback of a special form given by a pure spinor. For sources breaking bulk supersymmetry, we propose to replace the DBI action by the pullback of a more general polyform, which is no longer pure. This generalization provides corrections to the energy-momentum tensor which give a positive contribution to the cosmological constant. We find a de Sitter solution to all (bulk and world-volume) equations derived from this action. We argue it solves the equations derived from the standard source action. The paper also contains a review of solvmanifolds.

112 citations