Scielo RSS <![CDATA[Revista de la Unión Matemática Argentina]]> vol. 47 num. 1 lang. es <![CDATA[SciELO Logo]]> <![CDATA[II Encuentro de Geometría Diferencial: 6 al 11 de junio de 2005. La Falda, Sierras de Córdoba, Argentina]]> <![CDATA[Carnot manifolds]]> <![CDATA[4-step Carnot spaces and the 2-stein condition]]> We consider the 2-stein condition on k-step Carnot spaces S. These spaces are a subclass in the class of solvable Lie groups of Iwasawa type of algebraic rank one and contain the homogeneous Einstein spaces within this class. They are obtained as a semidirect product of a graded nilpotent Lie group N and the abelian group R. We show that the 2-stein condition is not satisfied on a proper 4-step Carnot spaces S. <![CDATA[An introduction to supersymmetry]]> This is a short introduction to supersymmetry based on the first of two lectures given at the II Workshop in Differential Geometry, La Falda, Córdoba, 2005. <![CDATA[The special geometry of Euclidian supersymmetry: a survey]]> This is a survey about recent joint work with Christoph Mayer, Thomas Mohaupt and Frank Saueressig on the special geometry of Euclidian supersymmetry. It is based on the second of two lectures given at the II Workshop in Differential Geometry, La Falda, Córdoba, 2005. <![CDATA[Special metrics in G<sub>2</sub> geometry]]> We discuss metrics with holonomy G2 by presenting a few crucial examples and review a series of G2 manifolds constructed via solvable Lie groups, obtained in [15]. These carry two related distinguished metrics, one negative Einstein and the other in the conformal class of a Ricci-flat metric, plus other features considered definitely worth investigating. <![CDATA[Dolbeault cohomology and deformations of nilmanifolds]]> In these notes I review some classes of invariant complex structures on nilmanifolds for which the Dolbeault cohomology can be computed by means of invariant forms, in the spirit of Nomizu's theorem for de Rham cohomology. Moreover, deformations of complex structures are discussed. Small deformations remain in some cases invariant, so that, by Kodaira-Spencer theory, Dolbeault cohomology can be still computed using invariant forms. <![CDATA[Taut submanifolds]]> This is a short, elementary survey article about taut submanifolds. In order to simplify the exposition, we restrict to the case of compact smooth submanifolds of Euclidean or spherical spaces. Some new, partial results concerning taut 4-manifolds are discussed at the end of the text. <![CDATA[Riemannian G-manifolds as Euclidean submanifolds]]> We survey on some recent developments on the study of Riemannian G-manifolds as Euclidean submanifolds. <![CDATA[On Complete Spacelike Submanifolds in the De Sitter Space With Parallel Mean Curvature Vector]]> The text surveys some results concerning submanifolds with parallel mean curvature vector immersed in the De Sitter space. We also propose a semi-Riemannian version of an important inequality obtained by Simons in the Riemannian case and apply it in order to obtain some results characterizing umbilical submanifolds and a product of submanifolds in the (n + p)-dimensional De Sitter space <img width=34 height=21 src="../../../../../img/revistas/ruma/v47n1/1a091x.png">. <![CDATA[Integrability of f-structures on generalized flag manifolds]]> Here we consider a generalized flag manifold <img width=76 height=19 src="/img/revistas/ruma/v47n1/1a100x.png">and a differential structure <img width=15 height=12 src="/img/revistas/ruma/v47n1/1a101x.png">which satisfy <img width=96 height=18 src="/img/revistas/ruma/v47n1/1a102x.png">these structures are called f-structures. Such structure <img width=15 height=12 src="/img/revistas/ruma/v47n1/1a104x.png">determines in the tangent bundle of <img width=11 height=12 src="/img/revistas/ruma/v47n1/1a105x.png">some <img width=60 height=19 src="/img/revistas/ruma/v47n1/1a106x.png">invariant distributions. Since flag manifolds are homogeneous reductive spaces, they certainly have combinatorial properties that allow us to make some easy calculations about integrability conditions for <img width=15 height=12 src="/img/revistas/ruma/v47n1/1a107x.png">itself and the distributions that it determines on <img width=14 height=12 src="/img/revistas/ruma/v47n1/1a108x.png">An special case corresponds to the case <img width=73 height=19 src="/img/revistas/ruma/v47n1/1a109x.png">, the unitary group, this is the geometrical classical flag manifold and in fact tools coming from graph theory are very useful. <![CDATA[On the Variety of Planar Normal Sections]]> In the present paper we present a survey of results concerning the variety <img width=46 height=19 src="/img/revistas/ruma/v47n1/1a110x.png">of planar normal sections associated to a natural embedding of a real flag manifold <img width=29 height=12 src="/img/revistas/ruma/v47n1/1a111x.png">. The results included are those that, we feel, better describe the nature of this algebraic variety of <img width=55 height=14 src="/img/revistas/ruma/v47n1/1a112x.png">. In particular we present results concerning its Euler characteristic showing that it depends only on <img width=52 height=12 src="/img/revistas/ruma/v47n1/1a113x.png">and not on the nature of <img width=18 height=12 src="/img/revistas/ruma/v47n1/1a114x.png">itself. Furthermore, when <img width=18 height=12 src="/img/revistas/ruma/v47n1/1a115x.png">is the manifold of complete flags of a compact simple Lie group, we present what is, in some sense, its dimension and a large class of submanifolds of <img width=55 height=15 src="/img/revistas/ruma/v47n1/1a116x.png">contained in <img width=46 height=19 src="/img/revistas/ruma/v47n1/1a117x.png">. <![CDATA[Connections compatible with tensors: A characterization of left-invariant Levi-Civita connections in Lie groups]]> Symmetric connections that are compatible with semi-Riemannian metrics can be characterized using an existence result for an integral leaf of a (possibly non integrable) distribution. In this paper we give necessary and sufficient conditions for a left-invariant connection on a Lie group to be the Levi-Civita connection of some semi-Riemannian metric on the group. As a special case, we will consider constant connections in <img width=22 height=14 src="/img/revistas/ruma/v47n1/1a120x.png">. <![CDATA[Spectral properties of elliptic operators on bundles of <img width=25 height=28 src="/img/revistas/ruma/v47n1/1a130x.png">-manifolds]]> We present some results on the spectral geometry of compact Riemannian manifolds having holonomy group isomorphic to <img width=19 height=19 src="/img/revistas/ruma/v47n1/1a131x.png">, <img width=108 height=15 src="/img/revistas/ruma/v47n1/1a132x.png">, for the Laplacian on mixed forms and for twisted Dirac operators.