Results 1  10
of
151
The Srní lectures on nonintegrable geometries with torsion
 Arch. Math. (Brno
, 2006
"... Abstract. This review article intends to introduce the reader to nonintegrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections ..."
Abstract

Cited by 58 (8 self)
 Add to MetaCart
(Show Context)
Abstract. This review article intends to introduce the reader to nonintegrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skewsymmetric torsion are exhibited as one of the main tools to understand nonintegrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a Gstructure as unifying principles. The General Holonomy Principle bridges over to parallel objects, thus motivating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenböck formulas for Dirac operators associated with torsion connections enable us to discuss spinorial field equations, such as those governing the common sector of type II superstring theory.
Some Remarks on G2Structures
 Proceedings of Gökova GeometryTopology Conference 2005, edited by
, 2006
"... Abstract. This article consists of loosely related remarks about the geometry of G2structures on 7manifolds, some of which are based on unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. After some preliminary background information about the group G2 and its repr ..."
Abstract

Cited by 51 (1 self)
 Add to MetaCart
(Show Context)
Abstract. This article consists of loosely related remarks about the geometry of G2structures on 7manifolds, some of which are based on unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. After some preliminary background information about the group G2 and its representation theory, a set of techniques is introduced for calculating the differential invariants of G2structures and the rest of the article is applications of these results. Some of the results that may be of interest are as follows: First, a formula is derived for the scalar curvature and Ricci curvature of a G2structure in terms of its torsion and covariant derivatives with respect to the ‘natural connection ’ (as opposed to the LeviCivita connection) associated to a G2structure. When the fundamental 3form of the G2structure is closed, this formula implies, in particular, that the scalar curvature of the underlying metric is nonpositive and vanishes if and only if the structure is torsionfree. These formulae are also used to generalize a recent result of Cleyton and Ivanov [3] about the nonexistence of closed Einstein G2structures (other than
Geometric model for complex nonKahler manifolds with SU(3) structure
 COMMUN. MATH. PHYS
, 2002
"... We propose a universal geometric construction of complex nonKähler manifolds with intrinsic SU(3) structure, used in supersymmetric string compactifications. All these manifolds are some T 2 fibrations over a CalabiYau base. We show that the conditions of N = 1 supersymmetry in the heterotic strin ..."
Abstract

Cited by 47 (0 self)
 Add to MetaCart
We propose a universal geometric construction of complex nonKähler manifolds with intrinsic SU(3) structure, used in supersymmetric string compactifications. All these manifolds are some T 2 fibrations over a CalabiYau base. We show that the conditions of N = 1 supersymmetry in the heterotic string theory specify a subclass of manifolds that we constructed, which generalizes the examples known in the literature. Moreover, many known examples of internal manifolds in type II string compactifications can also be described in our construction, although supersymmetry restrictions of the geometry are not known yet. Mathematically, we construct complex, Hermitian nonKähler n + 1folds with a holomorphically trivial canonical bundle fibering over CalabiYau nfolds. We show that one can lift Special Lagrangian submanifolds and fibrations from the base CalabiYau to Special Lagrangian (calibrated) submanifolds and fibrations upstairs. We discuss in detail the Recently, 6dimensional nonKähler manifolds with SU(3) structure have attracted considerable
Deformations of generalized calibrations and compact nonKähler manifolds with vanishing first Chern class
, 2002
"... ..."
Killing Spinor Equations In Dimension 7 And Geometry Of Integrable G_2Manifolds
, 2008
"... We compute the scalar curvature of 7dimensional G2manifolds admitting a connection with totally skewsymmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the ..."
Abstract

Cited by 41 (0 self)
 Add to MetaCart
We compute the scalar curvature of 7dimensional G2manifolds admitting a connection with totally skewsymmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3form field. In dimension n = 7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G2structure into a cocalibrated one of pure type W3.
Connections on Naturally Reductive Spaces, their Dirac OPERATOR AND HOMOGENEOUS MODELS IN STRING THEORY
, 2002
"... Given a reductive homogeneous space M = G/H endowed with a naturally reductive metric, we study the oneparameter family of connections ∇ t joining the canonical and the LeviCivita connection (t = 0, 1/2). We show that the Dirac operator D t corresponding to t = 1/3 is the socalled “cubic ” Dirac ..."
Abstract

Cited by 37 (11 self)
 Add to MetaCart
(Show Context)
Given a reductive homogeneous space M = G/H endowed with a naturally reductive metric, we study the oneparameter family of connections ∇ t joining the canonical and the LeviCivita connection (t = 0, 1/2). We show that the Dirac operator D t corresponding to t = 1/3 is the socalled “cubic ” Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new Ginvariant first order differential operator D on spinors and an eigenvalue estimate for the first eigenvalue of D 1/3. This geometric situation can be used for constructing Riemannian manifolds which are Ricci flat and admit a parallel spinor with respect to some metric connection ∇ whose torsion T ̸ = 0 is a 3form, the geometric model for the common sector of string theories. We present some results about solutions to the string equations and give a detailed discussion of some 5dimensional example.
On the holonomy of connections with skewsymmetric torsion
"... Abstract. We investigate the holonomy group of a linear metric connection with skewsymmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any nondegenerated 2form or any spinor. Suitable integral formulas allow us to pro ..."
Abstract

Cited by 34 (6 self)
 Add to MetaCart
(Show Context)
Abstract. We investigate the holonomy group of a linear metric connection with skewsymmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any nondegenerated 2form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skewsymmetric torsion. On the AloffWallach space N(1,1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any 7dimensional 3Sasakian manifold admits P 2parameter families of linear metric connections and spinorial connections defined by 4forms with parallel spinors. Contents
Canonical connections on paracontact manifolds
, 2008
"... The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A Dhomothetic transformation is determined as a special gauge transformation. The ηEinstein manifold are defined, it is prove that their scalar curvatu ..."
Abstract

Cited by 34 (4 self)
 Add to MetaCart
The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A Dhomothetic transformation is determined as a special gauge transformation. The ηEinstein manifold are defined, it is prove that their scalar curvature is a constant and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with a Dhomothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skewsymmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skewsymmetric and the defining vector field is Killing.