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Conformally parallel Spin structures on solvmanifolds

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In this paper we review the Spin (7) geometry in relation to solvmanifolds. Starting from a 7 - dimensional nilpotent Lie group N endowed with an invariant G2 structure, we present an example of a homogeneous conformally parallel Spin (7) metric on an associated solvmanifold. It is thought that this paper could lead to very interesting and exciting areas of research and new results in the direction of (locally conformally) parallel Spin (7) structures. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 166 – 178 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1105-36 Conformally parallel Spin(7) structures on solvmanifolds ∗ ˘ Selman UGUZ Department of Mathematics, Faculty of Science and Letters, Harran University, Osmanbey Campus, S ¸ anlıurfa, Turkey Received: 31.05.2011 • Accepted: 21.04.2013 • Published Online: 09.12.2013 • Printed: 20.01.2014 Abstract: In this paper we review the Spin(7) geometry in relation to solvmanifolds. Starting from a 7 -dimensional nilpotent Lie group N endowed with an invariant G2 structure, we present an example of a homogeneous conformally parallel Spin(7) metric on an associated solvmanifold. It is thought that this paper could lead to very interesting and exciting areas of research and new results in the direction of (locally conformally) parallel Spin(7) structures. Key words: Holonomy, G2 and Spin(7) manifolds, conformally parallel structures, solvmanifolds 1. Introduction The concept of the holonomy group for a Riemannian manifold was first defined by Cartan in 1923 and is known to be an efficient tool in the study of Riemannian manifolds [1]. The list of possible holonomy groups of irreducible, simply-connected, nonsymmetric Riemannian manifolds was given by Berger in 1955 [7]. The refinement of Berger’s list (as corrected later by Alekseevskii [3] and Gray-Brown [27]) includes the groups SO(n) in n -dimensions, U (n), SU (n) in 2n-dimensions, Sp(n), Sp(n)Sp(1) in 4n -dimensions, and 2 special cases, G2 holonomy in 7-dimensions and Spin(7) holonomy in 8 -dimensions. Manifolds with holonomy SO(n) constitute the generic case, all others are denoted as manifolds with “special holonomy”, and the last 2 cases are described as manifolds with “exceptional holonomy”. The existence problem of manifolds with exceptional holonomy was first solved by Bryant [11], complete examples were given by Bryant and Salamon [13], and the first