A note on closed G2 -structures and 3-manifolds
A note on closed G2 -structures and 3-manifolds
This article shows that given any orientable 3 -manifold X , the 7 -manifold T ∗X × R admits a closed G2 - structure φ = Re Ω − ω ∧ dt where Ω is a certain complex-valued 3 -form on T ∗X ; next, given any 2 -dimensional submanifold S of X , the conormal bundle N ∗S of S is a 3 -dimensional submanifold of T ∗X ×R such that φ|N∗S ≡ 0. A corollary of the proof of this result is that N ∗S×R is a 4 -dimensional submanifold of T ∗X×R such that φ|N∗S×R ≡ 0.
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- [1] Akbulut S, Salur S. Calibrated manifolds and gauge theory. J Reine Angew Math 2008; 625: 187214.
- [2] Akbulut S, Salur S. Mirror duality via G2 and Spin(7) manifolds. Prog Math 2010; 279: 121.
- [3] Arikan M, Cho H, Salur S. Existence of compatible contact structures on G2 -manifolds. Asian J Math 2013; 17: 321334.
- [4] Brown R, Gray A. Vector cross products. Comment Math Helv 1967; 42: 222236.
- [5] Bryant R. Metrics with exceptional holonomy. Ann Math 1987; 126: 526576.
- [6] Bryant R. Some remarks on G2 -structures. In: Proceedings of the G¨okova Geometry-Topology Conference. Somerville, MA, USA: International Press, 2005, pp. 75109.
- [7] Cabrera F, Monar M, Swann A. Classification of G2 -structures. J London Math Soc 1996; 53: 407416.
- [8] Cleyton R, Ivanov S. On the geometry of closed G2 -structures. Commun Math Phys 2007; 270: 5367.
- [9] da Silva A. Lectures on Symplectic Geometry. Lecture Notes in Mathematics. Berlin, Germany: Springer, 2001.
- [10] Fernandez M. An example of a compact calibrated manifold associated with the exceptional Lie group G2 . J Differ Geom 1987; 26: 367370.
- [11] Fernandez M. A family of compact solvable G2 -calibrated manifolds. Tohoku Math J 1987; 39: 287289.
- [12] Fernandez M, Gray A. Riemannian manifolds with structure group G2 . Ann Mat Pur Appl 1982; 132: 1945.
- [13] Fernandez M, Iglesias, T. New examples of Riemannian manifolds with structure group G2 . Rend Circ Mat Palermo 1986; 35: 276290.
- [14] Gray A. Vector cross products on manifolds. T Am Math Soc 1969; 141: 465504.
- [15] Harvey R, Lawson HB. Calibrated geometries. Acta Math 1982; 148: 48157.
- [16] Joyce D. Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford, UK: Oxford University Press, 2000.
- [17] Lawson HB, Michelsohn ML. Spin Geometry. Princeton, NJ, USA: Princeton University Press, 1989.
- [18] McDuff D, Salamon D. Introduction to Symplectic Topology. Oxford, UK, USA: Oxford University Press, 1998.
- [19] Milnor J. Spin structures on manifolds. Enseign Math 1963; 9: 198203.