Deformations in \(G_{2}\) manifolds. (English) Zbl 1143.53047
The authors study deformations of associative submanifolds inside a \(G_2\) manifold \(M^7\) with a calibration 3-form \(\varphi\). In the present paper all cited results, as well as the new ones about \(G_2\) manifolds are proved by using only the cross product operation (equivalently \(\varphi\)). The approach in their paper makes things easier and keeps the presentation elementary.
Reviewer: R. Iordanescu (Bucureşti)
References:
| [1] | Akbulut, S., Lectures on Seiberg-Witten invariants, Turkish J. Math., 20, 1329-1355 (1996) · Zbl 0871.57015 |
| [2] | S. Akbulut, S. Salur, Calibrated manifolds and gauge theory, math.GT/0402368; S. Akbulut, S. Salur, Calibrated manifolds and gauge theory, math.GT/0402368 · Zbl 1157.53027 |
| [3] | S. Akbulut, S. Salur, Mirror duality via \(G_2\mathit{Spin}(7)\) math.DG/0605277; S. Akbulut, S. Salur, Mirror duality via \(G_2\mathit{Spin}(7)\) math.DG/0605277 · Zbl 1211.53098 |
| [4] | Bryant, R. L., Metrics with exceptional holonomy, Ann. of Math., 126, 525-576 (1987) · Zbl 0637.53042 |
| [5] | R.L. Bryant, Some remarks on \(G_2\) math.DG/0305124, v3; R.L. Bryant, Some remarks on \(G_2\) math.DG/0305124, v3 · Zbl 1115.53018 |
| [6] | Harvey, F. R., Spinors and Calibrations (1990), Academic Press Inc.: Academic Press Inc. Boston, MA · Zbl 0694.53002 |
| [7] | Harvey, F. R.; Lawson, H. B., Calibrated geometries, Acta. Math., 148, 47-157 (1982) · Zbl 0584.53021 |
| [8] | Karigiannis, S., Deformations of \(G_2\) and \(Spin(7)\) structures on manifolds, Canad. J. Math., 57, 1012-1055 (2005) · Zbl 1091.53026 |
| [9] | McLean, R. C., Deformations of calibrated submanifolds, Comm. Anal. Geom., 6, 705-747 (1998) · Zbl 0929.53027 |
| [10] | Thomas, E., Postnikov invariants and higher order cohomology operations, Ann. of Math., 85, 184-217 (1967) · Zbl 0152.22002 |
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