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Deformations of G 2 and Spin(7) Structures

  • Spiro Karigiannis (a1)

Abstract

We consider some deformations of ${{G}_{2}}$ -structures on 7-manifolds. We discover a canonical way to deform a ${{G}_{2}}$ -structure by a vector field in which the associated metric gets “twisted” in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy ${{G}_{2}}$ . Similarly we consider analogous constructions for Spin(7)-structures on 8-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the Spin(7) case.

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References

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[1] Bonan, E., Sur des Variétés riemanniennes à groupe d’holonomie G2 ou Spin(7). C. R. Acad. Sci. Paris 262(1966), 127129.
[2] Brown, R. B. and Gray, A., Vector Cross Products. Comment.Math. Helv. 42(1967), 222236.
[3] Bryant, R. L., Metrics with Exceptional Holonomy. Ann. of Math. 126(1987), 525576.
[4] Bryant, R. L. and Salamon, S. M., On the Construction of Some Complete Metrics with Exceptional Holonomy. Duke Math. J. 58(1989), 829850.
[5] Cabrera, F. M., On Riemannian Manifolds with G2-Structure. Boll. Un.Mat. Ital. A (7) 10(1996), 99112.
[6] Cabrera, F. M., Monar, M. D. and Swann, A. F., Classification of G2-Structures. J. London Math. Soc (2) 53(1996), 407416.
[7] Fernández, M., A Classification of Riemannian Manifolds with Structure Group Spin(7). Ann.Mat. Pura Appl. (IV) 143(1986), 101122.
[8] Fernández, M., An Example of a Compact Calibrated Manifold Associated with the Exceptional Lie Group G2. J. Differential Geom. 26(1987), 367370.
[9] Fernández, M. and Gray, A., Riemannian Manifolds with Structure Group G2 . Ann. Mat. Pura Appl (4) 32(1982), 1945.
[10] Fernández, M. and Ugarte, L., Dolbeault Cohomology for G2-Manifolds. Geom. Dedicata 70(1998), 5786.
[11] Fernández, M., A Differential Complex for Locally Conformal Calibrated G2-Manifolds. Illinois J. Math. 44(2000), 363391.
[12] Gray, A., Vector Cross Products on Manifolds. Trans. Amer.Math. Soc. 141(1969), 463504; Errata to “Vector Cross Products onManifolds”. 148(1970), 625.
[13] Gray, A., Weak Holonomy Groups. Math. Z. 123(1971), 290300.
[14] Gray, A. and Hervella, L. M., The Sixteen Classes of Almost Hermitian Manifolds and Their Linear Invariants. Ann.Mat. Pura Appl. (4) 123(1980), 3558.
[15] Gukov, S., Yau, S. T. and Zaslow, E., Duality and Fibrations on G2 Manifolds. Turkish J. Math. 27(2003), 6197.
[16] Harvey, R., Spinors and Calibrations. Academic Press, San Diego, 1990.
[17] Hitchin, N., The Geometry of Three-Forms in Six and Seven Dimensions. J. Differential Geom. 55(2000), 547576.
[18] Hitchin, N., Stable Forms and Special Metrics. Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Contemp.Math. 288(2001).
[19] Joyce, D. D., Compact Riemannian 7-Manifolds with Holonomy G2. I. J. Differential Geom. 43(1996), 291328.
[20] Joyce, D. D., Compact Riemannian 7-Manifolds with Holonomy G2. II. J. Differential Geom. 43(1996), 329375.
[21] Joyce, D. D., Compact 8-Manifolds with Holonomy Spin(7). Invent.Math. 123(1996), 507552.
[22] Joyce, D. D., Compact Manifolds with Special Holonomy. Oxford University Press, Oxford, 2000.
[23] Karigiannis, S., Deformations of G2 and Spin(7)-structures on Manifolds. long version, http://arxiv.org/archive/math.DG/0301218.
[24] Karigiannis, S., On a system of PDE's related to G2 holonomy. in preparation.
[25] Salamon, S. M., Riemannian Geometry and Holonomy Groups. Longman Group UK Limited, Harlow, 1989.
[26] Yau, S. T., On the Ricci Curvature of a Compact Kähler Manifold and the Complex Monge–Ampère Equation I. Comm. Pure Appl. Math. 31(1978), 339411.
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Deformations of G 2 and Spin(7) Structures

  • Spiro Karigiannis (a1)

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