Illinois Journal of Mathematics

A differential complex for locally conformal calibrated $G_{2}$-manifolds

Marisa Fernández and Luis Ugarte

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We characterize $G_{2}$-manifolds that are locally conformally equivalent to a calibrated one as those $G_{2}$-manifolds $M$ for which the space of differential forms annihilated by the fundamental $3$-form of $M$ becomes a differential subcomplex of de Rham's complex. Special properties of the cohomology of this subcomplex are exhibited when the holonomy group of $M$ can be reduced to a subgroup of $G_{2}$. We also prove a theorem of Nomizu type for this cohomology which permits its computation for compact calibrated $G_{2}$-nilmanifolds.

Article information

Illinois J. Math., Volume 44, Issue 2 (2000), 363-390.

First available in Project Euclid: 19 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C38: Calibrations and calibrated geometries
Secondary: 53C10: $G$-structures 58J10: Differential complexes [See also 35Nxx]; elliptic complexes


Fernández, Marisa; Ugarte, Luis. A differential complex for locally conformal calibrated $G_{2}$-manifolds. Illinois J. Math. 44 (2000), no. 2, 363--390. doi:10.1215/ijm/1255984846.

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