## Illinois Journal of Mathematics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 19 October 2009

https://projecteuclid.org/euclid.ijm/1255984846

Digital Object Identifier
doi:10.1215/ijm/1255984846

Mathematical Reviews number (MathSciNet)
MR1775327

Zentralblatt MATH identifier
0949.53038

#### Citation

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. https://projecteuclid.org/euclid.ijm/1255984846