# nLab associative submanifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Definition

A 3-dimensional submanifold $\Sigma_3 \hookrightarrow X_7$ of a G2-manifold $X_7$ is called associative if it is a calibrated submanifold, hence if the canonical 3-form $\phi$ of the $G_2$-manifold (tangent space-wise the associative 3-form) restricts to the volume form on $\Sigma_3$.

Accordingly a 4-dimensional submanifold $\Sigma_4 \hookrightarrow X_7$ is called coassociative if it is a calibrated submanifold with respect to the coassociative 4-form $\star_X \phi$.

## References

• Ian Weiner, Associative 3-manifolds in $\mathbb{R}^7$, 2001 (pdf)

• Selman Akbulut, Sema Salur, Associative submanifolds of a $G_2$-manifold (pdf)

• Damien Gayet, Smooth moduli spaces of associative submanifolds (arXiv:1011.1744)

Discussion of wrapped branes on associative 3-cycles is in

Further discussion specifically of M5-branes wrapped on associative 3-cycle which are either the 3-sphere or a hyperbolic 3-manifold is in

Last revised on April 3, 2019 at 09:21:13. See the history of this page for a list of all contributions to it.