A $G_2$-structure on a manifold of dimension 7 is a choice of reduction of the structure group of the tangent bundle along the inclusion of G2 into $GL(7)$.
Given that $G_2$ is the subgroup of the general linear group on the Cartesian space $\mathbb{R}^7$ which preserves the associative 3-form on $\mathbb{R}^7$, a $G_2$ structre is a higher analog of an almost symplectic structure under lifting from symplectic geometry to 2-plectic geometry (Ibort).
A $G_2$-manifold is a manifold equipped with an “integrable” or “parallel” $G_2$-structure. This is equivalently a Riemannian manifold of dimension 7 with special holonomy group being the exceptional Lie group G2.
$G_2$-manifolds may be understood as 7-dimensional analogs of real 6-dimensional Calabi-Yau manifolds.
For $X$ a smooth manifold of dimension $7$ a $G_2$-structure on $X$ is a choice of differential 3-form $\omega \in \Omega^3(X)$ such that there is an atlas over which this 3-form locally identifies with the associative 3-form on the Cartesian space $\mathbb{R}^7$.
Equivalently, this is a choice of reduction of the structure group of the tangent bundle along the inclusion
A $G_2$-structure in particular implies an orthogonal structure, hence a Riemannian metric.
A manifold equipped with a $G_2$-structure $\omega$, def. 1, is called a $G_2$-manifold if $\omega$ is “parallel” or “integrable” in that
$d \omega = 0$
$d \star \omega = 0$
(where $d$ is the de Rham differential and $\star$ is the Hodge star operator of the canonical Riemannian metric of remark 1).
For instance (Joyce, p. 4).
The holonomy of the Levi-Civita connection on a $G_2$-manifold is contained in $G_2$.
There is a useful weakened notion of $G_2$-holonomy.
A 7-dimensional manifold is said to be of weak $G_2$-holonomy if it carries a 3-form $\omega$ with the relation of def. 2 generalized to
and hence
for $\lambda \in \mathbb{R}$. For $\lambda = 0$ this reduces to strict $G_2$-holonomy, by 2.
(See for instance (Bilal-Derendinger-Sfetsos).)
A 7-manifold admits a $G_2$-structure, def. 1, precisely if it admits a spin structure.
The canonical Riemannian metric $G_2$ manifold is Ricci flat. More generally a manifold of weak $G_2$-holonomy, def. 3, with weakness parameter $\lambda$ is an Einstein manifold with cosmological constant $\lambda$.
In string phenomenology models obtained from compactification of 11-dimensional supergravity/M-theory on $G_2$-manifolds (see for instance Duff) can have attractive phenomenological properties, see for instance the G2-MSSM.
classification of special holonomy manifolds by Berger's theorem:
Compact $G_2$-manifolds were first found in
Dominic Joyce, Compact Riemannian 7-manifolds with holonomy $G_2$, Journal of Differential Geometry vol 43, no 2 (Euclid)
Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press (2000)
Surveys include
Spiro Karigiannis, What is… a $G_2$-manifold (pdf)
Spiro Karigiannis, $G_2$-manifolds – Exceptional structures in geometry arising from exceptional algebra (pdf)
The relation to multisymplectic geometry/2-plectic geometry is mentioned explicitly in
(but beware of some mistakes in that article…)
For more see the references at exceptional geometry.
The following references discuss the role of $G_2$-manifolds in M-theory on G2-manifolds:
A survey of the corresponding string phenomenology for M-theory on G2-manifolds (see there for more) is in
Weak $G_2$-holonomy is discussed in