Definition 2.1 [03KE]

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Definition 2.1 Let $(M,g)$ be a Riemannian manifold. An oriented tangent $k$ -plane $V$ on $M$ is a vector subspace $V$ of some tangent space $T_{x}M$ to $M$ with $\mathop{\rm dim}V=k$ , equipped with an orientation. If $V$ is an oriented tangent $k$ -plane on $M$ then $g|_{V}$ is a Euclidean metric on $V$ , so combining $g|_{V}$ with the orientation on $V$ gives a natural volume form $\mathop{\rm vol}_{V}$ on $V$ , which is a $k$ -form on $V$ .

Now let $\varphi$ be a closed $k$ -form on $M$ . We say that $\varphi$ is a calibration on $M$ if for every oriented $k$ -plane $V$ on $M$ we have $\varphi|_{V}\leqslant\mathop{\rm vol}_{V}$ . Here $\varphi|_{V}=\alpha\cdot\mathop{\rm vol}_{V}$ for some $\alpha\in\mathbin{\mathbb{R}}$ , and $\varphi|_{V}\leqslant\mathop{\rm vol}_{V}$ if $\alpha\leqslant 1$ . Let $N$ be an oriented submanifold of $M$ with dimension $k$ . Then each tangent space $T_{x}N$ for $x\in N$ is an oriented tangent $k$ -plane. We say that $N$ is a calibrated submanifold if $\varphi|_{T_{x}N}=\mathop{\rm vol}_{T_{x}N}$ for all $x\in N$ .