Calibrated geometry

In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibration, meaning that:

• φ is closed, that is, dφ = 0, where d is the exterior derivative.
• φ has operator norm at most 1. That is, for any x ∈ M and any p-vector ${\displaystyle \xi \in \Lambda ^{p}T_{x}M}$, we have φ(ξ) ≤ vol(ξ), with volume defined with respect to the Riemannian metric g.

A main reason for defining a calibration is that it creates a distinguished set of "directions" (i.e. p-planes) in which φ is actually equal to the volume form, that is, the inequality above is an equality. For x in M, set G_x(φ) to be the subset of such planes in the Grassmannian of p-planes in T_xM. In cases of interest, G_x(φ) is always nonempty. Let G(φ) be the union of G_x(φ) for all ${\displaystyle x\in M}$, viewed as a subspace of the bundle of p-planes in TM.