Extrinsic geometry of calibrated submanifolds (Q6546574)

Let \((M, g)\) be a Riemannian \(n\)-manifold equipped with a calibrating \(k\)-form \(\alpha\), i.e., a closed \(k\)-form such that \(\alpha_x\) is a calibration for \((T_x M, g_x)\) for any \(x \in M\). We recall that a \(k\)-form \(\beta \in \Lambda^k \mathbb R^n\) is said to be a calibration if for any \(k\)-plane \(W \subset \mathbb R^n\), the restricted \(k\)-form \(\beta|_W \) satisfies the condition \(\beta|_W = \lambda_W {\cdot} \text{vol}_W\) for some real number \(\lambda_W \leq 1\), and that, if this is the case, a \(k\)-plane is said to be \(\beta\)-calibrated if \(\beta|_W = \text{vol}_W\). \N\NIn this paper, the authors focus on the calibrating \(k\)-forms \(\alpha\) that satisfy the following three conditons at each \(x \in M\): (1) \(\alpha\) is parallel with respect to the Levi-Civita connection, i.e., \(\nabla \alpha|_x = 0\); (2) there is a subgroup \(G_x \subset \operatorname{SO}(T_x M, g_x)\) of the orthogonal transformations that leaves \(\alpha_x \in \Lambda^k T_x M \) invariant and acts transitively on the set of the \(\alpha_x\)-calibrated \(k\)-planes; (3) the \(k\)-form \(\alpha_x\) is \(G_x\)-compliant, a technical condition on \(\mathfrak g_x = \operatorname{Lie}(G_x)\) and on the subalgebra \(\mathfrak h_x \subset \mathfrak g\) of the transformations fixing one of (hence, up to conjugation, all of) the \(\alpha_x\)-calibrated \(k\)-planes \(W\). Note that, by (1), all Lie algebras \(\mathfrak g_x\) (resp., \(\mathfrak h_x\)) can be assumed to be isomorphic to each other and hence all of them to be isomorphic to some fixed Lie algebra \(\mathfrak g\) (resp., \(\mathfrak h\)). \N\NThe main results of the paper concern the \(k\)-dimensional submanifolds \(L \subset M\) that are \(\alpha\)-calibrated, that is, such that each tangent space \(T_y L \subset T_y M\), \(y \in L\), is \(\alpha_y\)-calibrated. The authors prove that if \(\alpha\) satisfies the above conditions and if \(L\) is \(\alpha\)-calibrated, then (a) the infinitesimal holonomy algebra of the induced connection on the bundle \(TL \oplus_L (T L)^\perp \to L\) is a subalgebra of \(\mathfrak h\), and (b) at each point \(x\in M\) the second fundamental form \(A_x\) of \(L\) is determined by an appropriate corresponding element \(B_x\) of the Lie algebra \(\mathfrak g\). The results are illustrated by diverse non-trivial applications to Kähler, special Lagrangian, associative, coassociative and Cayley submanifolds.