Bundle Constructions of Calibrated Submanifolds in $\mathbb R^7$ and $\mathbb R^8$

We construct calibrated submanifolds of $\mathbb R^7$ and $\mathbb R^8$ by viewing them as total spaces of vector bundles and taking appropriate sub-bundles which are naturally defined using certain surfaces in $\mathbb R^4$. We construct examples of associative and coassociative submanifolds of $\mathbb R^7$ and of Cayley submanifolds of $\mathbb R^8$. This construction is a generalization of the Harvey-Lawson bundle construction of special Lagrangian submanifolds of $\mathbb C^{n}$.

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Published 1 January 2005