| Summary: | <p>In this thesis, we discuss some aspects of calibrated geometry in manifolds of exceptional holonomy. Manifolds of exceptional holonomy are Riemannian manifolds that are endowed with one of the following additional structures: a torsion-free G2-structure or a torsion-free Spin(7)-structure. G2 manifolds admit two special families of calibrated, hence volume minimizing, submanifolds: associative 3-folds and coassociative 4-folds. Spin(7) manifolds admit only one family of calibrated submanifolds: Cayley 4-folds. Understanding the geometry of such calibrated submanifolds is one of the key challenges in the study ofmanifolds with exceptional holonomy. </p>
<p>After recalling some basic notion on calibrated geometry and manifolds of exceptional holonomy, we define calibrated fibrations, and we prove a rigidity result for these objects under some linear condition. </p>
<p>Then, we describe the construction of two Cayley fibrations in the Bryant–Salamon Spin(7) manifold using a cohomogeneity one method. These are the first explicit examples of Cayley fibrations in a non-flat Spin(7) manifold and the fibres provide new examples of Cayley submanifolds. </p>
<p>Finally, we study the geometry of calibrated submanifolds in G2 manifolds that admit T2 × SU(2)-symmetry. We apply our results to C3 × S1 , to the Bryant–Salamon manifolds and to the manifolds recently constructed by Foscolo–Haskins–Nordström, where our analysis gives new large families of T2-invariant associatives. This is based on joint work with B. Aslan.</p>
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