In this master thesis we study calibrated geometries, a family of Riemannian or Hermitian manifolds with an associated differential form, φ. We show that it isuseful to introduce the concept of proper calibrated manifolds, which are in asense calibrated manifolds where the geometry is derived from the calibration. In particular, the φ-Grassmannian is considered in the case of proper calibratedmanifolds. The impact of proper calibrated manifolds as a model is studied, aswell as the usefulness of pluripotential theory as tools for the model. The specialLagrangian calibration is an example of an important calibration introduced byHarvey and Lawson, which leads to the definition of the special Lagrangian differentialequation. This partial differential equation can be formulated in threeand four dimensions as det(H(u)) = Δu, where H(u) is the Hessian matrix of some potential u. We prove the existence of solutions and some other propertiesof this nonlinear differential equation and present the resulting 6- and 8-dimensional manifolds defined by the graph {x + i<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cnabla" />u(x)}. We also considerthe physical applications of calibrated geometry, which have so far largely beenrestricted to string theory. However, we consider the manifold (M,g,F), whichis calibrated by the scaled Maxwell 2-form. Some geometrical properties of relativisticand classical electrodynamics are translated into calibrated geometry.

We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. We also relate various real forms of the Toda equations to minimal surfaces in quadrics of arbitrary signature. In the case of the Hitchin component for PSL(3,R) we provide a new proof of the relation to convex RP²-structures and hyperbolic affine spheres. For PSp(4,R) we prove such representations are the monodromy for a special class of projective structure on the unit tangent bundle of the surface. We prove these are isomorphic to the convex-foliated projective structures of Guichard and Wienhard. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, work which traces back to Cartan. Nurowski showed that there is an associated signature (2,3) conformal structure. We clarify this as a relationship between a parabolic geometry associated to the split real form of G₂ and a conformal geometry with holonomy in G₂. Moreover in terms of the conformal geometry we prove this distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation. The moduli space of deformations of a compact coassociative submanifold L in a G₂ manifold is shown to have a natural local embedding as a submanifold of H2(L,R). We consider G2-manifolds with a T^4-action of isomorphisms such that the orbits are coassociative tori and prove a local equivalence to minimal 3-manifolds in R^{3,3} = H²(T⁴,R) with positive induced metric. By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R^{3,3} and hence G₂-metrics from equations that are a set of affine Toda equations. The relation to semi-flat special Lagrangian fibrations and the Monge-Ampere equation is explained.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18444号 / 理博第4004号 / 新制||理||1577(附属図書館) / 31322 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 堤 誉志雄, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM