Riemannian holonomy groups is an area of Riemannian geometry, in the field of
Differential Geometry. The *holonomy group* Hol(*g*) of a Riemannian
manifold (*M*,*g*) determines the geometrical structures on* M*
compatible with *g*. Thus, Berger's classification of Riemannian holonomy
groups gives a list of interesting geometrical structures compatible with a
Riemannian metric. Most of the holonomy groups on Berger's list are
important in String Theory.

Given some class of mathematical objects, there is often a natural class of
subobjects living inside them, such as groups and subgroups for instance. The
natural subobjects of Riemannian manifolds (*M*,*g*) with special
holonomy are calibrated *submanifolds *--- lower-dimensional,
volume-minimizing submanifolds *N* in *M *compatible with the
geometric structures coming from the holonomy reduction. Calibrated submanifolds
are also important in String

Theory, as 'supersymmetric cycles' or 'branes'.

This lecture course is aimed at graduate students in Geometry, Mathematical
Physics, or String Theory, and should be accessible to new graduates. A major
focus of the course will be Calabi-Yau manifolds. It should be a good
preparation for Dr de la Ossa's graduate lecture course on Calabi-Yau Manifolds
and Mirror Symmetry in Hilary Term 07. The main prerequisites are knowledge of
Differentiable Manifolds and Lie Groups.

Synopsis

*Lectures 1-2: Background material.* Smooth manifolds. Tensors and exterior
forms. The exterior derivative and de Rham cohomology. Riemannian metrics.
Connections and curvature. Lie groups.

*Lectures 3-4: Riemannian holonomy groups.* The holonomy group of a
Riemannian metric. Relation with constant tensors. Berger's classification,
sketch of proof. Discussion of each case.

*Lectures 5-7: Complex and Kähler
geometry.* Complex manifolds and holomorphic functions. Complex submanifolds
of CP^{n} and complex
algebraic geometry. Holomorphic line bundles and sections. Kähler
metrics on complex manifolds. De Rham cohomology of Kähler
manifolds, the Kähler class.

*Lectures 8-10: The Calabi Conjecture and Calabi-Yau manifolds.* The
Conjecture and its proof. Ricci-flat Kähler
manifolds and Calabi-Yau manifolds. Finding examples of Calabi-Yau manifolds
using algebraic geometry. String Theory, and Mirror Symmetry of Calabi-Yau
3-folds.

*Lectures 11-12: The holonomy groups* Sp(*m*), Sp(*m*)Sp(1), *G*_{2}
*and* Spin(7). Brief introduction to the geometry of each holonomy group;
compact manifolds with these holonomy groups.

*Lectures 13-14: Introduction to calibrated geometry.* Submanifolds.
Minimal submanifolds of Riemannian submanifolds. Calibrations and calibrated
submanifolds. Natural calibrations on manifolds with special holonomy. Geometric
Measure Theory and calibrated currents.

*Lectures 15-16: Special Lagrangian submanifolds of Calabi-Yau manifolds.*
Symplectic manifolds and Lagrangian submanifolds. McLean's Theorem on
deformations of compact special Lagrangians. The SYZ Conjecture. Singularities
of special Lagrangians.