## The Srni lectures on non-integrable geometries with torsion

##
*Ilka Agricola*

**Address.**

Institut fuer Mathematik, Humboldt-Universitaet zu Berlin, Unter den Linden 6,

Sitz: John-von-Neumann-Haus, Adlershof, D-10099 Berlin, Germany

**E-mail. **

agricola@mathematik.hu-berlin.de

**Abstract.**

This review article intends to introduce the reader to non-integrable
geometric structures on Riemannian manifolds and
invariant metric connections with torsion, and to discuss
recent aspects of mathematical
physics---in particular superstring theory---where these naturally appear.

Connections with skew-symmetric torsion are exhibited as one of the main
tools to understand non-integrable geometries. To this aim a
a series of key examples is presented and successively dealt with
using the notions of intrinsic
torsion and characteristic connection of a $G$-structure as unifying
principles.
The General Holonomy Principle bridges over to parallel objects, thus
motivating the discussion of geometric stabilizers, with emphasis on
spinors and differential forms. Several Weitzenb\"ock formulas for
Dirac operators associated with torsion connections enable us
to discuss spinorial field equations, such as those governing the common
sector of type II superstring theory. They also provide the link to
Kostant's cubic Dirac operator.

**AMSclassification. **Primary 53-02 (C, D); Secondary 53C25-30, 53D5, 81T30.

**Keywords. **Metric connection with torsion; intrinsic torsion; $G$-structure;
characteristic connection; superstring theory; Strominger model; parallel
spinor; non-integrable geometry; integrable geometry; Berger's holonomy
theorem; naturally reductive space; hyper-K\"ahler manifold with torsion;
almost metric contact structure; $G_2$-manifold; $\Spin(7)$-manifold;
$\SO(3)$-structure; $3$-Sasakian manifold.