## The Srní lectures on non-integrable geometries with torsion.(English)Zbl 1164.53300

Summary: 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 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 Weitzenböck 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.

### MSC:

 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53D15 Almost contact and almost symplectic manifolds 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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