This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connec- tions 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 moti- vating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenbock formulas for Dirac operators associ- ated 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.

/pdf/the-srni-lectures-on-non-integrable-geometries-with-torsion-2xae9dqhgo.pdf

The Srní lectures on non-integrable geometries with torsion

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.

The Srni lectures on non-integrable geometries with torsion

A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolds. All conformal transformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact structure with vanishing torsion of the Biquard connection are explicitly described. A `3-Hamiltonian form' of infinitesimal conformal automorphisms of quaternionic contact structures is presented.

Quaternionic Contact Einstein Structures and the Quaternionic Contact Yamabe Problem

Conformal quaternionic contact curvature and the local sphere theorem

A complete solution to the quaternionic contact Yamabe problem on the seven-dimen- sional sphere is given. Extremals for the Sobolev inequality on the seven-dimensional Heisenberg group are explicitly described and the best constant in theL 2 Folland-Stein embedding theorem is determined.

https://www.ems-ph.org/fulltext/10.4171/JEMS/222

Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem

We determine the best (optimal) constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group and the non-negative functions for which equality holds

/pdf/the-optimal-constant-in-the-l2-folland-stein-inequality-on-2xltb92r4q.pdf

The optimal constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group

A complete solution to the quaternionic contact Yamabe problem on the seven dimensional sphere is given. Extremals for the Sobolev inequality on the seven dimensional Hesenberg group are explicitly described and the best constant in the $L^2$ Folland-Stein embedding theorem is determined.

Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem

https://www.degruyter.com/view/j/crll.2004.2004.issue-567/crll.2004.012/crll.2004.012.pdf

Ivan Minchev

Papers

Quaternionic Contact Einstein Structures and the Quaternionic Contact Yamabe Problem

Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem

The optimal constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group

Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem

Quaternionic Kaehler and hyperKaehler manifolds with torsion and twistor spaces