The Srni lectures on non-integrable geometries with torsion

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 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.

61 pages

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• Superstrings with Intrinsic Torsion
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• Geometric Model for Complex Non-Kaehler Manifolds with SU(3) Structure
• Geometry of Hyper-Kähler Connections with Torsion
• The intrinsic torsion of SU(3) and G_2 structures
• (Weak) G_2 Holonomy from Self-duality, Flux and Supersymmetry
• A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues
• Generalised $G_2$-manifolds
• Supersymmetric Domain Walls from Metrics of Special Holonomy
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• The Weyl Character Formula, the half-spin representations, and equal rank subgroups
• Cohomogeneity-one G2-structures
• On the geometry of closed G2-structure
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• Almost Hermitian Geometry on Six Dimensional Nilmanifolds
• Almost Hermitian 6-Manifolds Revisited
• M-theory on manifolds of G2 holonomy: the first twenty years
• Special almost Hermitian geometry
• Almost Hermitian Structures with Parallel Torsion
• M-Theory on Seven Manifolds with G-fluxes
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• Distinguished dimensions for special Riemannian geometries

Cited by in corpus (49)

• Hidden Symmetries of Dynamics in Classical and Quantum Physics
• The Ricci tensor of SU(3)-manifolds
• Instantons and Killing spinors
• Generalized hidden symmetries and the Kerr-Sen black hole
• 6-dimensional nearly Kaehler manifolds of cohomogeneity one
• 3-Sasakian manifolds in dimension seven, their spinors and G_2 structures
• G2-structures and octonion bundles
• Geometric algebra techniques in flux compactifications (II)
• A survey on geometry of warped product submanifolds
• Foliated eight-manifolds for M-theory compactification
• Generalizations of 3-Sasakian manifolds and skew torsion
• Distinguished connections on $(J^2=\pm 1)$-metric manifolds
• Geometric algebra techniques in flux compactifications
• Dirac-harmonic maps with torsion
• On the G_2 bundle of a Riemannian 4-manifold
• Hidden Symmetries for Ellipsoid-Solitonic Deformations of Kerr-Sen Black Holes and Quantum Anomalies
• Spin structures on compact homogeneous pseudo-Riemannian manifolds
• Homotheties and topology of tangent sphere bundles
• A Deformation of Sasakian Structure in the Presence of Torsion and Supergravity Solutions
• Adapted connections on metric contact manifolds
• Weighted metrics on tangent sphere bundles
• Geodesic lines in Nearly Kähler S^3\timesS^3
• Variations of gwistor space
• Special Riemannian geometries and the Magic Square of Lie algebras
• Six-dimensional nearly Kaehler manifolds of cohomogeneity one (II)
• The moduli space of Type~A surfaces with torsion and non-singular symmetric Ricci tensor
• A Berger-type theorem for metric connections with skew-symmetric torsion
• On the characteristic connection of gwistor space
• Spinors of real type as polyforms and the generalized Killing equation
• Harmonic maps with torsion
• On the rescaled Riemannian metric of Cheeger Gromoll type on the cotangent bundle
• A note on generalized Dirac eigenvalues for split holonomy and torsion
• Twistorial eigenvalue estimates for generalized Dirac operators with torsion
• Spin(9) and almost complex structures on 16-dimensional manifolds
• A note on the uniqueness of the canonical connection of a naturally reductive space
• $Spin(7)$ Compactifications and 1/4-BPS Vacua in Heterotic Supergravity
• Self-duality and associated parallel or cocalibrated ${\mathrm{G}}_2$ structures
• Symmetric brackets induced by connections with totally skew-symmetric torsion on skew-symmetric algebroids
• Abelian Hermitian geometry
• Cones of G manifolds and Killing spinors with skew torsion
• Warped-like product manifolds with exceptional holonomy groups
• SO(3)-Structures on 8-manifolds
• An integral formula for $G_2$-structures
• Harmonic $SU(3)$- and $G_2$-structures via spinors
• Isometries and curvatures of tangent sphere bundles
• An optimal inequality on warped product semi-slant submanifolds of nearly Kaehler manifolds
• An integral formula for Riemannian $G$-structures with applications to almost Hermitian and almost contact structures
• Einstein manifolds with skew torsion
• Hidden symmetries and Killing spinor of 5-dimensional minimal gauged supergravity solutions