We introduce and study a notion of 'Sasaki with torsion structure' (st) as an odd-dimensional analogue of Kähler with torsion geometry (KT). These are normal almost contact metric manifolds that admit a unique compatible connection with 3-form torsion. Any odd-dimensional compact Lie group is shown to admit such a structure; in this case, the structure is left-invariant and has closed torsion form. We illustrate the relation between st structures and other generalisations of Sasaki geometry, and we explain how some standard constructions in Sasaki geometry can be adapted to this setting. In particular, we relate the st structure to a KT structure on the space of leaves and show that both the cylinder and the cone over an st manifold are KT, although only the cylinder behaves well with respect to closedness of the torsion form. Finally, we introduce a notion of 'G-moment map'. We provide criteria based on equivariant cohomology ensuring the existence of these maps and then apply them as a tool for reducing st structures

Conti, D., Madsen, T. (2012). The odd side of torsion geometry. ANNALI DI MATEMATICA PURA ED APPLICATA, 193(4), 1041-1067 [10.1007/s10231-012-0314-6].

The odd side of torsion geometry

CONTI, DIEGO;
2012

Abstract

We introduce and study a notion of 'Sasaki with torsion structure' (st) as an odd-dimensional analogue of Kähler with torsion geometry (KT). These are normal almost contact metric manifolds that admit a unique compatible connection with 3-form torsion. Any odd-dimensional compact Lie group is shown to admit such a structure; in this case, the structure is left-invariant and has closed torsion form. We illustrate the relation between st structures and other generalisations of Sasaki geometry, and we explain how some standard constructions in Sasaki geometry can be adapted to this setting. In particular, we relate the st structure to a KT structure on the space of leaves and show that both the cylinder and the cone over an st manifold are KT, although only the cylinder behaves well with respect to closedness of the torsion form. Finally, we introduce a notion of 'G-moment map'. We provide criteria based on equivariant cohomology ensuring the existence of these maps and then apply them as a tool for reducing st structures
Articolo in rivista - Articolo scientifico
Connections with torsion, Almost contact metric, Sasakian, Moment map, Kähler with torsion
English
2012
193
4
1041
1067
reserved
Conti, D., Madsen, T. (2012). The odd side of torsion geometry. ANNALI DI MATEMATICA PURA ED APPLICATA, 193(4), 1041-1067 [10.1007/s10231-012-0314-6].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/47602
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