Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIntrinsic torsion in quaternionic contact geometry
187
0
0.0
(
0
)
Authors
Diego Conti
Ask ChatGPT about the research
No Arabic abstract
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in seven dimensions, when this condition is equivalent to integrability in the sense of Duchemin. We prove that the choice of a reduction to Sp(n)H^* (or equivalently, a complement of the qc distribution) yields a unique K-connection satisfying natural conditions on torsion and curvature. We show that the choice of a compatible metric on the qc distribution determines a canonical reduction to Sp(n)Sp(1) and a canonical Sp(n)Sp(1)-connection whose curvature is almost entirely determined by its torsion. We show that its Ricci tensor, as well as the Ricci tensor of the Biquard connection, has an interpretation in terms of intrinsic torsion.
rate research
Read More
We discuss the construction of Sp(2)Sp(1)-structures whose fundamental form is closed. In particular, we find 10 new examples of 8-dimensional nilmanifolds that admit an invariant closed 4-form with stabiliser Sp(2)Sp(1). Our constructions entail the notion of SO(4)-structures on 7-manifolds. We present a thorough investigation of the intrinsic torsion of such structures, leading to the construction of explicit Lie group examples with invariant intrinsic torsion.
We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.
We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.
We introduce and study a notion of `Sasaki with torsion structure (ST) as an odd-dimensional analogue of Kahler 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 generalizations of Sasaki geometry, and 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.
We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold is a Calabi-Eckmann manifold. Moreover we show that a complete, simply connected, normal metric contact pair manifold such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, then the manifold is the product of globally $phi$-symmetric spaces and fibers over a locally symmetric space endowed with a symplectic pair.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
