scholarly journals Symplectic Foliations and Generalized Complex Structures

2014 ◽  
Vol 66 (1) ◽  
pp. 31-56 ◽  
Author(s):  
Michael Bailey
Keyword(s):  
Symplectic Form ◽  
Fibre Bundle ◽  
Complex Structure ◽  
Sufficient Conditions ◽  
Affirmative Answer ◽  
Generalized Complex Structures ◽  
Generalized Complex Structure ◽  
Generalized Complex ◽  
Necessary And Sufficient ◽  
Obstruction Class

AbstractWe answer the natural question: when is a transversely holomorphic symplectic foliation induced by a generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and only if our question has an affirmative answer. We first study a component of this obstruction, which gives the condition that the leafwise cohomology class of the symplectic form must be transversely pluriharmonic. As a consequence, under certain topological hypotheses, we infer that we actually have a symplectic fibre bundle over a complex base. We then show how to compute the full obstruction via a spectral sequence. We give various concrete necessary and sufficient conditions for the vanishing of the obstruction. Throughout, we give examples to test the sharpness of these conditions, including a symplectic fibre bundle over a complex base that does not come from a generalized complex structure, and a regular generalized complex structure that is very unlike a symplectic fibre bundle, i.e., for which nearby leaves are not symplectomorphic.

scholarly journals VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

2018 ◽  
Vol 61 (3) ◽  
pp. 588-607 ◽  
Author(s):  
Honglei Lang ◽  
Yunhe Sheng ◽  
Aïssa Wade
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Contact Structures ◽  
Complex Structures ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Generalized Complex Structures ◽  
Generalized Complex Structure ◽  
Generalized Geometry ◽  
Generalized Complex

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

scholarly journals Generalized almost even-Clifford manifolds and their twistor spaces

2021 ◽  
Vol 8 (1) ◽  
pp. 96-124
Author(s):  
Luis Fernando Hernández-Moguel ◽  
Rafael Herrera
Keyword(s):  
Twistor Space ◽  
Complex Structure ◽  
Twistor Spaces ◽  
Recent Interest ◽  
Generalized Complex Structure ◽  
Generalized Complex ◽  
Space Construction ◽  
Clifford Structures

Abstract Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.

On the Relation of Real and Complex Lie Supergroups

2015 ◽  
Vol 58 (2) ◽  
pp. 281-284 ◽  
Author(s):  
Matthias Kalus
Keyword(s):  
Complex Structure ◽  
Sufficient Conditions ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
New Approach ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Lie Supergroups ◽  
Necessary And Sufficient

AbstractA complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complexHarish-Chandra superpairs. A universal complexi ûcation of a real Lie supergroup is constructed

Remarks about the Kaluza-Klein metric on tangent bundle

2019 ◽  
Vol 16 (03) ◽  
pp. 1950040
Author(s):  
Murat Altunbas ◽  
Lokman Bilen ◽  
Aydin Gezer
Keyword(s):  
Tangent Bundle ◽  
Complex Structure ◽  
Sufficient Conditions ◽  
Complex Structures ◽  
Almost Complex ◽  
Compatible Almost Complex Structure ◽  
Almost Kähler ◽  
Necessary And Sufficient ◽  
Curvature Tensors ◽  
Kaluza Klein

The paper is concerned with the Kaluza–Klein metric on the tangent bundle over a Riemannian manifold. All kinds of Riemann curvature tensors are computed and some curvature properties are given. The compatible almost complex structure is defined on the tangent bundle, and necessary and sufficient conditions for such a structure to be integrable are described. Then, the condition is given under which the tangent bundle with these structures is almost Kähler. Finally, almost golden complex structures are defined on this setting and some results related to them are presented.

scholarly journals Relations between Non-Compact Transformation Groups and Compact Transformation Groups

1971 ◽  
Vol 44 ◽  
pp. 97-117
Author(s):  
Hsin Chu
Keyword(s):  
Fundamental Group ◽  
Transformation Group ◽  
Fibre Bundle ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Transformation Groups ◽  
Almost Periodic ◽  
Minimal Set ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

In this paper certain relations between non-compact transformation groups and compact transformation groups are studied. The notion of re-ducibility and separability of transformation groups is introduced, several necessary and sufficient conditions are established: (1) A separable transformation group to be locally weakly almost periodic, (2) A reducible and separable transformation group to be a minimal set and (3) A reducible and separable transformation group to be a fibre bundle. As applications we show, among other things, that (1) for certain reducible transformation groups its fundamental group is not trivial which is a generalization of a result in [4].

Quasilinear Hamiltonian systems

1987 ◽  
Vol 106 (3-4) ◽  
pp. 195-204
Author(s):  
Jan S. Rogulski
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Scalar Product ◽  
Sufficient Conditions ◽  
Hamiltonian Form ◽  
Quasilinear Systems ◽  
Necessary And Sufficient ◽  
Evolution Systems ◽  
General Method

SynopsisWe consider quasilinear systems of 2N partial differential equations with 2N unknown functions depending on n + 1 variables as evolution systems on the space L2(Rn, RN) × L2(Rns, RN) endowed with a symplectic form induced by the standard scalar product on L2(Rn, RN). The necessary and sufficient conditions for such a system to be a Hamiltonian system are derived. The main purpose of this paper is to propose a straightforward link between the symplectic approach formulated by Chernoff, Hughes and Marsden and the multisymplectic formulations of evolution systems created by Kijowski and developed by Gawedzki and Kondracki. A general method of constructing the multisymplectic form and the Hamiltonian form for these systems is given.

scholarly journals Abelian Complex Structures and Generalizations

2021 ◽  
Vol 8 (1) ◽  
pp. 247-266
Author(s):  
Yat Sun Poon
Keyword(s):  
Deformation Theory ◽  
Complex Structure ◽  
Complex Structures ◽  
Generalized Complex Structure ◽  
Generalized Complex

Abstract After a review on the development of deformation theory of abelian complex structures from both the classical and generalized sense, we propose the concept of semi-abelian generalized complex structure. We present some observations on such structure and illustrate this new concept with a variety of examples.

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