This article shows that given any orientable 3 -manifold X , the 7 -manifold T * X × R admits a closed G2 -where Ω is a certain complex-valued 3 -form on T * X ; next, given any 2 -dimensional submanifold S of X , the conormal bundle N * S of S is a 3 -dimensional submanifold of T * X × R such that φ|N * S ≡ 0 .A corollary of the proof of this result is that N * S×R is a 4 -dimensional submanifold of T * X ×R such that φ| N * S×R ≡ 0 .
Abstract. In this article, we treat G 2 -geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G 2 -structure; in particular, we discuss existence and make a number of identifications of the spaces of Hamiltonian structures associated to the two multisymplectic structures associated to an integrable G 2 -structure. Along the way, we prove some results in multisymplectic geometry that are generalizations of results from symplectic geometry.
In this article, we study an almost contact metric structure on a G 2 -manifold constructed by Arikan, Cho and Salur in [2] via the classification of almost contact metric structures given by Chinea and Gonzalez [14]. In particular, we characterize when this almost contact metric structure is cosymplectic and narrow down the possible classes in which this almost contact metric structure could lie. Finally, we show that any closed G 2 -manifold admits an almost contact metric 3-structure by constructing it explicitly and characterize when this almost contact metric 3-structure is 3-cosymplectic.
We introduce G 2 -vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G 2 -structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces X G 2 of G 2 -vector fields and X Roc of Rochesterian vector fields are Lie subalgebras of the Lie algebra of vector fields with the standard Lie bracket. We also define, in analogy with the Poisson bracket on smooth real-valued functions from symplectic geometry, a bracket operation on the space of Rochesterian 1-forms Ω 1Roc associated to the space of Rochesterian vector fields and prove, despite the lack of a Jacobi identity, a relationship between this bracket and diffeomorphisms which preserve G 2 -structures.
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