On normal almost contact structures

Almost paracontact and parahodge structures on manifolds

“…Proof, The proof is similar to that of Theorem 6 in [4]. Let X be a tangent vector at peM, and let us denote by hX (resp.…”

“…(1.2) is also proved analogously as in the almost contact case (cf. [4]). φ leaves horizontal subspaces invariant, and if X is a horizontal vector, then from (2.3) we have π*φX -Iπ*X, which implies (1.3).…”

AN ABSTRACT MORIMOTO THEOREM FOR GENERALIZEDF-STRUCTURES

“…where the admissibility translates into the condition B 0 , C 0 ∈ GL(2l, R).In particular, the matrix Ψ can 0 = 0 Id −Id 0 yields the admissible triple used in the original work of Morimoto [12] and in some of its generalizations [13], [6], [7].…”

“…However, generalizing a classical construction of Morimoto, one can introduce a third SGF-structure Ψ on M 1 × M 2 in such a way that J 1 ⊕ J 2 ⊕ Ψ is a generalized almost complex structure. Extending a theorem of Morimoto [12] to the generalized setting, Gomez and Talvacchia [6] proved the existence of a canonical SGF-structure Ψ for which J 1 ⊕ J 2 ⊕ Ψ is a generalized complex structure if and only if J 1 , J 2 and Ψ are generalized CRF-structures and the natural framing of L Ψ ⊕L Ψ normalizes both J 1 and J 2 .…”

“…In this paper we abstract the features that make Morimoto's construction [12] work into the concept of (adaptable) Morimoto datum defined out of: 1) mutually orthogonal split structures E 1 , E 2 , E ′ 1 and E ′ 2 ; 2) SGF-structures J 1 on E 1 , J 2 on E 2 , Ψ on E ′ 1 ⊕ E ′ 2 and 3) global framings V 1 for E ′ 1 and V 2 for E ′ 2 . Our main results is an Abstract Morimoto Theorem stating that in presence of an adaptable Morimoto datum, J 1 ⊕J 2 ⊕Ψ is a generalized CRFstructure if and only if (J 1 , V 1 ) and (J 2 , V 2 ) are normal pairs, an abstraction of the concept of normal generalized almost contact structure introduced in [1].…”