Positive and negative 3-K-contact structures

Abstract

Abstract. The aim of this paper is to give a characterization of 3-K-contact and quasi 3-K-contact manifolds.

Quaternionic contact manifolds with a closed fundamental 4-form

“…For the last part of the above Theorem we have adopted the definition that a 4n + 3-dimensional (pseudo) Riemannian manifold (M, g M ) of signature either (4n + 3, 0) or (4n, 3) has a 3-Sasakian structure if the cone metric t 2 g M + ǫdt 2 on M × R is a hyperkähler metric of signature (4n + 4, 0) (positive 3-Sasakian structure, ǫ = 1) or (4n,4) (negative 3-Sasakian structure, ǫ = −1), respectively, see [21,24,18] for the negative case. In other words, the cone metric has holonomy contained in Sp(n + 1) (see [7]) or in Sp(n, 1) (see [1]), respectively.…”

From complex contact structures to real almost contact 3-structures

“…We also observe that, if Z is a projective contact manifold, such that b 2 (Z) ≥ 2, then Z = P(T * M ), for some projective manifold M , see for instance [15,Corollary 4]. Thus, in view of Remark 1.4, from Theorem 1 we have that the relationship between projective contact manifolds and almost 3-contact metric manifolds goes beyond the well-known interplay between twistor spaces of positive quaternionic Kähler manifolds and 3-Sasakian manifolds, see for instance [24], [31], [23], [10], [45], [25].…”

The Horizontal Heat Kernel on the Quaternionic Anti-De Sitter Spaces and Related Twistor Spaces

“…Thus, T 1 , T 2 , T 3 are the three Reeb vector fields of α and are also Killing vector fields on AdS 4n+3 (H). In this way, AdS 4n+3 (H) is a negative 3-K contact structure (see [30]).…”

“…Anti-de Sitter spaces play a fundamental role in mathematical physics since they appear as exact solutions of Einstein's field equations for an empty universe with a negative cosmological constant; see for instance [25] and the references therein for a pedagogical account on the importance of those spaces and the celebrated AdS/CFT correspondence. Anti-de Sitter spaces also appear as model spaces in Sasakian geometry: the complex anti de-Sitter space is the model space of a negative Sasakian manifold (see [9]) and the quaternionic anti-de Sitter fibration studied in this paper can be thought as the model space of a negative 3-Sasakian manifold (see [19,30]).…”