Submanifolds in Manifolds with Metric Mixed 3-Structures

Abstract

Mixed 3-structures are odd-dimensional analogues of paraquaternionic structures. They appear naturally on lightlike hypersurfaces of almost paraquaternionic hermitian manifolds. We study invariant and anti-invariant submanifolds in a manifold endowed with a mixed 3-structure and a compatible (semi-Riemannian) metric. Particular attention is given to two cases of ambient space: mixed 3-Sasakian and mixed 3-cosymplectic.Mathematics Subject Classification (2010). Primary 53C15; Secondary 53C50, 53C40, 53C12.

Metrics of Kaluza–Klein type on the anti‐de Sitter space

“…Thus, Hence, the vector fields are both either space‐like or time‐like and this forces the causal character of ξ 1 to be the opposite. Following the recent paper , we may then distinguish between positive and negative mxed metric 3 ‐structures , according to whether are both space‐like, or both time‐like vector fields. As remarked in , it is not known wether a mixed 3‐structure always admits both positive and negative compatible pseudo‐Riemannian metrics.…”

“…Following the recent paper , we may then distinguish between positive and negative mxed metric 3 ‐structures , according to whether are both space‐like, or both time‐like vector fields. As remarked in , it is not known wether a mixed 3‐structure always admits both positive and negative compatible pseudo‐Riemannian metrics.…”

An Almost Contact Structure on $G_2$-Manifolds

“…As with almost contact metric structures, various types of almost contact metric 3-structures have been of particular interest to several authors, see e. g., [11], [13], [20]. In particular, a 3-cosymplectic structure is an almost contact metric 3-structure (φ i , ξ i , η i , g), i = 1, 2, 3, where each of the almost contact metric structures is cosymplectic.…”

“…Various conditions can be placed on the almost contact metric structure (φ, ξ, η, g) to give various classes of almost contact metric structures such as cosymplectic, Sasakian and Kenmotsu geometries. These types of almost contact metric structures, and several others, have been studied extensively by many authors, see, e. g., [3], [4], [6], [11], [12], [13], [18], [20], [21], [24], [26], [27], [28], [29], [30] and [31]. The classification of the various types of almost contact metric structures was undertaken by Chinea and Gonzalez [14] wherein they classified almost contact metric structures based on the covariant derivative of a certain differential 2-form associated to the almost contact metric structure.…”