Semi-Riemannian hypersurfaces in manifolds with metric mixed 3-structures

Abstract

The mixed 3-structures are the counterpart of paraquaternionic structures in odd dimension. A compatible metric with a mixed 3-structure is necessarily semi-Riemann and mixed 3-Sasakian manifolds are Einstein. We investigate the differential geometry of the semi-Riemannian hypersurfaces of co-index both 0 and 1 in a manifold endowed with a mixed 3-structure and a compatible metric.

Submanifolds in Manifolds with Metric Mixed 3-Structures

“…Several examples of manifolds endowed with metric mixed 3-structures are given in [9,11]: R 4n+3 2n+1 admits a positive mixed 3-cosymplectic structure, R 4n+3 2n+2 admits a negative mixed 3-cosymplectic structure, the unit pseudo-sphere S 4n+3 2n+1 and the real projective space P 4n+3 2n+1 (R) are the canonical examples of manifolds with positive mixed 3-Sasakian structures, while the unit pseudo-sphere S 4n+3 2n+2 and the real projective space P 4n+3 2n+2 (R) can be endowed with negative mixed 3-Sasakian structures.…”

“…Proof. If M is a mixed 3-cosymplectic manifold then the assertion is a direct consequence of (11). On the other hand, if M is a mixed 3-Sasakian manifold, then using ( 4) and ( 12) we obtain for any N ∈ Γ(D ⊕ D ⊥ ):…”

“…This concept has been refined in [3], where the authors have introduced positive and negative metric mixed 3-structures. The differential geometry of the semi-Riemannian hypersurfaces of co-index both 0 and 1 in such manifolds has been recently investigated in [11]. In the present paper, we discuss non-degenerate invariant and anti-invariant submanifolds in manifolds endowed with metric mixed 3-structures, the relevant ambients being mixed 3-Sasakian and mixed 3-cosymplectic.…”

“…Remark 6.2. If M is mixed 3-cosymplectic, then (11) directly implies the integrability of ξ on M . Proposition 6.3.…”

“…Proof. (i) If M is mixed 3-cosymplectic, then using (11) we obtain for any X, Y ∈ Γ(D) and α = 1, 2, 3:…”