In the context of paracontact geometry, η-Ricci solitons are considered on manifolds satisfying certain curvature conditions: (ξ, ·) R · S = 0, (ξ, ·) S · R = 0, (ξ, ·) W 2 · S = 0 and (ξ, ·) S · W 2 = 0. We prove that on a para-Kenmotsu manifold (M, ϕ, ξ, η, g), the existence of an η-Ricci soliton implies that (M, g) is quasi-Einstein and if the Ricci curvature satisfies (ξ, ·) R · S = 0, then (M, g) is Einstein. Conversely, we give a sufficient condition for the existence of an η-Ricci soliton on a para-Kenmotsu manifold.
Torse-forming ?-Ricci solitons are studied in the framework of almost paracontact metric ?-Einstein manifolds. By adding a technical condition, called regularity and concerning with the scalars provided by the two ?-conditions, is obtained a reduction result for the parallel symmetric covariant tensor fields of order two.
Properties of invariant, anti-invariant and slant isometrically immersed submanifolds of metallic Riemannian manifolds are given with a special view towards the induced Σ-structure. Examples of such metallic manifolds are also given.
In this paper, we study the existence of proper warped product submanifolds in metallic (or Golden) Riemannian manifolds and we discuss about semi-invariant, semi-slant and, respectively, hemi-slant warped product submanifolds in metallic and Golden Riemannian manifolds. Also, we provide some examples of warped product submanifolds in Euclidean spaces.2010 Mathematics Subject Classification. 53B20, 53B25, 53C42, 53C15. Key words and phrases. Metallic Riemannian structure, Golden Riemannian structure, warped product submanifold, bi-slant submanifold, semi-invariant submanifold, semi-slant submanifold, hemi-slant submanifold. 1 2 CRISTINA E. HRETCANU AND ADARA M. BLAGA of a Kähler manifold and he found some properties of warped product submanifolds of the form M ⊥ × f M θ ([27]).Semi-invariant submanifolds in locally product Riemannian manifolds were studied in ([25], [2]). Semi-slant submanifolds in locally Riemannian product manifolds were studied by M. Atçeken which found that, in a locally Riemannian product manifold does not exist any warped product semi-slant submanifold of the forma proper slant submanifold and M ⊥ is an anti-invariant submanifold, but he found some examples of warped product semi-slant submanifolds of the form M θ × f M T and of the form M θ × f M ⊥ ([1]). Warped product submanifolds of the form M θ × f M T and M θ × f M ⊥ in Riemannian product manifolds were also studied by F. R. Al-Solamy and M. A. Khan ([28]). Warped product pseudo-slant (named also hemi-slant) submanifolds of the form M θ × f M ⊥ , where M θ and M ⊥ are proper slant and, respectively, anti-invariant submanifolds, in a locally product Riemannian manifold were studied by S. Uddin et al. in ([30]). Recently, warped product bi-slant submanifolds in Kähler manifolds were studied by S. Uddin et al. and some examples of this type of submanifolds in complex Euclidean spaces were constructed ([31]). Moreover, L. S. Alqahtani et al. have shown that there is no proper warped product bi-slant submanifold other than pseudo-slant warped product in cosymplectic manifolds ([4]).The authors of the present paper studied some properties of invariant, antiinvariant and slant submanifolds ([7]), semi-slant submanifolds ([19]) and, respectively, hemi-slant submanifolds ([18]) in metallic and Golden Riemannian manifolds and they obtained integrability conditions for the distributions involved in these types of submanifolds. Moreover, properties of metallic and Golden warped product Riemannian manifolds were presented in the two previews works of the authors ([6], [9]).In the present paper, we study the existence of proper warped product bi-slant submanifolds in locally metallic Riemannian manifolds. In Sections 2 and 3, we remind the main properties of metallic and Golden Riemannian manifolds and of their submanifolds. In Section 4, we discuss about slant and bi-slant submanifolds (with their particular cases: semi-slant and hemi-slant submanifolds) in locally metallic (or Golden) Riemannian manifolds. In Section 5, we find some properties of...
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