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We prove that if the f-sectional curvature at any point of a ( 2 n + s ) -dimensional metric f-contact manifold satisfying the ( &kappa, &mu, ) nullity condition with n &gt, 1 is independent of the f-section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric f-contact manifold satisfying the ( &kappa, ) nullity condition is of constant f-sectional curvature if and only if &mu, = &kappa, + 1 and we give an explicit expression for the curvature tensor field in such a case. Finally, we present some examples.

We prove a classification theorem of the generalized Sasakian space forms which are realized as real hypersurfaces in complex space forms.

In this paper, we introduce the notion of $$*$$-slant submanifold as that slant submanifold whose second fundamental form satisfies the equality case of an inequality between its mean curvature and its scalar curvature. In addition to that, we give several interesting examples about these submanifolds. Finally, we obtain the Ricci curvature for a $$*$$-slant submanifold depending on the mean curvature vector and we give lower and upper bounds for the Ricci curvature.

In this paper we define slant submanifolds in neutral almost contact pseudo-metric manifolds, with motivations and examples. We also provide some natural examples of the ambient spaces.

The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.

In this paper, we introduce the notion of bi-slant submanifolds of a para Hermitian manifold. They naturally englobe CR, semi-slant, and hemi-slant submanifolds. We study their first properties and present a whole gallery of examples. Ministerio de Economía y Competitividad (MINECO). España European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER) Plan Andaluz de Investigación, Desarrollo e Innovación (PAIDI) Instituto de Matemáticas de la Universidad de Sevilla (IMUS)

We determine Kenmotsu spaces which are also either D’Atry spaces (that is, all of whose local geodesic symmetries are volume-preserving) or C-spaces (that is, their Jacobi operators have constant eigenvalues along the corresponding geodesics). We prove that both D’Atry and C-type Kenmotsu spaces are of constant sectional curvature-1.

In this paper, we extend the notion of generalized Sasakian space form to the semi-Riemannian setting. We consider several interesting cases and we give examples of them all. We also study their structures.

In this paper, we introduce the notion of slant submanifolds of a para-Hermitian manifold. We study their first properties and present a whole gallery of examples.

We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces with Boeckx invariant $I \leq -1$. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct., Comment: 14 pages

In this paper we will show that a Lagrangian, Lorentzian surface $M^2_1$ in a complex pseudo space form $\widetilde M^2_1 (4c)$ is pseudo-isotropic if and only if $M$ is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic., Comment: 15 pages