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We study hypersurfaces isometrically immersed in a trans-S-manifolds in order to find out under what conditions they could inherit the structure of the ambient manifold and so, to obtain new examples of such trans-S-manifolds. Mainly, we investigate this situation depending the behaviour of the second fundamental form of the immersion.

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.

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.

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and study of generalized (\kappa,\mu,\nu)-space forms and of the necessary and sufficient conditions for them to be conformally flat.

B?zier curves and surfaces are two very useful tools in Geometric Modeling, with many applications. In this paper, we will offer a new method to provide approximations of regular curves and surfaces by B?zier ones, with the corresponding examples.

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 prove that every contact metric ( κ , μ ) -space admits a canonical η -Einstein Sasakian or η -Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κ and μ for which such metrics are Sasaki–Einstein and paraSasaki–Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric ( κ , μ ) -structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some geometric properties of ( κ , μ ) -spaces related to the existence of Einstein–Weyl and Lorentzian–Sasaki–Einstein structures.

We study the Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.

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

We define a new association between graphs and orthonormal bases of even-dimensional Euclidean vector spaces endowed with an special isomorphism motivated by the recently introduced theory of submanifolds associated with graphs. We provide several interesting examples and we analyze the shape of such graphs by proving some general results.

Generalized (\kappa ,\mu)-space forms are introduced and studied. We examine in depth the contact metric case and present examples for all possible dimensions. We also analyse the trans-Sasakian case., Comment: 20 pages, several changes have been done in this version

We present a characterization theorem for the Maslov form in certain non-invariant slant submanifolds of S-space-forms to be closed and, from it, we deduce a topological obstruction for these types of non-invariant slant immersions. Moreover, we also give conditions for an anti-invariant submanifolds of an S-manifold, tangent to the structure vector fields, to have closed and conformal Maslov form.

The study of generalized Sasakian space forms is continued in this paper. The behavior of such spaces under generalized D-conformal deformations is analyzed. As a consequence, new examples of generalized Sasakian space forms are given.

We introduce and study generalized S-space-forms with two structure vector fields. We also present several examples of these manifolds such as certain hypersurfaces of Sasakianspace-forms, principal toroidal bundles and warped products. Moreover, we investigate generalized S-space-forms endowed with an additional structure and we obtain some obstructions for them to be S-manifolds.

We study non-anti-invariant slant submanifolds of generalized S-space-forms with two structure vector felds in order to know if they inherit the ambient structure. In this context, we focus on totally geodesic, totally umbilical, totally ƒ-geodesic and totally ƒ-umbilical non-anti-invariant slant submanifolds and obtain some obstructions. Moreover, we present some new interesting examples of generalized S-space-forms.

We establish an interesting link between differential geometry and graph theory by defining submanifolds weakly associated with graphs. We prove that, in a local sense, every submanifold satisfies such an association, and other general results. Finally, we study submanifolds associated with graphs either in low dimensions or belonging to some special families. Junta de Andalucía Ministerio de Educación y Ciencia Fondo Europeo de Desarrollo Regional

In this paper, contact metric and trans-Sasakian generalized Sasakian-space-forms are deeply studied. We present some general results for manifolds with dimension greater than or equal to 5, and we also pay a special attention to the 3-dimensional cases.

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

In the present work, we briefly summarize our contributions to the study of CR submanifolds of (locally conformal almost) Kaehler manifolds and normal CR submanifolds of S-manifolds.

B.-Y. CHEN’S INEQUALITY FOR SUBMANIFOLDS OF GENERALIZED SPACE FORMS

We study some special types of slant submanifolds of S-manifolds related to the second fundamental form of the immersion: totally f-geodesic and f-umbilical, pseudo-umbilical and austere submanifolds. We also give several ex- amples of such submanifolds.

Generalized Sasakian-space-forms are introduced and studied. Many examples of these manifolds are presented, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations. New results on generalized complex-space-forms are also obtained.

In this note we show a bosonic conformal string theory on U (1)-bundles over AdS 3 . To this end, we first look for r -elastic helices in the space AdS 3 , because they generate solutions of the motion equation on backgrounds P which are principal circle-bundles over AdS 3 endowed with the standard metric or generalized Kaluza–Klein metrics. In fact, we reduce the search of U (1)-symmetric string configurations on P to the search of r -elastic curves in the orbit space AdS 3 .

We establish a canonical cobomology class on a proper slant submanifold of a Sasakian-spacefom and give some geometrical obstructions to such immersions. moreover, we combine both topological and geometrical properties in order to show other interesting obstructions to slant immersions in contact geometry, involving the scalar curvature, the Ricci curvatures and a certain intrinsic function.

We define and study both bi-slant and semi-slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold. We prove a characterization theorem for semi-slant submanifolds and we obtain integrability conditions for the distributions which are involved in the definition of such submanifolds. We also study an interesting particular class of semi-slant submanifolds.

An indefinite Riemannian manifold is called neutral it its index is equal to one half of its dimension and an indefinite Kaehler manifold is called neutral Kaehler if its complex index is equal to the half of its complex dimension. In the first part of this article, we extend the notion of slant surfaces in Lorentzian Kaehler surfaces to slant submanifolds in neutral Kaehler manifolds; moreover, we characterize slant submanifolds with parallel canonical structures. By applying the results obtained in the first part we completely classify slant surfaces with parallel mean curvature vector and minimal slant surfaces in the Lorentzian complex plane in the second part of this article.

We study slant submanifolds of S-manifolds with the smallest dimension, specially minimal submanifolds and establish some relations between them and anti-invariant submanifolds in S-manifolds, similar to those ones proved by B.-Y. Chen for slant surfaces and totally real surfaces in Kaehler manifolds. Plan Andaluz de Investigación (Junta de Andalucía)

Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not.