SLANT IMMERSIONS OF COMPLEX SPACE FORMS AND CHEN＇S INEQUALITY

A submanifold in a complex space form is called slant it it has constant Wirtinger angles. B, Y, Chen and Y. Tazawa proved that there do not exist minimal proper slant surfaces in CP^2 and CH^2. So it seems that the slant immersion has some interesting properties. The authors have great interest to consider slant immersions satisfying some additional conditions, such as unfull first normal bundles or Chen‘s equality holding. They prove that there do not exist n-dimensional Kaehlerian slant immersion sin CP^n and CH^n with unfull first normal bundles. Next, it is seen that every Kaehlerian slant submanifold satisfying an equality of Chen is minimal which is similar to that of Lagrangian immersions. But in contrast, it is shown that a large class of slant immersions do not exist thoroughly. Finally, they give an application of Chen‘s inequality to general slant immersions in a complex projective space, which generalizes a result of Chen.     

Existence and Uniqueness Theorem for Slant Immersions and Its Applications

A slant immersion is an isometric immersion from a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. In this paper we establish the existence and uniqueness theorem for slant immersions into complex-space-forms. By applying this result, we prove in this paper several existence and nonexistence theorems for slant immersions. In particular, we prove the existence theorems for slant surfaces with prescribed mean curvature or with prescribed Gaussian curvature. We also prove the non-existence theorem for flat minimal proper slant surfaces in non-flat complex space forms.     

The Maslov form in non-invariant slant submanifolds of S-space-forms

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.     

Topological obstructions for submanifolds in low codimension

We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for \(\delta \)-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second fundamental form.     

Orbits of SU(2)-representations and minimal isometric immersions

In contrast to all known examples, we show that in the case of minimal isometric immersions of into the smallest target dimension is almost never achieved by an -equivariant immersion. We also give new criteria for linear rigidity of a fixed minimal isometric immersion of into . The minimal isometric immersions arising from irreducible SU(2)-representations are linearly rigid within the moduli space of SU(2)-equivariant immersions. Hence the question arose whether they are still linearly rigid within the full moduli space. We show that this is false by using our new criteria to construct an explicit SU(2)-equivariant immersion which is not linearly rigid. Various authors [GT], [To3], [W1] have shown that minimal isometric immersions of higher isotropy order play an important role in the study of the moduli space of all minimal isometric immersions of into . Using a new necessary and sufficient condition for immersions of isotropy order , we derive a general existence theorem of such immersions. Received: 13 May 1999 / in final form: 13 July 1999     

Tensor product surfaces of euclidean plane curves

Recently B.Y. CHEN initiated the study of the tensor product immersion of two immersions of a given Riemannian manifold [3]. In [6] the particular case of tensor product of two Euclidean plane curves was studied. The minimal one were classified, and necessary and sufficient conditions for such a tensor product to be totally real or complex or slant were established. In the present paper we study for tensor product of Euclidean plane curves the problem of B.Y. CHEN: to what extent do the properties of the tensor product immersion f ? h of two immersions f, h determines the immersions f, h ? [3]     

Lagrangian immersions and Bott periodicity

We interpret the Bott Periodicity Theorem for the unitary group as a statement about a geometrical construction for Lagrangian immersions.     

Induced generalized <Emphasis Type="Italic">S</Emphasis>-space-form structures on submanifolds

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.     

Warped Product Bi-slant Immersions in Kaehler Manifolds

A submanifold M of an almost Hermitian manifold \((\widetilde{M},g,J)\) is called slant, if for each point \(p\in M\) and \(0\ne X\in T_p M\), the angle between JX and \(T_p M\) is constant (see Chen in Bull Aust Math Soc 41:135–147, 1990). Later, Carriazo (in: Proceedings of the ICRAMS 2000, Kharagpur, 2000) defined the notion of bi-slant immersions as an extension of slant immersions. In this paper, we study warped product bi-slant submanifolds in Kaehler manifolds and provide some examples of warped product bi-slant submanifolds in some complex Euclidean spaces. Our main theorem states that every warped product bi-slant submanifold in a Kaehler manifold is either a Riemannian product or a warped product hemi-slant submanifold.     

Compactness theorems and an isoperimetric inequality for critical points of elliptic parametric functionals

We consider parametric variational double integrals with elliptic Lagrangians F depending on the surface normal and prove a compactness theorem for -critical immersions. As a key ingredient for the relevant a priori estimates we use F. Sauvigny's F-conformal parameters adapted to the parametric integrand F. As a by-product of our analysis we obtain an isoperimetric inequality for -critical immersions generalizing the classical isoperimetric inequality for minimal surfaces. Received November 19, 1999 / Accepted February 4, 2000 / Published online July 20, 2000     

Generic 2-surfaces and 2-2 foliations in general relativity

Summary Special points of spacelike and timelike 2-surfaces are defined by means of algebraic relations between the second order invariants of their immersion. Generic immersions are then defined and by means of direction fields constructed over the surface, the least number of such points is related to global properties of the surface. Moreover it is proved that every compact spacelike 2-surface has at least one point where the normal curvature is zero. Consequences are drawn for generic 2-spheres and trapped surfaces. The theorems are generalised to 2-2 foliations and fibrations. Entrata in Redazione il 9 ottobre 1973.     

Semi-parallel surfaces in Euclidean space

Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersions. Using results of Backes on Euclidean Jordan triple systems, the totally geodesic immersions are shown to be the only minimal semi-parallel immersions into a Euclidean space. Semi-parallel immersions of surfaces into E^m are studied and a classification of semi-parallel immersions with pointwise planar normal sections of surfaces in E^m is given.Research Assistant of the National Fund of Scientific Research     

The geometry of closed conformal vector fields on Riemannian spaces

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.     

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

We introduce concepts of minimal immersions and bandlimited (Paley-Wiener) immersions of combinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of bandlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown that our minimal immersions can be used to perform interpolation, smoothing and approximation of immersions of graphs into Euclidean spaces. It is proved that under certain conditions minimal immersions converge to bandlimited immersions. Explicit expressions of minimal immersions in terms of eigenmaps are given. The results can find applications for data dimension reduction, image processing, computer graphics, visualization and learning theory.     

Conformal energy in four dimension

The conformal energy for 4-manifolds using the Paneitz operator is introduced in this article. The conformal invariance of the energy functional allows us to find a sharp lower bound in terms of the conformal volume. We also demonstrate certain obstruction to existence of minimal immersions to spheres using the fourth order curvature invariance associated to the operator. Received: 17 April 1999 / Revised version: 23 March 2002 / Published online: 5 September 2002     

Homotopy forg functions and generalized retraction

Summary We introduce here a generalization of the usual concept of homotopy for continuous functions, because the traditional concept is not adequate to deal with some important geometrical situations. Entrata in Redazione il 2 settembre 1969.     

Higher dimensional links are singular slice

We show that for all links of embedded n-spheres in are singular slice, i.e. bound pairwise disjoint (but not embedded) n+1-disks in . The proof relies on a careful analysis of immersions in codimension two, that allows us to work in a nilpotent setting. Received October 21, 1999 / Accepted October 12, 2000 / Published online March 12, 2001     

Affine planar geodesic immersions with maximal codimension

This work gives a classification theorem for affine immersions with planar geodesics in the case where the codimension is maximal. Vrancken classified parallel affine immersions in this case and obtained, among others, generalized Veronese submanifolds. In this work it is shown that the immersions with planar geodesics are the same as the parallel ones in the considered case. A geometric interpretation of parallel immersions is also given： The affine immersions with pointwise planar normal sections （with respect to the equiaffine transversal bundle） are parallel. This result is verified for surfaces in R4 and for immersions with the maximal codimension.     

Bimeasures in Banach spaces

In this paper we study integration with respect to a bimeasure with finite semivariation. The bimeasures as well as the functions to be integrated, take on their values in Banach spaces. Entrata in Redazione il 25 settembre 1998 e, in versione riveduta, il 25 giugno 1999.     

Embedding problems: Geometric and topological aspects

A survey of results obtained after 1976 on questions of the isometric immersions of Riemannian spaces in a Euclidean space, the immersions and embeddings of differential manifolds, and the immersions with minimal absolute curvature is presented.