Some slant submanifolds of <Emphasis Type="Italic">S</Emphasis>-manifolds

Summary 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 examples of such submanifolds.     

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.Mathematics Subject Classifications (2000): 53C15, 53C40, 53B20.     

Semi-Parallel, Semi-Symmetric Immersions and Chen’s Equality

Recently, B. Y. Chen introduced a new invariant δ(n₁,n₂,…,n_k) of a Riemannian manifold and proved a basic inequality between the invariant and the extrinsic invariant if, where H is the mean curvature of an immersion Mⁿ in a real space form R^m(ε) of constant curvature ε. He pointed out that such inequality also holds for a totally real immersion in a complex space form. The immersion is called ideal (by B. Y. Chen) if it satisfies the equality case of such inequality identically. In this paper we classify ideal semi-parallel immersions in an Euclidean space if their normal bundle is flat, and prove that every ideal semi-parallel Lagrangian immersion in a complex space form is totally geodesic, moreover this result also holds for ideal semi-symmetric Lagrangian immersions in complex projective space and hyperbolic space.     

Non-Existence of Stable Currents in Hypersurfaces

Let M^m be a compact hypersurface in the Euclidean space E^m+1. In this paper, we study the non-existence of stable integral currents in M^m and its immersed submanifolds. Some vanishing theorems concerning the homology groups of these manifolds are established.AMS Subject Classification (1991): 49Q15 53C40 53C20     

Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map

For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative on the Grassmann bundle ^ := G_m(T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N Rⁿ this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ G_m(T_pN) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving ) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way. Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21.     

Contact CR-Warped Product Submanifolds in Sasakian Space Forms

Recently, B.-Y. Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. Afterwards, I. Hasegawa and the present author obtained a sharp inequality for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically immersed in Sasakian manifolds. In this paper, we improve the above inequality for contact CR-warped products in Sasakian space forms. Some applications are derived. A classification of contact CR-warped products in spheres, which satisfy the equality case, identically, is given.Mathematics Subject Classifications (2000). 53C40, 53C25.     

Classification of Casorati ideal Lagrangian submanifolds in complex space forms

In this paper, using optimization methods on Riemannian submanifolds, we establish two improved inequalities for generalized normalized δ-Casorati curvatures of Lagrangian submanifolds in complex space forms. We provide examples showing that these inequalities are the best possible and classify all Casorati ideal Lagrangian submanifolds (in the sense of B.-Y. Chen) in a complex space form. In particular, we generalize the recent results obtained in G.E. Vîlcu (2018) [34].     

<Emphasis Type="Italic">p</Emphasis>-Ideals in <Emphasis Type="Italic">p</Emphasis>-Regular Semirings

In this paper, we introduce p-ideals in semirings. A new form of regularity, which is compatible with p-ideals, is defined. Our aim is to explore the possibilities of establishing an ideal theory in semirings, going alongside the existing literature of semiring theory.AMS Subject Classification (2000): Primary 16Y60     

A pinching theorem for extrinsically symmetric submanifolds of Euclidean space

We show that a compact connected manifold which can be immersed into ^m with almost parallel second fundamental form, admits an extrinsically symmetric immersion into ^m.Mathematics Subject Classification (2000): 53C20, 53C24, 53C30, 53C35, 53C40, 53C42Acknowledgement I am most grateful to J.–H. Eschenburg, P. Ghanaat and E. A. Ruh for valuable discussions and helpful remarks. This work was supported by the Swiss National Science Foundation Grants 20-67619.02 and PBFR-106367.     

Hamiltonian Stationary Lagrangian Surfaces in <Emphasis Type="Italic">ℂP</Emphasis><Superscript>2</Superscript>

In this paper we use a new equivalent condition of Hamiltonian stationary Lagrangian surfaces in ℂP² to show that any Hamiltonian stationary Lagrangian torus in ℂP² can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspace of a certain loop Lie algebra, i.e., is of finite type. Mathematics Subject Classifications (2000): Primary 53C40; Secondary 53C42, 53D12     

Pseudo-parallel Lagrangian submanifolds are semi-parallel

We prove a conjecture formulated by Pablo M. Chacon and Guillermo A. Lobos in [P.M. Chacon, G.A. Lobos, Pseudo-parallel Lagrangian submanifolds in complex space forms, Differential Geom. Appl. 27 (1) (2009) 137–145, doi:10.1016/j.difgeo.2008.06.014] stating that every Lagrangian pseudo-parallel submanifold of a complex space form of dimension at least 3 is semi-parallel. We also propose to study another notion of pseudo-parallelity which is more adapted to the Kaehlerian setting.     

Averaging of Legendrian Submanifolds of Contact Manifolds

We give a procedure to ‘average’ canonically C¹-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby. Mathematics Subject Classification (2000): 53D10.     

Totally Real Submanifolds in a Quaternion Space Form

In this paper, we prove a theorem for n-dimensional totally real minimal submanifold immersed in quaternion space form.     

Totally geodesic maps into metric spaces

We prove that a totally geodesic map between a Riemannian manifold and a metric space can be represented as the composite of a totally geodesic map from a Riemannian manifold to a Finslerian manifold and a locally isometric embedding between metric spaces. As a corollary, we obtain the homotheticity of a totally geodesic map from an irreducible Riemannian manifold to an Alexandrov space of curvature bounded above. This is a generalization of the case between Riemannian manifolds. Mathematics Subject Classification (2000): 53C20, 53C22, 53C24 Received: 14 March 2002; in final form: 6 May 2002 / / Published online: 24 February 2003     

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     

A Second Main Theorem of Nevanlinna Theory for Meromorphic Functions on Complex Submanifolds in C<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show a second main theorem of Nevalinna theory for meromorphic functions on complex submanifolds in C ⁿ . This has a similar form to the classical one and has a remainder term including Ricci curvature. We also give a concrete computation of the remainder term in the case of nonsingular algebraic submanifolds. Partially supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science.     

Almost Extrinsically Homogeneous Submanifolds of Euclidean Space

Consider a closed manifold M immersed in R^m. Suppose that the trivial bundle M × R^m = T M ⊗ ν M is equipped with an almost metric connection ~ ∇ which almost preserves the decomposition of M × R^m into the tangent and the normal bundle. Assume moreover that the difference Γ = ∂~∇ with the usual derivative ∂ in R^m is almost ~∇-parallel. Then M admits an extrinsically homogeneous immersion into R^m. Mathematics Subject Classifications (2000): 53C20, 53C24, 53C30, 53C42, 53C40.     

<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

复射影空间中迷向Kaehler子流形

讨论了复射影空间中迷向Kaehler流形,运用活动标架法获得关于截面曲率,Ricci曲率和第二基本形式模长的Pinching定理,将相关结果作了一定的推广.     

Normal <Emphasis Type="Italic">CR</Emphasis> structures on S

We classify the normal CR structures on S³ and their automorphism groups. Together with [2], this closes the classification of normal CR structures on compact 3-manifolds. We give a criterion to compare two normal CR structures, and we show that the underlying contact structure is, up to diffeomorphism, unique. Mathematics Subject Classification (2000): 53A40, 53C25, 53D10. Received: 9 April 2002; in final form: 8 May 2002 // Published online: 20 March 2003