Submanifolds In Complex Space Forms Research Articles

An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature

In this paper, we establish an optimal inequality involving normalized δ-Casorati curvature δC(n−1) of Lagrangian submanifolds in n-dimensional complex space forms. We derive a very singular and unexpected result: the lower bounds of the normalized δ-Casorati curvatures δC(n−1) and δCˆ(n−1) in terms of dimension, the holomorphic sectional curvature, the normalized scalar curvature and the squared mean curvature of the submanifold, are different, in contrast to all previous results obtained for several classes of submanifolds in many ambient spaces. We also investigate the equality case of the inequality and prove that a Casorati δC(n−1)-ideal Lagrangian submanifold of a complex space form without totally geodesic points is an H-umbilical Lagrangian submanifold of ratio 4. Some examples are discussed in the last part of the paper, showing that the constants in the inequality obtained in this work are the best possible.

On the generalized Wintgen inequality for Legendrian submanifolds in Sasakian space forms

The generalized Wintgen inequality was conjectured by De Smet, Dillen, Verstraelen and Vrancken in 1999 for submanifolds in real space forms. It is also known as the DDVV conjecture. It was proven recently by Lu (2011) and by Ge and Tang (2008), independently. The present author established a generalized Wintgen inequality for Lagrangian submanifolds in complex space forms in 2014. In the present paper we obtain the DDVV inequality, also known as generalized Wintgen inequality, for Legendrian submanifolds in Sasakian space forms. Some geometric applications are derived. Also we state such an inequality for contact slant submanifolds in Sasakian space forms.

$\delta^{\sharp}(2,2)$-Ideal Centroaffine Hypersurfaces of Dimension $5$

The notion of an ideal submanifold was introduced by Chen at the end of the last century. A survey of recent results in this area can be found in his book [9]. Recently, in [10], an optimal collection of Chen's inequalities was obtained for Lagrangian submanifolds in complex space forms. As shown in [2], these inequalities have an immediate counterpart in centroaffine differential geometry. Centroaffine hypersurfaces realising the equality in one of these inequalities are called ideal centroaffine hypersurfaces. So far, most results in this area have only been related with $3$- and $4$-dimensional $\delta^{\sharp}(2)$-ideal centroaffine hypersurfaces. The purpose of this paper is to classify $\delta^{\sharp}(2,2)$-ideal hypersurfaces of dimension~$5$ in centroaffine differential geometry.

Tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms

The notion of Lagrangian H-umbilical submanifolds was introduced by Chen (Isr J Math 99:69–108, 1997), and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanifolds, which are defined as submanifolds such that the bitension field of the inclusion map has vanishing tangential component. The normal bundle of a round hypersphere in $$\mathbb {R}^n$$ can be immersed as a tangentially biharmonic Lagrangian H-umbilical submanifold in $$\mathbb {C}^n$$ . Motivated by this fact, we classify tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms.

On Strongly Minimal Kähler Surfaces in $${\mathbf{C}^3}$$ C 3 and the Equality $${scal(p) = 4 \inf sec(\pi ^r)}$$ s c a l ( p ) = 4 inf s e c ( π r )

Pursuing an idea motivated by a question of S.-S. Chern from 1968 on the existence of intrinsic Riemannian obstructions to minimality [Chern, S.-S.: Minimal submanifolds in a Riemannian manifold (1968)], an important study of the very idea of curvature was deepened after 1993 by B.-Y. Chen, then by other authors. In the last two decades, B.-Y. Chen’s fundamental inequalities have been investigated by many authors in the context of various geometric structures. In this work, we start by presenting B.-Y. Chen’s fundamental inequality for Kahler submanifolds in complex space forms, and we recall why the case of Kahler surfaces in $${\mathbf{C}^3}$$ satisfying $${scal(p) = 4 \inf sec(\pi ^r)}$$ appears naturally and is important. Then we provide several characterizations of strongly minimal complex surfaces in the complex three dimensional space. We focus our study on the question of finding further examples of strongly minimal Kahler surfaces, as the question of a complete classification of these geometric objects is still open.

TWO OPTIMAL INEQUALITIES FOR ANTI-HOLOMORPHIC SUBMANIFOLDS AND THEIR APPLICATIONS

The $CR$ $\delta$-invariant for $CR$-submanifolds was introduced by B.-Y. Chen in a recent article \cite{C2012}. In this paper, we prove two new optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the $CR$ $\delta$-invariant. Moreover, we obtain some classification results for certain anti-holomorphic submanifolds in complex space forms which satisfy the equality case of either inequality.

On the generalized Wintgen inequality for Lagrangian submanifolds in complex space forms

The normal scalar curvature conjecture, also known as the DDVV conjecture, was formulated by De Smet et al. (1999). It was proven recently by Lu (2011) and by Ge and Tang (2008) independently.We obtain the DDVV inequality, also known as the generalized Wintgen inequality, for Lagrangian submanifolds in complex space forms. Some applications are given. Also we state such an inequality for slant submanifolds in complex space forms.

Pseudo-parallel CR submanifolds of a complex space form

We classify pseudo-parallel proper CR submanifolds of a non-flat complex space form with semi-flat normal connection under the condition that the dimension of the holomorphic tangent space is greater than 2.

Lagrangian submanifolds in complex projective space with parallel second fundamental form

From the Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classiflcation of Lagrangian submanifolds with parallel second fundamental form. In 1980’s, H. Naitoh completely classifled the Lagrangian submanifolds with parallel second fundamental form and without Euclidean factor in complex projective space, by using the theory of Lie groups and symmetric spaces. He showed that such a submanifold is always locally symmetric and is one of the symmetric spaces: SO(k + 1)=SO(k)(k ‚ 2), SU(k)=SO(k)(k ‚ 3), SU(k)(k ‚ 3), SU(2k)=Sp(k)(k ‚ 3), E6=F4. In this paper, we completely classify the Lagrangian submanifolds in complex projective space with parallel second fundamental form by an elementary geometrical method. We prove that such a Lagrangian submanifold is either totally geodesic or the Calabi product of a point with a lower dimensional Lagrangian submanifold with parallel second fundamental form, or the Calabi product of two lower dimensional Lagrangian submanifolds with parallel second fundamental form, or one of the standard symmetric spaces: SU(k)=SO(k)(k ‚ 3), SU(k)(k ‚ 3), SU(2k)=Sp(k)(k ‚ 3), E6=F4. As the arguments are of a local nature, at the same time, due to the correspondence between C-parallel Lagrangian submanifolds in Sasakian space forms and parallel Lagrangian submanifolds in complex space forms, we can also give a complete classiflcation of all Cparallel submanifolds of S 2n+1 equipped with its standard Sasakian structure. 2000 Mathematics Subject Classiflcation: Primary 53B25; Secondary 53C42

AN IMPROVED CHEN-RICCI INEQUALITY FOR KAEHLERIAN SLANT SUBMANIFOLDS IN COMPLEX SPACE FORMS

B. Y. Chen proved in [4] an optimal inequality for Lagrangian submanifolds in complex space forms in terms of the Ricci curvature and the squared mean curvature, well-known as the Chen-Ricci inequality. Recently, the Chen-Ricci inequality was improved in [7,11] for Lagrangian submanifolds in complex space forms. In this article we extend the improved Chen-Ricci inequality to Kaehlerian slant submanifolds in complex space forms. We also investigate the equality case of the inequality.

Biminimal Lagrangian H-umbilical submanifolds in complex space forms

All biminimal Lagrangian surfaces of nonzero constant mean curvature in 2-dimensional complex space forms have been determined in Sasahara (Differ Geom Appl 27:647–652, 2009). In this paper, we completely determine biminimal Lagrangian H-umbilical submanifolds of nonzero constant mean curvature in complex space forms of dimension ≥ 3.

Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections

Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections

Improved Chen–Ricci inequality for curvature-like tensors and its applications

We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.

Optimal general inequalities for Lagrangian submanifolds in complex space forms

Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. In this paper we establish general inequalities for Lagrangian submanifolds in complex space forms. We also provide examples showing that these inequalities are the best possible. Moreover, we provide simple non-minimal examples which satisfy the equality case of the improved inequalities.

Ricci curvature of Lagrangian submanifolds in complex space forms

As we showed in [5] and [6], the basic inequalities, involving Riemannian invariants of a Lagrangian submanifold immersed in a complex space form, can be improved using optimization methods. Also in [1] is showed that the improved Chen’s inequality from [5] is optimal. In this paper we find another proof for a Chen’s inequality, regarding the Ricci curvature [2] and we improve this inequality in the Lagrangian case. Mathematics subject classification (2010): 53C21, 53C24, 53C25, 49K35.

Geometric Inequalities for Purely Real Submanifolds in Complex Space Forms

As a generalisation of Kaehlerian slant submanifolds in Kaehler manifolds (i.e., proper slant submanifolds with the canonical endomorphism P parallel, ∇P = 0) one considers purely real submanifolds with ∇P = 0. The class of purely real submanifolds with ∇P = 0 also contains totally real submanifolds, in particular Lagrangian submanifolds. We obtain Chen-like inequalities for purely real submanifolds in complex space forms, i.e., relationships between intrinsic and extrinsic invariants of such submanifolds, involving the scalar curvature and Chen first invariant, respectively, and the squared mean curvature and the holomorphic sectional curvature of the ambient space.

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.

Codimension reduction and second fundamental form of CR submanifolds in complex space forms

Considering n -dimensional real submanifolds M of a complex space form ( M ¯ n + p , g ¯ , J ) which are CR submanifolds of CR dimension n − m 2 , we study the condition h ( F X , Y ) + h ( X , F Y ) = 0 on the structure tensor F naturally induced from the almost complex structure J of the ambient manifold and on the second fundamental form h of submanifolds M .

Pseudo-parallel Lagrangian submanifolds in complex space forms

In this work we study pseudo-parallel Lagrangian submanifolds in a complex space form. We give several general properties of pseudo-parallel submanifolds. For the 2-dimensional case, we show that any minimal Lagrangian surface is pseudo-parallel. We also give examples of non-minimal pseudo-parallel Lagrangian surfaces. Here we prove a local classification of the pseudo-parallel Lagrangian surfaces. In particular, semi-parallel Lagrangian surfaces are totally geodesic or flat. Finally, we give examples of pseudo-parallel Lagrangian surfaces which are not semi-parallel.

Warped product decompositions of real space forms and Hamiltonian-stationary Lagrangian submanifolds

A Lagrangian submanifold of a Kaehler manifold is said to be Hamiltonian-stationary (or H -stationary for short) if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In this article, we present some simple relationship between warped product decompositions of real space forms and Hamiltonian-stationary Lagrangian submanifolds. We completely classify H -stationary Lagrangian submanifolds in complex space forms arisen from warped product decompositions. More precisely, we prove that there exist two such families of H -stationary Lagrangian submanifolds in C n , two families in C P n , and twenty-one families in C H n . As immediate by-product we obtain many new families of Hamiltonian-stationary Lagrangian submanifolds in complex space forms.