Submanifolds Of Complex Projective Space Research Articles

On minimal Lagrangian submanifolds in complex space forms with semi-parallel second fundamental form

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension [Formula: see text] in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases [Formula: see text]. Second, supposing further that the scalar curvature is constant for [Formula: see text], we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with [Formula: see text], we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined.

Oxidation, reduction and semi-classical limit for quantum matrix geometries

Matrix configurations define noncommutative spaces endowed with extra structure including a generalized Laplace operator, and hence a metric structure. Made dynamical via matrix models, they describe rich physical systems including noncommutative gauge theory and emergent gravity. Refining the construction in [25], we construct a semi-classical limit through an immersed submanifold of complex projective space based on quasi-coherent states. We observe the phenomenon of oxidation, where the resulting semi-classical space acquires spurious extra dimensions. We propose to remove this artifact by passing to a leaf of a carefully chosen foliation, which allows to extract the geometrical content of the noncommutative spaces. This is demonstrated numerically via multiple examples.

A normal line congruence and minimal ruled Lagrangian submanifolds in [formula omitted

We characterize Lagrangian submanifolds in complex projective space for which each parallel submanifold along normal geodesics with respect to a unit normal vector field is Lagrangian, by using a normal line congruence of the Lagrangian submanifold to complex 2-plane Grassmannian and quaternionic Kähler structure. As a special case, we can construct minimal ruled Lagrangian submanifolds in complex projective space from an austere hypersurface in sphere with non-vanishing Gauss-Kronecker curvature.

Remarks on the abelian convexity theorem

This paper contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf–Ness functions. This is applied to give short proofs of the Atiyah–Guillemin–Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic submanifold of complex projective space. Finally we give an application to the action on the probability measures.

On the Strict Arnold Chord Property and Coisotropic Submanifolds of Complex Projective Space

Let be a contact form on a manifoldM, and L M a closed Legendrian submanifold. I prove that L intersects some characteristic for at least twice if all characteristics are closed and of the same period, and embeds nicely into the product of R2n and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most /2. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.

Variational formulas of higher order mean curvatures

In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional $$\mathcal{M}_{2p}$$ of a submanifold M n in a general Riemannian manifold N n+m for $$p = 0,1, \ldots ,\left[ {\tfrac{n} {2}} \right]$$ . As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional $$\mathcal{M}_{2p}$$ , called relatively 2p-minimal submanifolds, for all p. At last, we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms, as well as a special variational problem.

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

Riemannian submersions, δ-invariants, and optimal inequality

In this study, we establish a sharp relation between δ-invariants and Riemannian submersions with totally geodesic fibers. By using this relationship, we establish an optimal inequality involving δ-invariants for submanifolds of the complex projective space CPm(4) via Hopf’s fibration \({\pi:S^{2m+1}\to CP^{m}(4)}\) . Moreover, we completely classify submanifolds of complex projective space which satisfy the equality case of the inequality.

Willmore Lagrangian submanifolds in complex projective space

Willmore Lagrangian submanifolds in complex projective space

The scalar curvature of CR submanifolds of maximal CR dimension of complex projective space

We treat n-dimensional compact minimal submanifolds of complex projective space when the maximal holomorphic tangent subspace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces.

Inequalities of eigenvalues for the Dirac operator on compact complex spin submanifolds in complex projective spaces

For a compact complex spin manifold M with a holomorphic isometric embedding into the complex projective space, the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator, which depend on the data of an isometric embedding of M. Further, from the inequalities of eigenvalues, the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space.

Stability and rigidity of special Lagrangian cones over certain minimal Legendrian orbits

Special Lagrangian cones in complex Euclidean spaces are obtained as cones over compact minimal Legendrian submanifolds in the odd dimenisonal standard hypersphere. The notion of the stability, the Legendrian stability and the rigidity of special Lagrangian cones were recently introduced and investigated by D. Joyce, M. Haskins etc. In this paper we determine explicitly the stability-index, the Legendrian-index, and the rigidity of special Lagrangian cones over compact irreducible symmeric spaces of type $A$ obtained as minimal Legendrian orbits and over a minimal Legendrian $\mathit{SU}(2)$-orbit. We obtain the examples of stable and rigid special Lagrangian cones in higher dimensions. Moreover we discuss a relationship of these properties with the Hamiltonian stability of minimal Lagrangian submanifolds in complex projective spaces.

Submanifolds with splitting tangent sequence

A theorem of Van de Ven states that a projective submanifold of complex projective space whose holomorphic tangent bundle sequence splits holomorphically is necessarily a linear subspace. Note that the sequence always splits differentiably but in general not holomorphically. We are interested in generalizations to the case, when the ambient space is a homogeneous manifold different from projective space: quadrics, Grassmannians or abelian manifolds, for example. Split submanifolds are closely related to totally geodesic submanifolds.

Volume minimization and estimates for certain isotropic submanifolds in complex projective spaces

In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic RP 2m in CP n. Then it minimizes volume among the isotropic submanifolds in the same Z/2 homology class in CP n (but not among all submanifolds in this Z/2 homology class). Also the totally geodesic RP 2m−1 minimizes volume in its Hamiltonian deformation class in CP n. As a corollary we’ll give estimates for volumes of Lagrangian submanifolds in complete intersections in CP n.

Levi Form of CR Submanifolds of Maximal CR Dimension of Complex Space forms

Considering the Levi form on CR submanifolds of maximal CR dimension of complex space forms, we prove that on some remarkable real submanifolds of complex projective space the Levi form can never vanish and we determine all such submanifolds in the case when the ambient manifold is a complex Euclidean space.

Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces

A compact minimal Lagrangian submanifold immersed in a Kahler manifold is called Hamiltonian stable if the second variation of its volume is nonnegative under all Hamiltonian deformations. We study compact Hamiltonian stable minimal Lagrangian submanifolds with parallel second fundamental form embedded in complex projective spaces. Moreover, we completely determine Hamiltonian stability of all real forms in compact irreducible Hermitian symmetric spaces, which were classified previously by M. Takeuchi.

Certain application of an integral formula to CR submanifold of complex projective space

Certain application of an integral formula to CR submanifold of complex projective space

Higher rank curved Lie triples

A substantial proper submanifold M of a Riemannian symmetric space S is called a curved Lie triple if its tangent space at every point is invariant under the curvature tensor of S, i.e. a sub-Lie triple. E.g. any complex submanifold of complex projective space has this property. However, if the tangent Lie triple is irreducible and of higher rank, we show a certain rigidity using the holonomy theorem of Berger and Simons: M must be intrinsically locally symmetric. In fact we conjecture that M is an extrinsically symmetric isotropy orbit. We are able to prove this conjecture provided that a tangent space of M is also a tangent space of such an orbit.

Spin^c Structures and Scalar Curvature Estimates

In this note, we look at estimates for the scalar curvature k of a Riemannian manifold M which are related to spin^c Dirac operators: We show that one may not enlarge a Kaehler metric with positive Ricci curvature without making k smaller somewhere on M. We also give explicit upper bounds for min(k) for arbitrary Riemannian metrics on certain submanifolds of complex projective space. In certain cases, these estimates are sharp: we give examples where equality is obtained.