Lagrangian submanifolds in complex projective space CP n

We first prove a basic theorem with respect to the moving frame along a Lagrangian immersion into the complex projective space CP ⁿ . Applying this theorem, we study the rigidity problem of Lagrangian submanifolds in CP ⁿ .     

On some real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?H ⁿ ,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H ₁ ²ⁿ⁺¹ .Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ?H ⁿ .     

关于复射影空间的三维全实极小子流形

本文给出复射影空间中三维紧致全实极小子流形的Ricci曲率和数量曲率的鞭些拼挤定理.特别是证得:若M³是CP³的紧致全实极小子流形且它的Ricci曲率大于1/6,则M³是全测地的.     

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \mathfrak{D}^ \bot-parallel structure Jacobi operator

Regarding the generalized Tanaka-Webster connection, we considered a new notion of \(\mathfrak{D}^ \bot\) -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G ₂(? ^m+2) and proved that a real hypersurface in G ₂(? ^m+2) with generalized Tanaka-Webster \(\mathfrak{D}^ \bot\) -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ?P ⁿ in G ₂(? ^m+2), where m = 2n.     

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to minus n+2 hyperplanes.Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accommodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of h^n,o(V) spheres Sⁿ.     

A characterization of a certain real hypersurface of type (A<Subscript>2</Subscript>) in a complex projective space

In the class of real hypersurfaces M ^2n?1 isometrically immersed into a nonflat complex space form \(\widetilde {{M_n}}\left( c \right)\) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ?P ⁿ (c) or a complex hyperbolic space ?H ⁿ (c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in \(\widetilde {{M_n}}\left( c \right)\), we consider a certain real hypersurface of type (A₂) in ?P ⁿ (c) and give a geometric characterization of this Hopf manifold.     

On Fubini-Study form and Reidemeister torsion

For a general chain complex (C_*,_∂*), one can associate the Reidemeister torsion of it. We prove the relation between Reidemeister torsion and Fubini-Study 2-form ωFS of the complex projective space CPn.     

Double solid twistor spaces: the case of arbitrary signature

In a recent paper ([9]) we constructed a series of new Moishezon twistor spaces which are a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on n CP ² for arbitrary n≥3, which can be regarded as a generalization of the twistor spaces of ‘double solid type’ on 3CP ² studied by Kreußler, Kurke, Poon and the author. Similarly to the twistor spaces of ‘double solid type’ on 3CP ², projective models of the present twistor spaces have a natural structure of double covering of a CP ²-bundle over CP ¹. We explicitly give a defining polynomial of the branch divisor of the double covering, whose restriction to fibers is degree four. If n≥4 these are new twistor spaces, to the best of the author’s knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from [9], the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces.     

Randers metrics on CP2

The complex projective space CP² is a classical example of Einstein metric in Riemannian geometry. Moreover, beside this property, it has other interesting geometrical properties: it is a symmetric space, and a C manifold. We would like to know whether there is an Einstein metric of Randers type on CP² with similar properties. Based on some the generalization of Zermelo navigation problem for Finsler manifolds we construct such Randers metric on CP² and study some of its geometrical properties.     

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 CP ^m (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.     

The Hamiltonian dynamics over real hypersurfaces in Kaehler manifolds

Let X be a Kaehler manifold with complex dimension n. Let ωX be its Kaehler form. Let M be a strongly pseudo convex real hypersurface in X. For this hypersurface, the deformation theory of CR structures is successfully developed. And we find that H¹(M,T^′) (the T^′-valued Kohn-Rossi cohomology) is the Zariski tangent space of the versal family. In this paper, the geometrical meaning of H¹(M,O) is studied, and we propose to study displacements of the real hypersurface, which preserves the type of the differential form, ωX, over CR structures, on M, infinitesimally.     

Compact minimal hypersurfaces with index one in the real projective space

Let M ⁿ be a compact (two-sided) minimal hypersurface in a Riemannian manifold . It is a simple fact that if has positive Ricci curvature then M cannot be stable (i.e. its Jacobi operator L has index at least one). If is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.?We prove that if is the real projective space , obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface obtained as the product of two spheres of dimensions n ₁, n ₂ and radius R ₁, R ₂, with and . Received: June 6, 1998     

Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space

We prove that there do not exist CR submanifolds Mⁿ of maximal CR dimension of a complex projective space \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex projective subspace \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) of \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) such that M is a real hypersurface of \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\).     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.     

Nonexistence of Levi-degenerate hypersurfaces of constant signature in CP n

Let M be a smooth hypersurface of constant signature in CP ⁿ , n≥3. We prove the regularity for on M in bidegree (0,1). As a consequence, we show that there exists no smooth hypersurface in CP ⁿ , n≥3, whose Levi form has at least two zero-eigenvalues.     

Note on equivariant harmonic maps in complex projective spaces

We give a simple criterion for equivariant harmonic maps into complex projective spaces CP ⁿ . As an application of the criterion, we give examples of equivariant harmonic cylinders. We also give examples of non-equivariant harmonic cylinders as perturbations of equivariant harmonic cylinders.     

On spectral geometry of minimal parallel submanifolds

We consider the problem of characterizing some minimal submanifolds using the spectrum⁰Spec of the Laplace-Beltrami operator acting on fucntions. In particular we characterize then-dimensional compact minimal totally real parallel submanifolds immersed in the complex projective spaceCP ⁿ, 3≤n≤6, by their⁰Spec in the class of all compact totally real minimal submanifolds ofCP ⁿ. Moreover, we characterize the Clifford torus by its⁰Spec in the class of all compact minimal submanifolds of the Euclidean sphereS ⁿ⁺¹(1). Authors supported by funds of the University of Lecce and the M.U.R.S.T.     

Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.