Narrow Results

Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of Kähler manifolds, we prove a combinatorial formula for powers of the complex Laplacian and use it to derive an explicit graph-theoretic formula for the numerics in heat coefficients as a linear combination of metric jets based on Polterovich’s formula.

This paper proves that the universal covering of a compact Kähler manifold with small positive sectional curvature in a certain sense is contractible.

In this note, we will show that the fundamental group of any negatively δ-pinched manifold cannot be the fundamental group of a quasi-compact Kähler manifold. As a consequence of our proof, we also show that any nonuniform lattice in F_4(-20) cannot be the fundamental group of a quasi-compact Kähler manifold. The corresponding result for uniform lattices was proved by Carlson and Hernández [3]. Finally, we follow Gromov and Thurston [6] to give some examples of negatively δ-pinched manifolds of finite volume which, as topological manifolds, admit no hyperbolic metric with finite volume under any smooth structure. This shows that our result for δ-pinched manifolds is a nontrivial generalization of the fact that no nonuniform lattice in SO(n,1)(n≥3) is the fundamental group of a quasi-compact Kähler manifold [21].

Let M be a compact complex manifold with a positive holomorphic line bundle L, and K be its canonical line bundle. We give some sufficient conditions for the non-vanishing of H⁰(M, K + L). We also show that the criterion can be applied to interesting classes of examples including all compact locally hermitian symmetric spaces of non-compact type, Mostow–Siu [10] surfaces, Kähler threefolds given by Deraux [3] and examples of Zheng [17].

We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kähler manifolds.

Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball ${𝔹}^m(R_0)$ in ${ℂ}^m$ ($0<R_0\le +\infty$). The first purpose of this paper is to establish non-integrated defect relations for linearly non-degenerate meromorphic mappings intersecting a family of hyperplanes in subgeneral position. Also, the defect relation for the mappings with only one hypersurface, which is the sum of some hyperplanes, is considered. Our second aim is to establish a uniqueness theorem for two meromorphic mappings sharing such family of hyperplanes. Our results generalize and improve the previous results in this area.

In this paper, we give some results on the algebraic dependence of meromorphic mappings from a complete Kähler manifold whose universal covering is biholomorphic to a ball in ${ℂ}^m$ into ${\mathbb{P}}^n(ℂ)$ sharing few hyperplanes in subgeneral position with truncated multiplicities to level $p$, where all zeros with multiplicities greater than certain values do not need to be counted. Our results are generalizations and improvements of the previous results in recent times.

Let $n\ge 3$. In this paper, we show that for any abelian subgroup $G$ of $s_n$ the crystallographic group $B_n/[P_n,P_n]$ has Bieberbach subgroups ${\mathrm{\Gamma }}_G$ with holonomy group $G$. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group ${\mathrm{\Gamma }}_G$. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of ${\mathrm{\Gamma }}_G$ and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold ${{𝒳}}_{{\mathrm{\Gamma }}_G}$ with fundamental group the Bieberbach group ${\mathrm{\Gamma }}_G\le B_n/[P_n,P_n]$.

Consider the Kähler–Ricci flow with finite time singularities over any closed Kähler manifold. We prove the existence of the flow limit in the sense of current toward the time of singularity. This answers affirmatively a problem raised by Tian in [New results and problems on Kähler–Ricci flow, Astérisque322 (2008) 71–92] on the uniqueness of the weak limit from sequential convergence construction. The notion of minimal singularity introduced by Demailly in the study of positive current comes up naturally. We also provide some discussion on the infinite time singularity case for comparison. The consideration can be applied to more flexible evolution equation of Kähler–Ricci flow type for any cohomology class. The study is related to general conjectures on the singularities of Kähler–Ricci flows.

As a generalization of semi-invariant submersions, we introduce conformal semi-invariant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which arise from the definition of a conformal submersion and show that there are certain product structures on the total space of a conformal semi-invariant submersion. Moreover, we also check the harmonicity of such submersions and find necessary and sufficient conditions of a conformal semi-invariant submersion to be totally geodesic.

Let X be strongly q-convex domain of an n-dimensional Kähler manifold M and E be a holomorphic vector bundle over M. Then, if E satisfies certain positivity conditions, we prove vanishing theorems for the -cohomology groups of X with values in E.

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrödinger framework from this perspective and provide a description of the Weyl–Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

As a generalization of isometric immersions and Riemannian submersions, Riemannian maps were introduced by Fischer [Riemannian maps between Riemannian manifolds, Contemp. Math.132 (1992) 331–366]. It is known that a real valued Riemannian map satisfies the eikonal equation which provides a bridge between physical optics and geometrical optics. In this paper, we introduce invariant and anti-invariant Riemannian maps between Riemannian manifolds and almost Hermitian manifolds as a generalization of invariant immersions and totally real immersions, respectively. Then we give examples, present a characterization and obtain a geometric characterization of harmonic invariant Riemannian maps in terms of the distributions which are involved in the definition of such maps. We also give a decomposition theorem by using the existence of invariant Riemannian maps to Kähler manifolds. Moreover, we study anti-invariant Riemannian maps, give examples and obtain a classification theorem for umbilical anti-invariant Riemannian maps.

The aim of this paper is to improve Hamiltonian energy equations for the complex jet bundles using Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated in Hamiltonian energy equations and applications to complex bundle structures. The improved Hamiltonian energy equations have been applied to the presented example in order to test its performance. Moreover, we have presented a new concept of velocity and time dimensions for energy movement equations. Results showed that imaginary Hamiltonian values are opposed to real Hamiltonian values in 2n dimensions. Finally, this study showed a physical application and interpretation of velocity and time dimensions in Hamiltonian energy equations for given examples.

In this study, we present a new analogue of Euler–Lagrange and Hamilton equations on an almost Kähler model of a Finsler manifold. Also, we give some corollaries about the related mechanical systems and equations.

This study proposes to improve the Lagrangian energy equations for instructed complex jet bundles on Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated and applications to complex bundle structures are carried out. The energy equations have been applied to the numerical example in order to test its performance. Moreover, velocity and time dimensions for energy movement equations have been presented as a new concept. This study shows some physical applications of those equations and interpretations are made. Results show that the imaginary coordinates are same as real coordinates. One of the interesting conclusion of this study is that the Lagrangian energy of a movement particle is static when the particle moves in a large velocity.

We study the pseudoduality transformations in two-dimensional N = (2, 2) sigma models on Kähler manifolds. We show that structures on the target space can be transformed into the pseudodual manifolds by means of (anti)holomorphic preserving mapping. This map requires that torsions related to individual spaces and riemann connection on pseudodual manifold must vanish. We also consider holomorphic isometries which puts additional constraints on the pseudoduality.

We study the Yang–Mills functional for principal fiber bundles with structure group a compact Lie group K over a Kähler manifold. In particular, we analyze the absolute minimizers for this functional and prove that they are exactly the Einstein K-connections. By means of the structure of the Yang–Mills functional at an absolute minimum, we prove that the characteristic classes of a principal K-bundle which admits an Einstein connection satisfy two inequalities. One of them is a generalization of the Bogomolov inequality whereas the other is an inequality related to the center of the structure group. Therefore, this way we offer a new and natural proof of the Bogomolov inequality that helps understanding its origin. Finally, in view of the Hitchin–Kobayashi correspondence we prove that every (poly-)stable principal K_ℂ-bundle has to satisfy this generalized Bogomolov type inequality.

In this paper we present the induced representation of SO(2N) canonical transformation group and introduce coset variables. We give a derivation of the time-dependent Hartree–Bogoliubov (TDHB) equation on the Kähler coset space from the Euler–Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of time-dependent Riccati–Hartree–Bogoliubov equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO(2N) matrix based on the nonlinear Möbius transformation together with the geodesic flow on the manifold.