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We denote by $ \Delta _\nu $ Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues $ \beta _m, \ m\in \mathbb {Z}_+ $ β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of $ \Delta _\nu $ Δ ν . Using a suitable polynomial decomposition of the multiplicity of each $ \beta _m $ β m , we write down a trace formula for the heat operator associated with $ \Delta _\nu $ Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as $ t\searrow 0^+ $ t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with $ \Delta _\nu $ Δ ν . [ABSTRACT FROM AUTHOR]

Recently, Pipoli and Sinestrari [Pipoli, G. and Sinestrari, C., Mean curvature flow of pinched submanifolds of ℝℙn, Comm. Anal. Geom., 25, 2017, 799–846] initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space ℝℙm. The purpose of this paper is to develop the work due to Pipoli and Sinestrari, and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space. Namely, the authors prove that if the initial submanifold in ℝℙm satisfies a suitable pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as t → ∞. Consequently, they obtain a differentiable sphere theorem for submanifolds in the complex projective space. [ABSTRACT FROM AUTHOR]

Let M be a real hypersurface of a complex projective space. For any operator B on M and any nonnull real number k, we can define two tensor fields of type (1,2) on M, B F (k) and B T (k) . We will classify real hypersurfaces in complex projective space for which B F (k) and B T (k) either take values in the maximal holomorphic distribution D or are parallel to the structure vector field ξ , in the particular case of B = A , where A denotes the shape operator of M. We also introduce the concept of A F (k) and A T (k) being D -recurrent and classify real hypersurfaces such that either A F (k) or A T (k) are D -recurrent. [ABSTRACT FROM AUTHOR]

We denote by Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues β m ,   m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of Δ ν . Using a suitable polynomial decomposition of the multiplicity of each β m , we write down a trace formula for the heat operator associated with Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with Δ ν . [ABSTRACT FROM AUTHOR]

We define a Euler characteristic χ (X , G) for a finite cell complex X with a finite group G acting cellularly on it. Then, each K i (X) (a complex vector space with basis the i-cells of X) is a representation of G, and we define χ (X , G) to be the alternating sum of the representations K i (X) , as elements of the representation ring R (G) of G. By adapting the ordinary proof that the alternating sum of the dimensions of the chain complexes is equal to the alternating sum of the dimensions of the homology groups, we prove that there is another definition of χ (X , G) with the alternating sum of the representations H i (X) , again as elements of the representation ring R (G) . We also show that the character of this virtual representation χ (X , G) , with respect to a given element g, is just the ordinary Euler characteristic of the fixed-point set by this element. Finally, we give a topological proof of a version of Artin's induction theorem. More precisely, we show that, if G is a group with an irreducible representation of dimension greater than 1, then each character of G is a linear combination with rational coefficients of characters induced up from characters of proper subgroups of G. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

When we consider an error model in a quantum computing system, we assume a parametric model where a prepared qubit belongs. Keeping this in mind, we focus on the evaluation of the amount of information we obtain when we know the system belongs to the model within the parameter range. Excluding classical fluctuations, uncertainty still remains in the system. We propose an information quantity called purely quantum information to evaluate this and give it an operational meaning. For the qubit case, it is relevant to the facility location problem on the unit sphere, which is well known in operations research. For general cases, we extend this to the facility location problem in complex projective spaces. Purely quantum information reflects the uncertainty of a quantum system and is related to the minimum entropy rather than the von Neumann entropy. [ABSTRACT FROM AUTHOR]

We consider real hypersurfaces M in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull real number k and any symmetric tensor field of type (1,1) L on M we can define a tensor field of type (1,2) on M, L (k) F, related to both connections. We study symmetry and skewsymmetry of the tensor A (k) F associated to the shape operatorA of M [ABSTRACT FROM AUTHOR]

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces. [ABSTRACT FROM AUTHOR]

Let π : S 2 n + 1 → C P n be the Hopf map and let ϕ be a totally real immersion of a k (≥ 3) -dimensional simply connected manifold Σ into C P n . It is well known that there exists an isotropic lift ϕ ¯ into S 2 n + 1 preserving the second fundamental form. Using this isotropic lift, we obtain a vanishing theorem for of L p harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space provided the L k norm of the traceless second fundamental form Φ is sufficiently small. Moreover, we prove that if the L k norm of Φ is finite, then the dimension of L p harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space is finite. As consequences, we obtain a vanishing theorem and a finiteness result for L 2 harmonic 1-forms on a complete noncompact minimal Lagrangian submanifold in a complex projective space. [ABSTRACT FROM AUTHOR]

Quasitoric manifolds, introduced by Davis and Januskiewicz in their seminal paper in 1991, are topological generalizations of smooth complex projective spaces. In 1992, Banchoff and K¨uhnel constructed a 10-vertex equilibrium triangulations of CP2. We generalize this construction for quasitoric manifolds providing an algorithm and construct equilibrium triangulation of several quasitoric manifolds. In some cases our construction give vertex minimal equilibrium triangulations. [ABSTRACT FROM AUTHOR]

We show that ruled real hypersurfaces with constant mean curvature in the complex projective and hyperbolic spaces must be minimal. This provides their classification, by virtue of a result of Lohnherr and Reckziegel. [ABSTRACT FROM AUTHOR]

Abstract We study the automorphism group of curves and surfaces in CP 3 with respect to the conformal group, i.e. the group of all A ∈ P G L (4 , ℂ) commuting with the anti-holomorphic involution j defined by j ((z 0 : z 1 : z 2 : z 3)) = (− z ¯ 1 : z ¯ 0 : z ¯ 3 : − z ¯ 2). For some singular surfaces we check when this group is finite. Among the singular surfaces we handle there are: (1) certain cones; (2) surfaces X containing no line and with j (X) ≠ X ; (3) surfaces containing only finitely many, k , twistor lines with k ≥ 3. In many cases the proofs need results on conformal automorphisms of singular curves. [ABSTRACT FROM AUTHOR]

We prove that there do not exist CR submanifolds Mn of maximal CR dimension of a complex projective space Pn+p2(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 Pn+12(C) of Pn+p2(C) such that M is a real hypersurface of Pn+12(C). [ABSTRACT FROM AUTHOR]

In this paper we give explicit formulae for the determinants of the Laplacians on the n -dimensional complex projective spaces P n ( C ) ( n ≥ 1 ). This explicit computation of the determinants of the Laplacians Δ P n ( C ) is appearing for the first time in the Literature. We also discuss in clear terms other spectral invariants on P n ( C ) – the Minakshisundaram–Pleijel coefficients and Minakshisundaram–Pleijel zeta functions. [ABSTRACT FROM AUTHOR]

The equivariant CR minimal immersions from the round 3-sphere S3 into the complex projective space CPn have been classified by the third author explicitly (Li in J Lond Math Soc 68:223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant and that all geometric properties of S3=SU(2) endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra su(2), we establish an extended classification theorem for equivariant CR minimal immersions from the 3-sphere S3 into CPn without the assumption of constant sectional curvatures. [ABSTRACT FROM AUTHOR]

In this paper, we study the isolation phenomena of Einstein manifolds from the viewpoint of submanifolds theory. First, for locally strongly convex Einstein affine hyperspheres we prove a rigidity theorem and as its direct consequence we establish a unified affine differential geometric characterization of the noncompact symmetric spaces E6(−26)/F4 and SL(m,R)/SO(m), SL(m,C)/SU(m), SU∗ (2m)/Sp(m) for each m ≥ 3. Second and analogously, for Einstein Lagrangian minimal submanifolds of the complex projective space CPn(4) with constant holomorphic sectional curvature 4, we prove a similar rigidity theorem and as its direct consequence we establish a unified differential geometric characterization of the compact symmetric spaces E6/F4 and SU(m)/SO(m), SU(m), SU(2m)/Sp(m) for each m ≥ 3. [ABSTRACT FROM AUTHOR]

The present paper deals with the classification of a complete totally submanifold of a complex projective space by applying Bochner formula. [ABSTRACT FROM AUTHOR]

Let f : C → Pn(C) be a holomorphic curves with hyperorder strictly less than 1, and algebraically nondegenerate over the field P1c which consists of c-periodic meromorphic functions on C. Let {Qj}qj=1 be fixed or c-periodic slowly moving hypersurfaces with degree dj (j ∊ {1,...,q}) in (weakly) N-subgeneral position in Pn(C). In this paper, we prove a difference version of the second main theorem for f intersecting {Qj}qj=1 by using the Casorati determinant. A difference counterpart of the truncated second main theorem is also obtained. Our results extend the second main theorems for differences with fixed hyperplanes [9] or c-periodic slowly moving hyperplanes [10]. [ABSTRACT FROM AUTHOR]

On a real hypersurface M in a complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the k -th g-Tanaka–Webster connection. Associated to g-Tanaka–Webster connection we can define a differential operator of first order. We classify real hypersurfaces such that both the Lie derivative and this differential operator, either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the structure Jacobi operator of M . [ABSTRACT FROM AUTHOR]

The objective of this paper is to provide information on the set of periodic points of a continuous self-map defined in the following compact spaces: Sn (the n-dimensional sphere), Sn ×Sm (the product space of the n-dimensionalwith them-dimensional spheres),CPn (the n-dimensional complex projective space) andHPn (the n-dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces. [ABSTRACT FROM AUTHOR]

We consider real hypersurfaces M in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections and classify them when the covariant derivatives associated with both connections, either in the direction of the structure vector field or any direction of the maximal holomorphic distribution, coincide when applying to the Riemannian curvature tensor of the real hypersurface. [ABSTRACT FROM AUTHOR]

Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed. [ABSTRACT FROM AUTHOR]

We study parallel and symmetric second order tensor fields on CR submanifolds of maximal CR dimension of the complex projective space. Under some natural conditions, these tensors are scalar multiples of the metric; an example of submanifold satisfying these assumptions is the geodesic hypersphere in P n+1/2 (ℂ). The result is used for Ricci solitons on these CR submanifolds. [ABSTRACT FROM AUTHOR]

Pseudo Ricci symmetric real hypersurfaces of a complex projective space are classified and it is proved that there are no pseudo Ricci symmetric real hypersurfaces of the complex projective space CPn for which the vector field ξ from the almost contact metric structure(?, ξ, η, g) is a principal curvature vector field. [ABSTRACT FROM AUTHOR]

This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by Gleason to prove his celebrated theorem. In particular, the problem of associating quantum states with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical-like observables (i.e. real scalar functions on the projective space) representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical-like objects representing quantum ones result to be frame functions. The requirements of covariance and (convex) linearity play a central role in the proof of those theorems. A new characterization of classical-like observables describing quantum observables is presented, together with a geometric description of the -algebra structure of the set of quantum observables in terms of classical-like ones. [ABSTRACT FROM AUTHOR]

Irreducible isoparametric foliations of arbitrary codimension q on complex projective spaces ℂPn are classified, for (q, n) ≠ (1, 15). Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations, even of higher codimension. In fact, every irreducible isoparametric foliation on ℂPn is homogeneous if and only if n + 1 is prime. The main tool developed in this work is a method to study singular Riemannian foliations with closed leaves on complex projective spaces. This method is based on a certain graph that generalizes extended Vogan diagrams of inner symmetric spaces. [ABSTRACT FROM AUTHOR]

In this paper the notion of Lie-Codazzi type of a tensor field T of type (1,1) on real hypersurfaces, which generalizes the notion of Lie parallel, is introduced. Real hypersurfaces in non-flat complex space forms, whose structure Jacobi operator is of Lie-Codazzi type, are studied. More precisely, the non-existence of such real hypersurfaces is proved. [ABSTRACT FROM AUTHOR]

We prove a classification theorem for a submanifold of real codimension two of a complex projective space, which is not its totally geodesic complex hypersurface, under the condition h ( F X , Y ) + h ( X , F Y ) = 0 , where h is the second fundamental form of the submanifold and F is the endomorphism induced from the almost complex structure J on the tangent bundle of the submanifold. [ABSTRACT FROM AUTHOR]

In finite dimension (at least), Quantum Mechanics can be formulated as a proper Hamiltonian theory where a notion of phase space is given by the projective space P(H) constructed on the Hilbert space H of the considered quantum theory. It is well-known P(H) can be equipped with a structure of Kähler manifold, in particular we have a symplectic form and a Poisson structure; Quantum dynamics can be described in terms of a Hamiltonian vector field on P(H). In this paper, exploiting the notion and properties of so-called frame functions, I describe a general prescription for associating quantum observables to real functions on P(H), classical-like observables, and quantum states to probability densities on P(H), Liouville densities, in order to obtain a complete and meaningful Hamiltonian formulation of a finite-dimensional quantum theory. [ABSTRACT FROM AUTHOR]

The generalization of the Poincaré sphere to $N \ge 2$ modes is the ( $N-1$ )-dimensional complex projective space CP( $N-1$ ). There is a minimal set of $2N-2$ Stokes vector components that determine the coherent multimode optical field. These are obtained from the inverse stereographic projection of coordinate hyperplanes in CP( $N-1$ ) into a $2N-2$ sphere, just as in the $N=2$ case. We derive $N$ -mode analogs of Poole’s optical fiber polarization-mode dispersion (PMD) equations that involve only $2N-2$ independent variables. This is achieved by means of an explicit generalized coherent state representation of the optical field, which enables the components of the PMD vector to be expressed in terms of the optical state and its frequency derivatives. Poole’s equations describe mode coupling as a flow on CP( $N-1$ ). We give general constraints on the mode-coupling matrix and Stokes vector components. The group delay operator is shown to be a rank-2 perturbation of a diagonal matrix. [ABSTRACT FROM AUTHOR]

Let H ⊂ ℙn be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian G(n + 1, n). Assuming H has a global defining function, we prove H is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension 2n - 2 or dimension 2n - 4. If the singular set is of dimension 2n - 4, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ℙn with a meromorphic (rational of degree 1) first integral. In this case, H is in some sense simply a complex cone over an algebraic curve in ℙ1. Similarly if H has a degenerate singularity, then H is also algebraic. If the dimension of the singular set is 2n - 2 and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ℙ2 of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ℙ2. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold. [ABSTRACT FROM AUTHOR]

In the paper, we study the existence of holomorphic isometric immersions from nonhomogeneous Kähler–Einstein manifolds into infinite dimensional complex projective space. It can also be regarded as an application of explicit solutions of complex Monge–Ampère equations on some pseudoconvex domains. [ABSTRACT FROM AUTHOR]

In this paper the notion of *-Ricci soliton is introduced and real hypersurfaces in non-flat complex space forms admitting a *-Ricci soliton with potential vector field being the structure vector field ξ are studied. At the end of the paper discussion on this new notion and ideas for further work are presented. [ABSTRACT FROM AUTHOR]

On a real hypersurface M in complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka-Webster connection. We classify real hypersurfaces such that both the Lie derivative associated to the Levi-Civita connection and the kth g-Tanaka-Webster derivative in the direction of the structure vector field ξ coincide when we apply them to either the shape operator or the structure Jacobi operator of M. [ABSTRACT FROM AUTHOR]

We provide the quantum treatment of the relational quadrilateral. The underlying reduced configuration spaces are ℂℙ2 and the cone over this. We consider exact free and isotropic HO potential cases and perturbations about these. Moreover, our purely relational kinematical quantization is distinct from the usual one for ℂℙ2, which turns out to carry absolutist connotations instead. Thus, this paper is the first to note absolute-versus-relational motion distinctions at the kinematical rather than dynamical level. It is also an example of value to the discussion of kinematical quantization along the lines of Isham, 1984. The relational quadrilateral is the simplest RPM whose mathematics is not standard in atomic physics (the triangle and four particles on a line are both based on 핊2 and ℝ3 mathematics). It is far more typical of the general quantum relational N-a-gon than the previously studied case of the relational triangle. We consider useful integrals as regards perturbation theory and the peaking interpretation of quantum cosmology. We subsequently consider problem of time (PoT) applications of this: quantum Kuchař beables, the Machian version of the semiclassical approach and the timeless naïve Schrödinger interpretation. These go toward extending the combined Machian semiclassical-Histories-Timeless Approach of [ Int. J. Mod. Phys. D 23 (2014) 1450014] to the case of the quadrilateral, which will be treated in subsequent papers. [ABSTRACT FROM AUTHOR]

We characterize geodesic hyperspheres of radius r such that cot²(r) = 1/2 as the unique real hypersurfaces in complex projective space whose structure Jacobi operator satisfies a pair of conditions. [ABSTRACT FROM AUTHOR]

Abstract: Let be the space of degree holomorphic maps from a compact Riemann surface to . In the case and , the metric on was computed exactly by Speight. In this paper, the Ricci curvature tensor and the scalar curvature on are determined explicitly for . An exact direct computation of the Einstein–Hilbert action with respect to the metric on is made and shown to coincide with a formula conjectured by Baptista. [Copyright &y& Elsevier]

According to the famous theorem of Wei-Liang Chow (1949) each analytic subset of ℙm is a projective algebraic set. In the case of an analytic subset of ℂm there are different criteria in order for it to be algebraic. The present author proves three criteria in the opposite direction. [ABSTRACT FROM AUTHOR]

Abstract: We propose a conjugate gradient type optimization technique for the computation of the Karcher mean on the set of complex linear subspaces of fixed dimension, modeled by the so-called Grassmannian. The identification of the Grassmannian with Hermitian projection matrices allows an accessible introduction of the geometric concepts required for an intrinsic conjugate gradient method. In particular, proper definitions of geodesics, parallel transport, and the Riemannian gradient of the Karcher mean function are presented. We provide an efficient step-size selection for the special case of one dimensional complex subspaces and illustrate how the method can be employed for blind identification via numerical experiments. [Copyright &y& Elsevier]

The symmetric group $\operatorname{Sym}(d)$ acts on the Cartesian product ( S) by coordinate permutation, and the quotient space $(S^{2})^{d}/\operatorname{Sym}(d)$ is homeomorphic to the complex projective space ℂ P. We used the case d=2 of this fact to construct a 10-vertex triangulation of ℂ P earlier. In this paper, we have constructed a 124-vertex simplicial subdivision $(S^{2})^{3}_{124}$ of the 64-vertex standard cellulation $(S^{2}_{4})^{3}$ of ( S), such that the $\operatorname{Sym}(3)$-action on this cellulation naturally extends to an action on $(S^{2})^{3}_{124}$. Further, the $\operatorname{Sym}(3)$-action on $(S^{2})^{3}_{124}$ is 'good', so that the quotient simplicial complex $(S^{2})^{3}_{124}/\operatorname{Sym}(3)$ is a 30-vertex triangulation $\mathbb{C}P^{3}_{30}$ of ℂ P. In other words, we have constructed a simplicial realization $(S^{2})^{3}_{124} \to\mathbb{C} P^{3}_{30}$ of the branched covering ( S)→ℂ P. [ABSTRACT FROM AUTHOR]

We consider the symmetric product spaces of closed manifolds. We introduce some geometric invariants and the topological properties of symmetric product spaces via the symmetric invariant ones of product spaces and apply to Gromov-Witten invariants. We examine the symmetric product spaces of the complex projective line, their Gromov-Witten invariants and compute the generating series induced by their Gromov-Witten invariants. [ABSTRACT FROM AUTHOR]

A quantum system can be entirely described by the Kähler structure of the projective space ${\mathbb P}(\mathcal{H})$ associated to the Hilbert space $\mathcal{H}$ of possible states; this is the so-called geometrical formulation of quantum mechanics. In this paper, we give an explicit link between the geometrical formulation (of finite dimensional quantum systems) and statistics through the natural geometry of the space $\mathcal{P}_{n}^{\times}$ of non-vanishing probabilities $p\, {:}\, E_{n}\rightarrow {\mathbb R}_{+}^{*}$ defined on a finite set En: = {x1, ..., xn}. More precisely, we use the Fisher metric gF and the exponential connection ∇(1) (both being natural statistical objects living on $\mathcal{P}_{n}^{\times}$) to construct, via the Dombrowski splitting theorem, a Kähler structure on $T\mathcal{P}_{n}^{\times}$ which has the property that it induces the natural Kähler structure of a suitably chosen open dense subset of ℙ(ℂn). As a direct physical consequence, a significant part of the quantum mechanical formalism (in finite dimension) is encoded in the triple $(\mathcal{P}_{n}^{\times}, g_{F}, \nabla^{(1)})$. [ABSTRACT FROM AUTHOR]

In this paper we investigate the quantum cohomologies of symmetric products of Kähler manifolds. To do this we study the moduli space of product space and symmetric group action on it, Gromov-Witten invariant and relative Gromov-Witten invariant. Also we investigate the relations between symmetric invariant properties on the products space and the corresponding ones on the symmetric product. As an example we examine the symmetric product of k copies complex projective line ℙ1, which is the k-dimensional complex projective space ℙk. [ABSTRACT FROM AUTHOR]

Abstract: We characterize the Lefschetz periodic point free self-continuous maps on the following connected compact manifolds: the n-dimensional complex projective space, the n-dimensional quaternion projective space, the n-dimensional sphere and the product space of the p-dimensional with the q-dimensional spheres. [Copyright &y& Elsevier]

We study the noncommutative geometrical structures of quantum entangled states. We show that the space of a pure entangled state is a noncommutative space. In particular we show that by rewriting the coordinate ring of a conifold or the Segre variety we can get a q-deformed relation in noncommutative geometry. We generalized our construction into a multi-qubit state. We also in detail discuss the noncommutative geometrical structure of a three-qubit state. [ABSTRACT FROM AUTHOR]