Complex Projective Space Research Articles

New curvature characterizations for spherical space forms and complex projective spaces

In this paper, we introduce a new positivity notion for curvature of Riemannian manifolds and obtain characterizations for spherical space forms and the complex projective space C P n \mathbb {C}\mathbb {P}^n .

A new class of distances on complex projective spaces

The complex projective space P(Cn) can be interpreted as the space of all quantum pure states of size n. A distance on this space, interesting from the perspective of quantum physics, can be induced from a classical distance defined on the n-point probability simplex by the ‘earth mover problem’. We show that this construction leads to a quantity satisfying the triangle inequality, which yields a true distance on complex projective space belonging to the family of quantum 2-Wasserstein distances.

Purity and hybridness of two tensors on a real hypersurface in complex projective space

Abstract On a real hypersurface M M in complex projective space, we can define two tensor fields of type (1, 2), A F ( k ) {A}_{F}^{\left(k)} and A T ( k ) {A}_{T}^{\left(k)} , associated with the shape operator A A of the real hypersurface, for any nonnull real number k k . Following Tachibana [Analytic tensor and its generalization, Tohoku Math. J. 12 (1960), 208–221], we study the purity and hybridness of such tensors with respect to A A .

Rational homotopy type of projectivization of the tangent bundle of certain spaces

PurposeThe paper aims to determine the rational homotopy type of the total space of projectivized bundles over complex projective spaces using Sullivan minimal models, providing insights into the algebraic structure of these spaces.Design/methodology/approachThe paper utilises techniques from Sullivan’s theory of minimal models to analyse the differential graded algebraic structure of projectivized bundles. It employs algebraic methods to compute the Sullivan minimal model of P(E) and establish relationships with the base space.FindingsThe paper determines the rational homotopy type of projectivized bundles over complex projective spaces. Of great interest is how the Chern classes of the fibre space and base space, play a critical role in determining the Sullivan model of P(E). We also provide the homogeneous space of P(E) when n = 2. Finally, we prove the formality of P(E) over a homogeneous space of equal rank.Research limitations/implicationsLimitations may include the complexity of computing minimal models for higher-dimensional bundles.Practical implicationsUnderstanding the rational homotopy type of projectivized bundles facilitates computations in algebraic topology and differential geometry, potentially aiding in applications such as topological data analysis and geometric modelling.Social implicationsWhile the direct social impact may be indirect, advancements in algebraic topology contribute to broader mathematical knowledge, which can underpin developments in science, engineering, and technology with societal benefits.Originality/valueThe paper’s originality lies in its application of Sullivan minimal models to determine the rational homotopy type of projectivized bundles over complex projective spaces, offering valuable insights into the algebraic structure of these spaces and their associated complex vector bundles.

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.

On the Algebraic Geometry of Multiview

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras.

A Simons-type Integral Inequality for Minimal Surfaces with Constant Kähler Angle in Complex Projective Spaces

A Simons-type Integral Inequality for Minimal Surfaces with Constant Kähler Angle in Complex Projective Spaces

Equivariant harmonic maps of the complex projective spaces into the quaternion projective spaces

We classify equivariant harmonic maps of the complex projective spaces CPm into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective line, we have one parameter family of those maps. (This result is already shown in [2] and [4] in other ways). However, when m≧2, we will obtain the rigidity results.

Hofer–Zehnder capacity of disc tangent bundles of projective spaces

AbstractWe compute the Hofer–Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension. The disc bundle is taken with respect to the Fubini–Study resp. round metric, but we can obtain explicit bounds for any other metric. In the case of the complex projective space, we also compute the Hofer–Zehnder capacity for the magnetically twisted case, where the twist is proportional to the Fubini–Study form. For arbitrary twists, we can still give explicit upper bounds.

Dynamical branes on expanding orbifold and complex projective space

We construct some new dynamical p-brane solutions to gravity theories on curved backgrounds. We discuss the relations between dynamical branes, a new time-dependent solution on complex projective space CPn, and the static p-branes on the orbifold Cn/Zn. Published by the American Physical Society 2024

Extensions of extremal Kähler submanifolds of complex projective spaces

Extensions of extremal Kähler submanifolds of complex projective spaces

Almost complex torus manifolds - a problem of Petrie type

The Petrie conjecture asserts that if a homotopy C P n \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of C P n \mathbb {CP}^n . Petrie proved this conjecture in the case where the manifold admits a T n T^n -action. An almost complex torus manifold is a 2 n 2n -dimensional compact connected almost complex manifold equipped with an effective T n T^n -action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2 n 2n -dimensional almost complex torus manifold M M only shares the Euler number with the complex projective space C P n \mathbb {CP}^n , the graph of M M agrees with the graph of a linear T n T^n -action on C P n \mathbb {CP}^n . Consequently, M M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch χ y \chi _y -genus, Todd genus, and signature as C P n \mathbb {CP}^n , endowed with the standard linear action. Furthermore, if M M is equivariantly formal, the equivariant cohomology and the Chern classes of M M and C P n \mathbb {CP}^n also agree.

Algebraic dependences of meromorphic mappings sharing few moving hyperplanes with truncated multiplicity

In this article, we will prove an algebraic dependence theorem for meromorphic mappings into a complex projective space sharing few moving hyperplanes with different truncated multiplicity. Moreover, we also consider the weaker condition: \[\nu _{(f,{a_i}), \le k_i}\le \nu _{(g,{a_i}), \le k_i} \text{ instead of } \nu _{(f,{a_i}), \le k_i}= \nu_{(g,{a_i}), \le k_i}\] for some moving hyperplanes among the given moving hyperplanes. In order to implement this, besides using the technique reported by S. D. Quang in (Two meromorphic mappings having the same inverse images of some moving hyperplanes with truncated multiplicity, Rocky Mountain J. Math., vol. 52, no. 1, pp. 263–273, 2022) we have to separate the 2n + 2 moving hyperplanes from the given p+1 moving hyperplanes. After that, we count multiples of the intersection of the inverse images of the mappings f and g sharing these moving hyperplanes. Our result is an improvement of many previous results in this topic.

Complex submanifolds of indefinite complex space forms

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al.

Characterization of invariant complex Finsler metrics on the complex Grassmann manifold

Let P:=U(p+q)/U(p)×U(q) be the complex Grassmann manifold and F:T1,0P→[0,+∞) be an arbitrary U(p+q)-invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a Kähler-Berwald metric which is not necessary Hermitian quadratic. We also prove that F is Hermitian quadratic if and only if F is a constant multiple of the canonical U(p+q)-invariant Kähler metric on P. In particular on the complex projective space CPn=U(n+1)/U(n)×U(1), there exists no U(n+1)-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on P which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the U(p+q)-invariant Kähler metrics on P, nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases.

On the finiteness of meromorphic mappings from a complete Kähler manifold into complex projective spaces

On the finiteness of meromorphic mappings from a complete Kähler manifold into complex projective spaces

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.

The string topology coproduct on complex and quaternionic projective space

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.

Legendre magnetic flows for geodesic spheres in a complex projective space

Legendre magnetic flows for geodesic spheres in a complex projective space

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.