Moduli of Quaternionic Superminimal Immersions of 2-Spheres into Quaternionic Projective Spaces

The aim of this paper is to describe the moduli spaces of degree d quaternionic superminimal maps from 2-spheres to quaternionic projective spaces HPn. We show that such moduli spaces have the structure of projectivized fibre products and are connected quasi-projective varieties of dimension 2nd + 2n + 2. This generalizes known results for spaces of harmonic 2-spheres in S⁴.     

The Magnetic Geodesic Flow in a Homogeneous Field on the Complex Projective Space

We prove commutative integrability of the Hamilton system on the tangent bundle of the complex projective space whose Hamiltonian coincides with the Hamiltonian of the geodesic flow and the Poisson bracket deforms due to addition of the Fubini–Study form to the standard symplectic form.     

The Nullity of a Compact Minimal Real Hypersurface in a Quaternion Projective Space

We will prove that the nullities of compact minimal real hypersurfaces in a quaternion projective space B Pn are bounded from below by 4n, and those with nullity 4n must be minimal geodesic hyperspheres.     

四元数射影空间的全实伪脐子流形

给出了四元数射影空间的紧致全实伪脐子流形的关于第二基本形式长度的一个Ｐｉｎｃｈｉｎｇ定理．     

Quantum Mechanics in Riemannian Space: Different Approaches to Quantization of the Geodesic Motion Compared

We compare different approaches to the construction of the quantum mechanics of a particle in the general Riemannian space and space–time via quantization of motion along geodesic lines. We briefly review different quantization formalisms and the difficulties arising in their application to geodesic motion in a Riemannian configuration space. We then consider canonical, semiclassical (Pauli–De Witt), and Feynman (path-integral) formalisms in more detail and compare the quantum Hamiltonians of a particle arising in these models in the case of a static, topological elementary Riemannian configuration space. This allows selecting a unique ordering rule for the coordinate and momentum operators in the canonical formalism and a unique definition of the path integral that eliminates a part of the arbitrariness involved in the construction of the quantum mechanics of a particle in the Riemannian space. We also propose a geometric explanation of another main problem in quantization, the noninvariance of the quantum Hamiltonian and the path integral under configuration space diffeomorphisms.     

On the Twistor Spaces of Almost Hermitian Manifolds

We study Grassmannian bundles G_k(M) of analytical 2k-planes over an almost Hermitian manifold M²ⁿ, from the point of view of the generalized twistor spaces of [13], and with the method of the moving frame [9]. G₁(M⁴) is the classical twistor space. We find four distinguished almost Hermitian structures, one of them being that of [13], and discuss their integrability and Kählerianity. For n=2, we compute the corresponding Hermitian connections, and derive consequences about the corresponding first Chern classes.     

Congruence of hypersurfaces inS 6 and inC P n

The invariants needed to decide when a pair of hypersurfaces ofS ⁶ orCP ⁿ are respectivelyG ₂-congruent or holomorpic congruent are determined and this result is used to characterize the hypersurfaces of these spaces whose Hopf vector fields are also Killing fields.     

A Gradient Estimate for the Ricci–Kähler Flow

We show that for a complete solution to theRicci–Kähler flow where the curvature, the potential andscalar curvature functions and their gradients are bounded depending ontime, the absolute value of both the scalar curvature and the gradientsquared of a modified potential function are bounded byC/t.     

The Eigenspaces of the Bose-Mesner-Algebras of the Association Schemes Corresponding to Projective Spaces and Polar Spaces

In an earlier paper 7, some properties of the eigenspaces of the Bose-Mesner-algebras of association schemes are figured out, leaving open the problem of determining the eigenspaces. In the present paper, these eigenspaces and the eigenvalues are determined for projective spaces and for polar spaces. This allows characterizations of certain sets of subspaces of these geometries.     

The Deformation Spaces of Projective Structures on 3-Dimensional Coxeter Orbifolds

A real projective structure on a 3-orbifold is given by locally modeling the orbifold by real projective geometry. We present some methodology to study Coxeter groups which are fundamental groups of 3-orbifolds with representations in and deformation spaces. There are related examples by Benoist. These examples give us nontrivial deformation spaces of projective structures.     

Integration of Geodesic Flows on Homogeneous Spaces: The Case of a Wild Lie Group

We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces M with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space T ^* M based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.     

Some Remarks on the Geodesic Completeness of Compact Nonpositively Curved Spaces

We study the following question: does a compact nonpositively curved space have a totally geodesic core?     

Cotangent Bundle over Projective Space and the Manifold of Nondegenerate Null-Pairs

A nondegenerate null-pair of the real projective space consists of a point and of a hyperplane nonincident to this point. The manifold of all nondegenerate null-pairs carries a natural Kählerian structure of hyperbolic type and of constant nonzero holomorphic sectional curvature. In particular, is a symplectic manifold. We prove that is endowed with the structure of a fiber bundle over the projective space , whose typical fiber is an affine space. The vector space associated to a fiber of the bundle is naturally isomorphic to the cotangent space to . We also construct a global section of this bundle; this allows us to construct a diffeomorphism between the manifold of nondegenerate null-pairs and the cotangent bundle over the projective space. The main statement of the paper asserts that the explicit diffeomorphism is a symplectomorphism of the natural symplectic structure on to the canonical symplectic structure on .     

Mean Curvature Flow of Arbitrary Codimension in Complex Projective Spaces*

Recently, Pipoli and Sinestrari [Pipoli, G. and Sinestrari, C., Mean curvature flow of pinched submanifolds of CPⁿ, 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 CP^m. 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 CP^m 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.     

The Spectral Density Function for the Laplacian on High Tensor Powers of a Line Bundle

For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin–Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density function. We give an explicit computation of the spectral density function, by constructing certain quasimodes on the associated principle bundle.     

实Grassmann流形上的道路空间

Ｇ（ｎ，ｍ）表示Ｒ￣ｎ＋ｍ中全体ｎ维子空间所构成的实Ｇｒａｓｓｍａｎｎ流形。本文首先找到ｐ，ｑ∈Ｇ（ｎ，ｍ）沿任何测地线均不共轭的充要条件，因此连接这样两点的测地线有可数条。通过计算得到编号为（ｋ＿１，ｋ＿２，…，ｋ＿ｎ）的测地线指标λ（ｋ＿１，ｋ＿２…，ｋ＿ｎ）．最后根据Ｍｏｒｓｅ基本定理得到：设ｐ，ｑ是Ｇ（ｎ，ｍ），上沿任何测地线均不共轭的两点，则连接ｐ，ｑ的分段光滑道路空间同伦于一可数ＣＷ－复形，该复形中的胞腔可编号为（ｋ＿１，ｋ＿２，…，ｋ＿ｎ），ｋ＿ｉ为整数，且编号为（ｋ＿１，ｋ＿２，…，ｋ＿ｎ的胞腔的维数为λ（ｋ＿１，ｋ＿２…，ｋ＿ｎ）。     

On the Constant Type of Almost Hermitian Manifolds

We suggest a most natural generalization of the notion of constant type for nearly Kählerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of G ₁-manifolds of zero constant type coincides with the class of 6-dimensional G ₁-manifolds with a non-integrable structure. Finally, we prove that the class of normal G ₂-manifolds of nonzero constant type coincides with the class of 4-dimensional G ₂-manifolds with a nonintegrable structure.     

一四元数矩阵方程组的广义酉矩阵解

给出了四元数矩阵方程组[XmnAns=Bns,XnnCnt=Dnt]有广义酉矩阵解的充要条件及其解集结构。     

Periodic Rotating Waves of a Geodesic Curvature Flow on the Sphere

Given a zone on the unit sphere S² with periodic undulating boundaries, we consider the motion of a curve in this zone which is driven by its geodesic curvature. First, we give a necessary and sufficient condition for the existence of periodic rotating waves. Then we study how the average rotating speed of the periodic rotating wave depends on the geometry of the boundaries. We find that when the period of the boundaries tends to 0, the homogenization limit of the rotating speed depends only on the maximum slope of the domain boundaries.