复射影空间中迷向Kaehler子流形

讨论了复射影空间中迷向Kaehler流形,运用活动标架法获得关于截面曲率,Ricci曲率和第二基本形式模长的Pinching定理,将相关结果作了一定的推广.     

复射影空间CP^n＋p的Kaehler子流形

本文对复射影空间ＣＰｎ＋ｐ中具有非负全实双截面曲率的ｎ维（ｐ＜ｎ）紧致Ｋａｅｈｌｅｒ子流形作了完全分类．     

Caylay射影平面的全实曲面

It has been shown, under certain conditions on the Gauss curvature, every totally real surface of the Cayley projective plane with parallel mean curvature vector is either flat or totally geodesic.     

关于四元射影空间的正曲率全复子流形

本文讨论四元射影空间的全复子流形，证明了四元射影空间的正截面曲率紧致全复子流形一定是全测地的。     

复射影空间的拟共形平坦Kaehler完备子流形

研究复射影空间的拟共形平坦Kaehler完备子流形得到局部结构与关于数量曲率的拼挤常数．     

Kaehler流形的全脐实超曲面

给出了Ｋａｅｈｌｅｒ流形的全脐实超曲面的一个分类定理。     

射影平面上标号图的辅助图

刘彦佩教授给出了平面图的辅助图.推广平面图的辅助图到射影平面,给出标号图的射影平面性的如下刻画:给定标号图$G$对应的辅助图是平衡的当且仅当$G$是射影平面图.     

射影平面的模型和莫比乌斯带

射影几何研究图形在射影变换下的不变性,射影变换可以直观地看成是由连续施行若干次中心投影所得到的变换,为了使中心投影成为两平面的点之间的一一对应,我们必须把通常的欧氏平面加以拓广,添加无穷远点和无穷远直线,即对平面上的一族平行线添加一个无穷远点,且规定平面上所有无穷远点的集合为一条无穷远直线,这和经过拓广以后的平面,若对     

射影—空间通往平面的桥梁

平面是空间的一个元素.当我们选定一个平面作为认识空间各元素的关系的基础时,这个平面叫这个空间的基平面.于是,一些空间元素间的距离,或者线、面所成的角,可以通过射影的方式,把要求的数据,通过它们在基面上的影象而获得.直接把空间距离或角投射到平面上且不改变大小的射影,我们称为一次射影.1　求空间两点间的距离例1　线段AB、CD夹在两个平行平面α与β之间,ACα,BDβ,AB⊥α,AC=BD=5,AB=12,CD=13.E、F分别分AB、CD为1:2,求线段EF的长.分析　无论对于平面α还是β,E、F都是空间两点,它们好象是分别长在两棵树上的果子,不易…     

Einstein hypersurfaces of the Cayley projective plane

We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a certain radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces.     

Minimal embeddings in the projective plane

We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 153–163, 1997     

The width of quadrangulations of the projective plane

We show that every 4‐chromatic graph on n vertices, with no two vertex‐disjoint odd cycles, has an odd cycle of length at most . Let G be a nonbipartite quadrangulation of the projective plane on n vertices. Our result immediately implies that G has edge‐width at most , which is sharp for infinitely many values of n. We also show that G has face‐width (equivalently, contains an odd cycle transversal of cardinality) at most , which is a constant away from the optimal; we prove a lower bound of . Finally, we show that G has an odd cycle transversal of size at most inducing a single edge, where Δ is the maximum degree. This last result partially answers a question of Nakamoto and Ozeki.     

Generating even triangulations of the projective plane

We shall determine the 20 families of irreducible even triangulations of the projective plane. Every even triangulation of the projective plane can be obtained from one of them by a sequence of even‐splittings and attaching octahedra, both of which were first given by Batagelj 2 . © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 333–349, 2007     

Envelopes of splines in the projective plane

In this paper a family of curves—Riemannian cubics—inthe unit sphere and the real projective plane is investigated.Riemannian cubics naturally arise as solutions to variationalproblems in Riemannian spaces. It is remarkable to find thatan envelope of lines generated by a Riemannian cubic in onespace is (nearly) a Riemannian cubic in another space.     

A fake projective plane constructed from an elliptic surface with multiplicities (2,4)

Given any (2,4)-elliptic surface with nine smooth rational curves,eight (2)-curves and one (3)-curve,forming a Dynkin diagram of type [2,2][2,2][2,2][2,2,3],we show that a fake projective plane can be constructed from it by taking a degree 3 cover and then a degree 7 cover.We also determine the types of singular fibres of such a (2,4)-elliptic surface.     

Y‐rotation in k‐minimal quadrangulations on the projective plane

Let G be a quadrangulation on a surface, and let f be a face bounded by a 4‐cycle abcd. A face‐contraction of f is to identify a and c (or b and d) to eliminate f. We say that a simple quadrangulation G on the surface is k‐minimal if the length of a shortest essential cycle is k(≥3), but any face‐contraction in G breaks this property or the simplicity of the graph. In this article, we shall prove that for any fixed integer k≥3, any two k‐minimal quadrangulations on the projective plane can be transformed into each other by a sequence of Y‐rotations of vertices of degree 3, where a Y‐rotation of a vertex v of degree 3 is to remove three edges vv₁, vv₃, vv₅ in the hexagonal region consisting of three quadrilateral faces vv₁v₂v₃, vv₃v₄v₅, and vv₅v₆v₁, and to add three edges vv₂, vv₄, vv₆. Actually, every k‐minimal quadrangulation (k≥4) can be reduced to a (k?1)‐minimal quadrangulation by the operation called Möbius contraction, which is mentioned in Lemma 13. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 301–313, 2012