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1.
For a knotted surface in -space, its generic projection into -space has branch points as its singularities, and its successive projection into -space has fold points and cusps as its singularities. In this paper, we show that for non-orientable knotted surfaces, the numbers of branch points and cusps can be minimized by isotopy.

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2.
We show that if a non-orientable surface embedded in 4-space has a projection into 3-space with at most one triple point, then it is ambient isotopic to a connected sum of some unknotted projective planes and an embedded surface in 4-space with vanishing normal Euler number.

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3.
§1.IntroductionThesymbolsandthetechnicaltermsnotexplainedinthispapercanbefoundin[1]and[3].AgraphisoftendenotedbyG=(V,E),υ=|V|andε=|E|arecalleditsorderandsizerespectively.IthasbeenshownbyR.Duke[4]thatifaconnectedgraphGhasembeddings(or2-cellembeddingsinDuke’stermi…  相似文献
4.
We prove that for any orientable surface S and any non-negative integer k, there exists an integer fS(k) such that every graph G embeddable in S has either k vertex-disjoint odd cycles or a vertex set A of cardinality at most fS(k) such that G-A is bipartite. Such a property is called the Erd?s-Pósa property for odd cycles. We also show its edge version. As Reed [Mangoes and blueberries, Combinatorica 19 (1999) 267-296] pointed out, the Erd?s-Pósa property for odd cycles do not hold for all non-orientable surfaces.  相似文献
5.
We show that, for a closed non-orientable surfaceF, an automorphism ofH 1(F, ℤ) is induced by a homeomorphism ofF if and only if it preserves the (mod 2) intersection pairing. We shall also prove the corresponding result for punctured surfaces.  相似文献
6.
一个近-三角剖分嵌入是指一个曲面上的嵌入图使得几乎所有的面都是三角形,至多只有一个可能的例外.文中作者证明了如下结论:如果一个图G 在球面S0(或环面S1)上有近-三角剖分嵌入,那么G在每一个可定向曲面Sk有近-三角剖分嵌入,其中k=h,h+1,\cdots ,\lfloor\frac{\beta(G)}{2}\rfloor$, 而h=0(或1)并且β(G)是图G的Betti数.特别地,G是上可嵌入的.  相似文献
7.
Let be a closed orientable surface of genus , endowed with the structure of a Riemann manifold of constant negative curvature. For the universal covering , there is the notion of absolute, each of whose points determines an asymptotic direction of a bundle of parallel equidirected geodesics. In the paper it is proved that there is a set on the absolute having the cardinality of the continuum and such that if an arbitrary flow on has a semitrajectory whose covering has asymptotic direction defined by a point from , then this flow is not analytical and has infinitely many stationary points.  相似文献
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9.
一个近三角剖分嵌入是指一个图嵌入在一个曲面上,使得至多可能有一个面不是三角面。在本文中我们证明了如下结果:如果一个图G在某个可定向曲面S_h上有三角剖分嵌入,那么G在S_k上有一个近三角剖分嵌入,这里k=h,h 1,…,[β(G)/2],而β(G)是图G的Betti数。  相似文献
10.
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