The Gauss map for surfaces in 4-space
The Gauss map for surfaces in 4-space
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4 空间表面的高斯图
DOI：
10.1007/bf01450764
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发表时间：
1984
期刊：
影响因子：
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通讯作者：
J. Weiner
中科院分区：
文献类型：
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作者：
J. Weiner
给定一个连通的n维定向流形M“和一个浸没X：M”-R“+k，与它相关的是一个Gauss映射g：M”G(n，k)，其中G(n，k)是(n+k)空间中定向n平面的Grassmann流形.也就是说，与X有联系的是一个映射g，它为每个m~M分配与X(M)在X(M)处相切的有向n平面。这是推广三维空间中曲面的经典高斯映射的一种方法。另一方面，假设我们有一个映射g：m“-+G(n，k)。是否存在以g为高斯映射的浸没X：m“-.~”+k？G局部是高斯映射吗？g在多大程度上唯一地决定了X？本文研究了n=k=2情形下的这类问题，但仅在附加的限制条件下，假设g是Mz在G(2，2)中的浸入，得到的主要结果如下：在定理1中，我们给出了以g为高斯映射的X存在的纯代数必要条件。如果满足关于g的这个代数条件，则在某些“正则性”假设(一个秩条件)下，g的X的存在归结为求解一个偏微分方程组。如果这个系统是双曲型的或椭圆型的，我们就研究它。对于双曲情形，我们分别在定理2、3中证明了当X沿M(柯西问题)中的一条无处特征曲线表示时，以给定的g为其高斯映射的浸没X的存在唯一性。对于椭圆情形，我们在定理4中证明了以给定的g为其Gauss映射的X的存在性，如果M是一个闭圆盘，并且在定理5中我们得到了这样一个X的唯一性，如果X的值是沿着M中的任何圆弧指定的，其中一些结果是Aminov在最近的一篇论文[1]中得到的，但他的观点和方法与我们的有很大的不同。(作为我们的研究的副产品，我们在附录中证明了Chern和Spiier的一个结果的略微锐化。)如果~是M上的一个向量丛，我们设~m~M，表示m上的纤维.对于M，TM的切丛，我们记M，.对于(TM)，..IF是一个集合，f：-~S是一个映射，我们表示f对~的限制。由f，..此外，AK({)表示
Given a connected oriented manifold M" of dimension n and an immersion X: M"-~ R"+ k, there is associated to it a Gauss map g: M"~ G (n, k), where G (n, k) is the Grassmann manifold of oriented n-planes in (n+ k)-space. That is, there is associated to X a map g which assigns to each m~ M the oriented n-plane tangent to X (M) at X (m). This is one way in which one may generalize the classical Gauss map for surfaces in 3-space. Suppose on the other hand, we are given a map g: M"-+ G (n, k). Does there exist an immersion X: M"-.~"+ k with g as its Gauss map? Is even g locally a Gauss map and also to what extent does g uniquely determine X? In this paper, we study these questions for the case n= k= 2, however only under the additional restriction that g is supposed to be an immersion of M z into G (2, 2).The main results obtained are the following ones: In Theorem 1 we state a purely algebraic necessary condition on g for the existence of X with this g as its Gauss map. If this algebraic condition on g is fulfilled, the existence of X for g reduces-under certain" regularity" assumptions (a rank condition)-to solving a system of partial differential equations. We shall study this system if it is either of hyperbolic or of elliptic type. For the hyperbolic case we prove in Theorem 2 resp, 3 the existence resp, uniqueness of an immersion X with the given g as its Gauss map, if X is prescribed along a nowhere-characteristic curve in M (Cauchy problem). For the elliptic case we prove in Theorem 4 the existence of X with the given g as its Gauss map, if M is a closed disc, and in Theorem 5 we get for arbitrary M the uniqueness of such an X, if the values of X are prescribed along any arc in M, Some of these results are in a recent paper [1] by Aminov but his point of view and methods are very different from ours.(As a by-product of our investigations we prove in the Appendix a slight sharpening of a result of Chern and Spanier.) If~ is a vector bundle over M and m~ M we let~., denote the fibre over m. In the case of the tangent bundle of M, TM, we write M,. for (TM),.. IfS is a set and f:-~ S is a map, we denote the restriction of f to~,. by f,.. Also Ak ({) denotes the