A new proof of Hamburger's Index Theorem on umbilical points
A new proof of Hamburger's Index Theorem on umbilical points
复制标题
脐点汉堡指数定理的新证明
DOI：
10.3929/ethz-a-000915107
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
Hanspeter Scherbel
中科院分区：
文献类型：
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作者：
Hanspeter Scherbel
H.的指数定理Hamburger指出在ffi，3中的真实的解析曲面上的孤立脐点P的指数J不大于1。这证实了C. Caratheodory的分析情况：“存在至少两个脐点上的每一个椭圆是足够光滑的”。G.波尔试图给这个定理一个更简单的证明。T. Klotz意识到，有一个差距，波尔的证明，使一般情况下没有涵盖。她提供了额外的论据，一个完整的证明，但事实证明，她没有完全成功。我们用同样的几何方法来解决这个问题，并成功地给出了完整的证明。其主要步骤如下：命题J（P）< 1等价于W（B）> 0，其中W是平面曲线族B（g，ti）= x（g，i？）+ iy（g，t？）（p > 0，曲线参数·d mod 2 ir）。我们考虑由x和y的零点构成的Pu x级数。它们的形式是i？(p)= 00 $0 + r <<tA '，X，其中p = rP。对于我们确定t^的每一步，这个过程为我们提供了对基本曲线B（g固定）的一个越来越精确的近似。近似曲线可以通过原点。这意味着x和y至少有一个共同的零点f的形式i？0 + r> 0 + r> 0）。在这种情况下，我们对近似曲线应用“修改规则”。我们给出了这些修改曲线与通过原点的直线相交数的下界。一些零交叉（称为“奇异情况”）必须更详细地研究，即我们使用相应的Puiquix级数的下一个系数继续爆破过程。在有限的步骤之后，没有奇异情况留下，过程停止。通过对所有修正逼近曲线的交数求和，证明了W（B）> 0 .新证明的成功有两点至关重要：首先，我们为高阶逼近曲线制定了一个新的修改规则。其次，通过控制与这些曲线有关的常数，我们可以给出它们的交数的下界为-2。这最终导致汉堡指数定理的完整证明。
The Index Theorem of H. Hamburger states that the index J of an isolated umbilical point P on a real analytic surface in ffi,3 is not greater than 1. This confirms a conjecture of C. Caratheodory for the analytical case : "There exist at least two umbilical points on every ovaloid which is sufficiently smooth". G. Bol tried to give a simpler proof for this theorem. T. Klotz realized that there is a gap in Bol's proof so that the general case is not covered. She provided additional arguments for a complete proof but it turned out that she did not succeed completely. We use the same geometrical methods of blowing up certain singularities to solve this problem and succeed in giving a complete proof. The main steps are the following : The statement J(P) < 1 is equivalent to W(B) > 0 , where W is the winding number of a family of plane curves B(g, ti) = x(g, i?) + iy(g, t?) (p > 0, curve parameter •d mod 2ir) around the origin. We consider the Puiseux series of the zeros of x and y. They are of the form i?(p) = oo $o + r« £ tA',X with p = rP . A=0 For every step where we determine t^, this process provides us with a more and more refined approximation to parts of the base curve B (g fixed). The approximating curves may pass through the origin. This means that x and y have at least one common zero f of the form i?0 + r» £ txrx + o^) . A=o In this case we apply a "modification rule" to the approximating curves. We give a lower bound for the intersection number of these modified curves with a line through the origin. Some of the zero crossings (called "singular cases") have to be investigated in more detail, i.e. we continue the blowing up process using the next coefficient of the corresponding Puiseux series. After a finite number of steps, there are no singular cases left and the process stops. By summing up the intersection numbers of all modified approximating curves we show that W(B) > 0 . Two points are crucial to the success of the new proof : We first formulate a new modification rule for higher order approximating curves. Second, controlling some constants related to these curves, we are able to give a lower bound of -2 for their intersection numbers. This finally results in a complete proof of Hamburger's Index Theorem.