On curves with nonnegative torsion
On curves with nonnegative torsion
复制标题
在具有非负扭转的曲线上
DOI：
10.1007/s00013-015-0767-0
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发表时间：
2013
影响因子：
0.6
通讯作者：
Jeffrey L. Jauregui
中科院分区：
文献类型：
--
作者：
H. Bray;Jeffrey L. Jauregui
我们给出了关于$${\mathbb{R}^3}$$R3中曲线扭转的新结果和新证明。设$${\Gamma}$$γ是$${\mathbb{R}^3}$$R3中的一条光滑曲线，它是$${\mathbb{R}^2}$$R2中具有正曲率的简单闭曲线上的图。我们给出了一个新的证明：如果$$\Gamma}$$γ有非负(或非正)挠率，则$$\Gamma}$$γ有零挠率，从而位于平面上。此外，我们还证明了一个新的结果，即一条简单的闭平面曲线在没有任何曲率假设的情况下，不能扰动到一条常非零挠率的闭空间曲线。我们还证明了洛伦兹曲线在Lorentzian$$\mathbb{R}^{2，1}$$R2，1中的类似命题，这与时空中的时间平面和广义相对论中的质量有关的重要公开问题有关。
We provide new results and new proofs of results about the torsion of curves in $${\mathbb{R}^3}$$R3. Let $${\gamma}$$γ be a smooth curve in $${\mathbb{R}^3}$$R3 that is the graph over a simple closed curve in $${\mathbb{R}^2}$$R2 with positive curvature. We give a new proof that if $${\gamma}$$γ has nonnegative (or nonpositive) torsion, then $${\gamma}$$γ has zero torsion and hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve, without any assumption on its curvature, cannot be perturbed to a closed space curve of constant nonzero torsion. We also prove similar statements for curves in Lorentzian $${\mathbb{R}^{2,1}}$$R2,1 which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity.