Surfaces which contain helical geodesics in the 3-sphere
Surfaces which contain helical geodesics in the 3-sphere
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3 球体中包含螺旋测地线的曲面
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发表时间：
2004
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通讯作者：
S. Maeda
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作者：
Michiko Tamura;S. Maeda
螺旋曲线（或螺旋线）是 3 维空间形式 M(c) 中曲率和挠率均为常数的恒定曲率 c 的曲线。如果其曲率恒定且挠率为零，或者其曲率为零，则它分别简化为黎曼圆或测地线。如果曲率和挠率都是非零常数，则螺旋曲线被称为真螺旋。众所周知，欧几里得 3 空间 E 中的圆柱体包含这些曲线作为测地线。另一方面，虽然 E 中的螺旋线包含普通螺旋线，但它们不是测地线。此外，E 旋转面上的（子午线）圆并不总是测地线。基于这些事实，我们将 M(c) 中曲面 M 上的螺旋测地线表示为 M(c) 中的曲线为螺旋的曲线，将测地线表示为 M 上的曲线。在我们之前的论文[12]中，我们已经证明，E 中的恒定平均曲率的完整曲面，其上存在通过每个点的两条螺旋测地线，是平面、球体或圆柱体。在本文中，我们将[12]中获得的这种表征推广到非负曲率的黎曼空间形式。更准确地说，我们展示了 3 球体表面的以下结果。在本文中，我们假设 M(c) 中的所有曲面都是光滑且相连的。定理 设 M 是 3 球体 S 中恒定平均曲率的完整曲面。如果 M 上存在通过 M 的每个点的两条螺旋测地线，则 M 要么是大球体，要么是小球体，要么是圆上的 Hopf 环面。
A helical curve (or a helix) is a curve in 3-dimensional space form M(c) of constant curvature c whose both curvature and torsion are constants. It reduces to a Riemannian circle or a geodesic, if its curvature is constant and torsion is zero, or if its curvature is zero, respectively. A helical curve is said to be a proper helix if both curvaure and torsion are non zero constants. As is well known, circular cylinders in Euclidean 3-space E contain these curves as geodesics. On the other hand, although a helicoid in E contains ordinary helices, they are not geodecics. Furthermore, (meridian) circles on a surface of revolution in E are not always geodesics. Based on these facts, we mean by a helical geodesic on a surface M in M(c) a curve which is helical as a curve in M(c) and a geodesic as a curve on M . In our previous paper [12], we have shown that complete surfaces of constant mean curvature in E on which there exist two helical geodesics through each point are planes, spheres or circular cylinders. In this paper we generalize this characterization obtained in [12] to Riemannian space forms of non-negative curvature. More precisely we show the following result for surfaces in the 3-sphere. We assume that all surfaces in M(c) are smooth and connected in this paper. Theorem Let M be a complete surface of constant mean curvature in the 3-sphere S. If there exist two helical geodesics on M through each point of M , then M is either a great sphere, a small sphere, or a Hopf torus over a circle.