Submanifolds in euclidean space with simple geodesics
Submanifolds in euclidean space with simple geodesics
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具有简单测地线的欧几里得空间中的子流形
DOI：
10.1007/bf01475754
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发表时间：
1982
影响因子：
1.4
通讯作者：
S. Schirrmacher
中科院分区：
文献类型：
--
作者：
D. Ferus;S. Schirrmacher
在这种情况下，所有测地线要么都是直线，要么都是相同半径的圆。那么子流形局部是仿射子空间，或者是 1 阶标准嵌入式紧致黎曼对称空间。如果“简单”被解释为“恒定（第一）曲率”，其中该常数可能取决于各个测地线，并且如果我们将注意力限制在 E"+ 1 的超曲面上，那么它局部是一个球柱 Sk (r) x E"-k：从下面的公式 (2) 到 (4)，很容易得出第二个基本形式是协变常数，然后 [-3] 意味着该断言。最近研究了其他“简单性条件”，请参阅[12]，以及[2]中的类似问题。
In this case either all the geodesics are straight lines, or they are all circles of the same radius. Then the submanifold is locally an affine subspace, or a standard imbedded compact riemannian symmetric space of rank 1. If" simple" is interpreted as" of constant (first) curvature", where this constant may depend on the individual geodesic, and if we restrict our attention to a hypersurface of E"+ 1, then it is locally a spherical cylinder Sk (r) x E"-k: From the formulas (2) to (4) below it follows readily that the second fundamental form is covariantly constant, and then [-3] implies the assertion. More recently other" simplicity conditions" were studied, see [12], and also [2] for a similar problem.