Surfaces with a parallel isoperimetric section
Surfaces with a parallel isoperimetric section
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DOI：
10.1090/s0002-9904-1973-13219-7
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发表时间：
1973-05
影响因子：
1.3
通讯作者：
Bang‐Yen Chen
中科院分区：
文献类型：
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作者：
Bang‐Yen Chen
这一声明是陈[1]（也是丘[3]）的延续。我们将提出有关具有平行法截面的空间形式中曲面的附加定理。设M是m维黎曼流形R中具有诱导法联络D的曲面。对于M上的单位法截面（也就是说，M在R中的单位法向量场\令A\是关于ε的第二基本张量;如果我们具有DC = 0，那么ε称为平行截面;如果A\的迹是常数（分别为零），那么ε称为等周截面。（分别为极小截面）;如果A4的行列式处处不为零，则ε称为非退化截面;如果A4为零，则ε称为测地线截面;并且如果^不是处处与恒等变换成比例，则λ称为无脐部分。
This announcement is a continuation of Chen [1] (also, Yau [3]). We shall present additional theorems relating surfaces in a space form with a parallel normal section. Let M be a surface in an m-dimensional Riemannian manifold R with the induced normal connection D. For a unit normal section £ on M (that is, a unit normal vector field of M in R\ let Aç be the second fundamental tensor with respect to £; if we have DC = 0 identically, then £ is called a parallel section; if the trace of A% is constant (respectively, zero), then £ is called an isoperimetric section (respectively, minimal section) on M ; if the determinant of A4 is nowhere zero, then £ is called a nondegenerate section; if Aç vanishes identically, then £ is called a geodesic section; and if ^ is not proportional to the identity transformation everywhere, then Ç is called a umbilical-free section.