ON SUBMANIFOLDS WITH HARMONIC MEAN CURVATURE
ON SUBMANIFOLDS WITH HARMONIC MEAN CURVATURE
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关于具有调和平均曲率的子流形
DOI：
10.1090/s0002-9939-1995-1254831-7
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
Ó. Garay
中科院分区：
文献类型：
--
作者：
M. Barros;Ó. Garay
抽象的。给出了具有法丛中调和平均曲率向量场的Em中曲线的分类，并利用它得到了一些应用。1.设x：M”-Em是黎曼流形在欧氏空间中的等距浸入。用H和A分别表示（M，x）的平均曲率向量场和M的拉普拉斯算子。Beltrami的一个经典而著名的方程给出了H和A之间的一个很好的关系，即Ax =-nH。因此，欧氏空间中的极小子流形对应于调和子流形.满足AH = 0的子流形被B称为双调和子流形。Y.尘他还指出，双调和性意味着调和性（以及最小性）。这一猜想在某些特殊情况下已被证明是正确的。例如，B。Y. Chen [Ch]解决了E3中曲面的猜想。我也是。Dimitric [迪]这样做的曲线在EM表明，直线在EM是唯一的双调和曲线在EM。另一方面，如果一个认为一个圆在E2或更普遍的角
Abstract. The classification of curves in Em with harmonic mean curvaturevector field in the normal bundle is obtained and then it is used to obtain some applications. 1. IntroductionLet x : M" —► Em be an isometric immersion of a Riemannian manifoldin the Euclidean space. Denote by H and A the mean curvature vector fieldof (M, x) and the Laplacian of M respectively. A classical and well-knownequation of Beltrami gives a nice relation between H and A, namely Ax =-nH. Therefore minimal submanifolds in Euclidean space correspond withharmonic submanifolds. Submanifolds satisfying AH = 0 were called biharmonic submanifolds byB. Y. Chen. He also conjectured that biharmonicity implies harmonicity (andso minimality). This conjecture has been proved to be true in some special cases. For instance, B. Y. Chen [Ch] solved the conjecture for surfaces in E3. Also I. Dimitric [Di] did so for curves in Em by showing that straight lines in Emare the only biharmonic curves in Em .On the other hand if one considers a circle in E2 or more generally a Cornu