Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature
Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature
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DOI：
10.1007/s00025-011-0209-7
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发表时间：
2011-10
影响因子：
2.2
通讯作者：
Nobumitsu Nakauchi;H. Urakawa
中科院分区：
文献类型：
--
作者：
Nobumitsu Nakauchi;H. Urakawa
在本文中，我们证明，对于具有非正截面曲率的黎曼流形（N，h）的每个双调和子流形（M，g），如果，则（M，g）在（N，h）中最小，即，其中η是（M，g）在（N，h）中的平均曲率张量场。这个结果在以下条件下给出了肯定的答案：具有非正截面曲率的黎曼流形的每个双调和子流形都必须是极小值。在不完全黎曼流形 (M,g) 的情况下，该猜想被 Ou 和 Tang 的反例证明是错误的（《广义陈氏双调和子流形猜想是错误的》，预印本，2010）。
In this paper, we show that, for every biharmonic submanifold (M,g) of a Riemannian manifold (N,h) with non-positive sectional curvature, if, then (M,g) is minimal in (N,h), i.e.,, whereηis the mean curvature tensor field of (M,g) in (N,h). This result gives an affirmative answer under the conditionto the followinggeneralized Chen’s conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M,g) by a counter example of Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).