Examples of complete manifolds with positive Ricci curvature
Examples of complete manifolds with positive Ricci curvature
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DOI：
10.4310/jdg/1214439562
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发表时间：
1985
影响因子：
2.5
通讯作者：
Detlef Gromoll;Wolfgang Meyer
中科院分区：
文献类型：
--
作者：
Detlef Gromoll;Wolfgang Meyer
黎曼几何中一个长期存在的问题是：具有正 Ricci 曲率 Ric 的完备流形 M 是否也承认具有非负截面曲率 KΊ 的完备度量。通常认为这并不总是正确的，但不知道反例。对于维度 n = 3，答案实际上是肯定的（参见[6]、[16]）。请注意，当存在 Ric > 0 的度量时，有时已知 K > 0 会受到阻碍。简单的例子是非紧致情况下的 S X R' [5]，以及 k、I ^ 2 的非简单连通紧致情况下的 RP X RP，这是 Synge 引理 [4] 的结果。 K > 0 的完全流形的例子仍然相当稀少。不管怎样，它们都可以使用经典空间和等距群作用的商来获得（参见[3]以获得详细的参考文献列表）。还有几种其他方法可以生成 Ric > 0 的完整指标。 [14] 和 [15] 中处理了某些纤维束，[7] 中处理了一大类 Brieskorn 品种。最后，根据 Yau 的工作，Ric > 0 的凯勒度量存在于第一陈类 cλ > 0 的任何紧凯勒流形上（参见[17]）。有趣的例子是 CP 中的完全相交，特别是超曲面。特别是，CP 中的 ΛΓ3 面（四次）承认 Ricci 平面度量，但这是一个真正的边界线情况：由于 A 属不会消失，因此只要 Ric> 0，我们就有 Ric Ξ 0（参见[8]）。由此可见，K ^ 0 意味着 K = 0，这是不可能的。因此，至少可以在弱意义上区分条件 Ric ^ 0 和 K > 0。在本文中，我们提出了 Ric > 0 的新类完全流形。首先，我们构造非紧示例，其中许多示例不能携带 K ^ 0 的度量。这解决了非紧情况下的上述问题。
A long standing question in riemannian geometry has been: Does a complete manifold M with positive Ricci curvature Ric also admit a complete metric with nonnegative sectional curvature KΊ It is generally believed that this is not always true, but counterexamples were not known. The answer is actually affirmative for the dimension n = 3 (cf. [6], [16]). Note that K > 0 is sometimes known to be obstructed when a metric with Ric > 0 exists. Simple examples are S X R' in the noncompact case [5], and RP X RP in the nonsimply connected compact case for k, I ^ 2, as a consequence of Synge's Lemma [4]. Examples of complete manifolds with K > 0 remain fairly scarce. One way or another, they can all be obtained using classical spaces and quotients of isometric group actions (cf. [3] for a detailed list of references). There are several additional methods to produce complete metrics with Ric > 0. Certain fiber bundles were treated in [14] and [15], and a large class of Brieskorn varieties in [7]. Finally, by Yau's work, Kaehler metrics with Ric > 0 exist on any compact Kaehler manifold with first Chern class cλ > 0 (cf. [17]). Interesting examples arise as complete intersections in CP, notably hypersurfaces. In particular, the ΛΓ3-surface (quartic) in CP admits a Ricci flat metric, but this is a true border line case: Since the A -genus does not vanish, we have Ric Ξ 0 whenever Ric> 0 (cf. [8]). It follows that K ^ 0 would imply K = 0, which is impossible. Therefore one can distinguish at least between the conditions Ric ^ 0 and K > 0, in a weak sense. In this paper we present new classes of complete manifolds with Ric > 0. First of all we construct noncompact examples many of which cannot carry metrics with K ^ 0. This settles the above question in the noncompact case.