Recall that a submanifold L of a Riemannian manifold is totally umbilic if it curves equally in all directions, i.e., if there is some vector N ⊥ L such that the second fundamental tensor S_X of L in any direction X ⊥ L is given by S_X = < X, N > Id. In this note, we investigate some properties of k-dimensional Riemannian foliations F~k with totally umbilic leaves, which we call umbilic foliations for short. Notice that a Riemannian flow (k = 1) is always umbilic. It is to be expected that restrictions increase with k. In fact, we show that on manifolds M~n of positive sectional curvature, there are no umbilic foliations if k > (n-1)/2. This is a best possible estimate, since for example a Euclidean 3-sphere admits an abundance of Riemannian flows. Similarly, it turns out that on spaces of nonpositive curvature, an umbilic foliation of dimension n-1 is, when lifted to the universal cover, 'almost always' a foliation by horospheres or by hypersurfaces equidistant from a totally geodesic one.
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机译：回想一下，如果黎曼流形的子流形L在所有方向上均等地弯曲，即如果存在矢量N⊥L，则在任何方向X X L上的L的第二基本张量S_X由S_X =给出，则它是完全脐带的。 ID。在本文中，我们研究了具有全脐叶的k维黎曼叶面F〜k的一些特性，我们简称为脐叶。注意，黎曼流（k = 1）总是脐带的。可以预期，限制会随着k增加。实际上，我们表明，在k>（n-1）/ 2的情况下，在正截面曲率的歧管M〜n上，没有脐带叶。这是可能的最佳估计，因为例如欧几里得三球体允许大量的黎曼流。类似地，事实证明，在非正曲率的空间上，尺寸为n-1的脐带叶被提升到通用覆盖层时，几乎总是由球体或与全测地线等距的超曲面形成。
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