ALMOST FLAT MANIFOLDS
ALMOST FLAT MANIFOLDS
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DOI：
10.4310/jdg/1214434488
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发表时间：
1978-06
影响因子：
2.5
通讯作者：
M. Gromov
中科院分区：
文献类型：
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作者：
M. Gromov
1.1. 我们用V表示连通的^维完备黎曼流形，用d = d(V)表示V的直径，用c = c(V)和c~ = c~(V)分别表示V的截面曲率的上界和下界，设c = c(V) = max （| c1, | c~ |）。我们说F i是e-flat， e > 0，如果cd 0。b.每一个紧化零流形对于任何e- >0都有一个e-平坦度规。{如果一个流形允许幂零李群的传递作用，则称其为零流形；见4.5。)第二个例子表明，对于n > 3, e > 0，存在无穷多个具有不同基本群的e平^维流形。1.3. 定义归纳的ext(x) = exp (eXi_λ(x)\ exo(x) - x，并设e（ή） = exp (- exj (n))，其中j = 200。（我们在本文中处处都很慷慨，因为常数的真实值是未知的。）1.4. 主要定理。假设V是一个紧凑的“έ(n)”平面流形，π是它的基本群。则：(a)存在一个极大幂零正规因子nc π (b) ord(π ln) 0)，则它的基群π和π的每一子群都可以由3个元素生成。（ii）若d(V) -K，K>0，则N < 3ex2 （nK&）元可生成π；如果π是一个自由群，且KQ) 1 < e(n)，则π是由一个元素生成的。
1.1. We denote by V a connected ^-dimensional complete Riemannian manifold, by d = d(V) the diameter of V, and by c = c(V) and c~ = c~(V), respectively, the upper and lower bounds of the sectional curvature of V. We set c = c(V) = max (| c1, | c~ |). We say that F i s e-flat, e > 0, if cd 0. b. Every compact nil-manifold possesses an e-flat metric for any e > 0. {A manifold is called a nil-manifold if it admits a transitive action of a nilpotent Lie group; see 4.5.) The second example shows that for n > 3, e > 0 there are infinitely many e-flat ^-dimensional manifolds with different fundamental groups. 1.3. Define inductively ext(x) = exp (eXi_λ(x)\ exo(x) — x, and set e(ή) = exp (—eXj(n)), where j = 200. (We are generous everywhere in this paper because the true value of the constants is unknown.) 1.4. Main Theorem. Let V be a compact έ(n)-flat manifold, and π its fundamental group. Then: (a) There exists a maximal nilpotent normal divisor N C π (b) ord(πlN) 0), then its fundamental group π and every subgroup of π can be generated by 3 elements. (ii) If d(V) -K,K>0, then π can be generated by N < 3 ex2(nK&) elements; if π is a free group and KQ) 1 < e(n), then π is generated by one element.