A classification of BusemannG-surfaces which possess convex functions
A classification of BusemannG-surfaces which possess convex functions
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具有凸函数的 BusemannG 曲面的分类
DOI：
10.1007/bf02392724
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发表时间：
1982
期刊：
影响因子：
3.7
通讯作者：
Nobuhiro Innami
中科院分区：
文献类型：
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作者：
Nobuhiro Innami
定义在无边界完备黎曼流形M上的函数q 0称为凸函数，如果q 0是每个弧长参数化测地线上的一元凸函数. ~v是局部Lipschitz连续的，因此在M上是连续的。这是一个自然的问题，问在多大程度上存在的凸函数M暗示限制的拓扑结构M。在最近的工作[4]中，详细研究了具有局部非常数凸函数的M的拓扑。他们的一个结果给出了一个2维完备黎曼流形的分类定理，该流形允许局部非常数凸函数：它们要么是平面，要么是圆柱，要么是开的M6 bius带。Cohn-Vossen [3]的一个经典结果指出，具有非负高斯曲率的完备非紧二维黎曼流形同胚于平面、柱面或开M6 bius带。Cheeger-Gromoll在文献[2]中证明了：如果完备非紧黎曼流形具有非负截面曲率，则其上的每个Busemann函数都是凸的（且是局部非常数的）。H. Busemann推广了Cohn-Vossen在[1] pp. 292-294，证明了一个具有有限连通性和零余且角测度在：r处一致的非紧G-曲面是拓扑平面、柱面或M6 bius带。现在，本文的目的是证明以下内容：
A flmction q0 defined on a complete Riemannian manifold M without boundary is said to be convex if q0 is a one variable convex function on each arc-length parametrized geodesic. ~v is locally Lipschitz cont inuous and hence continuous on M. It is a natural question to ask to what extent the existence of a convex function on M implies restrictions to the topology of M. In a recent work [4], the topology of M with locally nonconstant convex functions has been studied in detail. One of their results gives a classification theorem of 2-dimensional complete Riemannian manifolds which admit locally nonconstant convex functions: they are diffeomorphic to either a plane, a cylinder, or an open M6bius strip. A classical result of Cohn-Vossen [3] states that a complete noncompact Riemannian 2-dimensional manifolds with nonnegative Gaussian curvature is homeomorphic to a plane, a cylinder, or an open M6bius strip. Moreover , Cheeger-Gromoll have proved in [2] that if a complete noncompact Riemannian manifold has nonnegative sectional curvature, then every Busemann function on it is convex (and locally nonconstant). H. Busemann generalized Cohn-Vossen 's result in [1] pp. 292-294, proving that a noncompact G-surface with finite connectivi ty and zero excess whose angular measure is uniform at :r is topologically a plane, a cylinder, or a M6bius strip. Now, the purpose of the present paper is to prove the following: