The Schwarzian derivative and conformal mapping of Riemannian manifolds
The Schwarzian derivative and conformal mapping of Riemannian manifolds
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黎曼流形的施瓦茨导数和共形映射
DOI：
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发表时间：
1992
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通讯作者：
Henrich Cheng
中科院分区：
文献类型：
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作者：
Rune W. Berg;Ming;Hsueh;M. Hsiao;Henrich Cheng
S f t，-，]-FT，If 2 t，，-FT，]Schwarzian在许多复杂分析领域是重要的(例如，参见O.Lehto ill的新书)，但它首先也是最重要的是通过它与M6bius变换的联系而发生的。基本事实是az+b(1.1)S(F)0当且仅当bc：/：0 cz+d‘和(1.2)S(Fo H)S(H)当且仅当M_6bius。方程(1.2)是两个解析函数合成的施瓦兹函数的一般公式的特例，公式(1.3)S(F)=S(H)+(S(F)o h)(h‘)对我们的工作将是重要的推广。设(M，g)是n>；2维黎曼流形，V表示度量g()的黎曼联络。对于光滑函数40：m->；R，我们定义一个张量
S f t,,-,] -fT,I f 2 t,,--fT,] The Schwarzian is important in many areas of complex analysis (see, for example, the recent book of O. Lehto ILl) but it occurs first and foremost through its connection with M6bius transformations. The basic facts are az+b (1.1) S(f) 0 if and only iff(z) ad bc :/: 0 cz+d’ and (1.2) S(fo h) S(h) if and only iff is M6bius. Equation (1.2) is a special case of a general formula for the Schwarzian of the composite of two analytic functions, which reads (1.3) S(fo h)= S(h) + (S(f)o h)(h’) A generalization of (1.3) will be important for our work. Let (M, g) be a Riemannian manifold of dimension n > 2 and let V denote the Riemannian connection for the metric g ( ). For a smooth function 40: M ---> R we define a tensor