On Some Properties of Nonpositively Curved Alexandrov Spaces

Abstract

[[abstract]]本計劃的主要目的是研究下面的問題並由此問題 的解答了解亞力山大空間的幾何性質: 假設 (M,d) ,(N,d’) 分別為 Alexandrov 空間， 其中M 代表實數軸R， d與d’ 分別為M 與N 空間 上的距離函數。假設函數 f 為定義在實數軸的凸 函數，令距離函數 e代表由 d, d’ 與 f 所誘導的 距離函數。 一個很自然的問題是如果(M,d) 與(N,d’)同時為 非正的Alexandrov 空間，則由這二個空間所產生 的積空間當其距離函數為e’時是否仍為非正 Alexandrov 空間 ? Reshetnyak 教授提出一個類似的問題。假設 f (t ) = cos Kt 定義在一個閉區間 ] 2 , 2 [ K K I p p = - 上，並且假設(N,d’) 為一個 有上界K Alexandrov 空間，那麼空間I 與空間N的 積空間是否仍是一個有上界的Alexandrov 空間。[[abstract]]The purpose of this project is to investigate the following problems : Suppose that (M,d) and (N,d’) are Alexandrov spaces, where M is real line R, d and d’ are distance function on M and N respenctively. Let function f be a convex function define on real line. Let distance function e be induced from d, d’ and f. It is easy to define the meaning of induced metric from that of Riemannian manifolds. The precise definition of induced distance function will be investigated. The natural question one may ask is that if both (M.d) and (N,d’) are nonpositive Alexandrov spaces. Is their product space with distance function e still a nonpositive Alexandrov space ? A similar problem raised by Professor Reshetnyak is that suppose that f (t ) = cos Kt is define on the interval ] 2 , 2 [ K K I p p = - . Suppose that (N,d’) is Alexandrov space with curvature bounded above by K. Is their product space (I with N) still a space of curvature bounded from above