在本篇論文中，我們學習在一個維度大於3的正純曲率流形上執行0-surgery，所得到的新流形也保有距離函數和正純曲率的性質。主要的技巧是要利用一條曲線來建構一個頸狀的管子接上原有的流形，使得這個管子在一開始的部分有原有流形上的距離函數，在最後的部分有 的乘積距離。我們利用Gauss formula得到管子和流形上的截面曲率的關係式，進而得到這兩者間的純曲率關係式。最後我們將此關係式化簡為一個微分方程式，得到此方程式的解，則能保證我們的確可以找到這樣的曲線來建構出這個管子，使得在此管子上保有正純曲率的性質。

In this thesis, we study the 0-surgery on a manifold M of dimension . The main theorem can be stated in the following way. If   obtained from a Riemannian manifold M with positive scalar curvature, ,by performing 0-surgery can still have a metric with positive scalar curvature. The main technique we use is to construct a neck   according to a given curve   in the way that X has Riemannian metric on M in the beginning and product metric of the form   in the end. We using the Gauss formula to simplify the relation of sectional curvature between X and  . Then we also has the relation of scalar curvature between X and  . Finally, the problem can be reduced to a differential equation. We can find the solution of that differential equation. So that we can construct the curve   such that X has positive scalar curvature.

Chapter 1. Introduction	1

Chapter 2. Manifold	3

Chapter 3. Curvature	5

Chapter 4. Main Theorem	14

Reference	                  19

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[2]  John Oprea, Differential Geometry and its applications, Upper Saddle River, NJ : Prentice Hall, c1997.
[3]  William M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, New York : Academic Press, 1975.
[4]  Manfredo Perdigão do Carmo, Riemannian Geometry, Boston : Birkhäuser 1992.