L25A
　　　1. Theorem: Gauss-Bonnet Theorem for Orientable Compact Surface
　　　2. Example: Sphere with Radius r
　　　3. Example: Convex Surface in R^3
　　　4. Example: Polar Cap
 
L25B
　　　1. Example: Polar Cap (cont.)
　　　2. Euler Characteristic Number and Genus
　　　3. Theorem: Diffeomorphic Surfaces Have the Same Euler Characteristic
　　　　Number and Two Compact Oriented Surfaces with the Same Euler
　　　　Characteristic Number Are Diffeomorphic
 
L25C
　　　1. Theorem: A compact Oriented Surface with Positive Gauss　Curvature
             Is Diffeomorphic to a Standard Sphere
　　　2. Four Color Map Theorem