differential geometry

Project Description:

the following is suggested project for differential geometry. your projects must be written in word. you must begin with a definition of the thing you are interested in. do not just list problems and solutions.
to read see book the second edition of differential geometry and its applications.

surfaces with constant gauss and mean curvatures this project will prove the following theorem:
if a surface m has constant gauss curvature and constant mean curvature, then m is a plane, a right circular cylinder or a sphere. note that compactness is not assumed here. the plan is to first show that the principal curvatures k1, k2 are constant. then, there are two cases: k1 = k2 and k1 > k2. the second case will require an analysis of the corresponding case of liebmann’s theorem to show k = 0. then, by taking a patch given by lines of curvature, show that e = 1, f = 0, g = 1 and ℓ = k1, m = 0, n = 0. then apply the fundamental theorem of surfaces (which must be looked up). filling in all these details constitutes the project.

similar is not the same of this theorem in chapter 3 page 130 in second edition of differential geometry and its applications.
this mean you can look at it just to see the idea. so please proof this required theorem very clear and correctly.
please see pdf file how to write the project to proof the theorem.
i want to write 8 to 10 pages.
please include any references used in project.

thank you.
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