Math 423 Differential Geometry Fall 2006
Homework 8: More on Curvature
Due Thursday Nov. 9
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1. Prove that โยฏxuโยฏxvU=โยฏxvโยฏxuU
2. Prove that a surface of revolution with K= 0 is part of a cone or a plane. Hint:
model your proof after our proof that a surface of revolution with H= 0 is part
of a catenoid or a plane.
3. Compute the Gaussian curvature of the sphere of radius Rusing the formula from
Gaussโs Theorem Egregium:
K=โ1
2โEG ๎โ
โv ๎Ev
โEG ๎+โ
โu ๎Gu
โEG ๎๎.
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Bonus Questions for those taking the course for four credits or for those who
want more of a challenge.
4. Show that the Uterms in the equations xuuv โxuvu = 0 and xvvu โxuvu = 0 give
LvโMu=M๎Gu
2GโEu
2E๎+EvH=LEv
2E+M๎Gu
2GโEu
2E๎+NEv
2G
and
NuโMv=M๎Ev
2EโGv
2G๎+GuH=LGu
2E+M๎Ev
2EโGv
2G๎+NGu
2G.
What do these equations say in the case when H= 0 and the parameterization
satisfies E=G?
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