我問的這題目是出自
do Carmo, M.P. Differential geometry of curves and surfaces ,
Exercise 1 , Page 454 .
底下是題目的敘述 , 我的問題是在 是否b小題有錯 ?
(大於小於對象相反了?)
(Stoker's Remark on Efimov's theorem)
Let S be a complete geometric surface . Assume that the
Gaussian curvature K≦δ<0 . show that there is no isometric
3
immersion φ:S-->R such that absolute value of the mean curvature
H is bounded . The following outline may useful :
2
a) Assume such a φ exists and consider the Gauss map N:φ(S)-->S
2
, where S is the unit sphere. Since K≠0 every where , N induces
2
a new metric ( , ) on S by requiring that N。φ:S-->S be a local
isometry . Choose coordinates on S so that images by φ of the
coordinate curves are lines of curvature of φ(S) .
Show that the coefficients of the new metric in this
coordinate system are
2 2
g =(k1) E , g = 0 , g =(k2) G
11 12 22
where E,F(=0),and G are the coefficients of the initial metric
in the same system .
2 2
b) show that there exists a constant M >0 such that (k1) ≦M , (k2) ≦M ,
Use that fact the initial metric is complete to conclude that
new metric is also complete .
證明過程中,引進新的 Riemannian metric,在這 metric下,曲面 S 的
curvature都是 1 , 只要證明了 S 是 complete (new metric) ,
那就可以使用 Bonnet's theorem (Page352,443) 說明 S 是 compact .
證明 S is complete過程中 ( 如同 Exercise 3 (b) , Page 455 ,
T.K. Milnor's proof of Hilbert's Theorem ) 令
2 2
g = E (du) ＋ G (dv)
1
2 2 2 2
g =(k1) E (du) ＋ (k2) G (dv)
2
只要能證存在一個正數 c>0 使的 cg ≧ g
2 1
那麼 S is complete就得證 , 所以我認為 大於小於 對象相反了 ～”～
而從題目給的條件,我也只能證明這條件
2 2
(k1) ≧M , (k2) ≧M
(k1 , k2 : principal curvature )
請教各位大大 , 是否題目如同我所說的錯誤
還是我哪方面有出錯 ?
c) Use part b to show that S is compact ; hence , it has points with
positive curvature , a contradiction .
在此,先謝謝各位寶貴的意見 ！！
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※ 編輯: keroro321 來自: 59.112.236.27 (03/29 14:50)
※ 編輯: keroro321 來自: 59.112.236.27 (03/29 14:52)