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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