Problem from Do Carmo

[Zac9908]

I don't know how to solve this problem from Do Carmo, Differential Geometry, p. 216:

Let [FONT=MathJax_Math-italic]SS be a regular surface without umbilical pints. Prove: If the Gauss map N:S→S2 satisfies[/FONT]
[FONT=MathJax_Main]
⟨[FONT=MathJax_Math-italic]d[FONT=MathJax_Math-italic]N[FONT=MathJax_Math-italic]p[FONT=MathJax_Main]([FONT=MathJax_Math-italic]v[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]N[FONT=MathJax_Math-italic]p[/FONT][/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]v[FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]⟩[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]λ[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]p[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]⟨[/FONT][FONT=MathJax_Math-italic]v[FONT=MathJax_Main]1[/FONT][/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]v[FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main]⟩

[/FONT][/FONT][/FONT][/FONT][/FONT]for all [FONT=MathJax_Math-italic]p[FONT=MathJax_Main]∈[FONT=MathJax_Math-italic]S[/FONT][/FONT]p∈S and all [FONT=MathJax_Math-italic]v[FONT=MathJax_Main]1[FONT=MathJax_Main],[FONT=MathJax_Math-italic]v[FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math-italic]T[FONT=MathJax_Math-italic]p[/FONT][/FONT][FONT=MathJax_Math-italic]S[/FONT][/FONT][/FONT][/FONT][/FONT]v1,v2∈TpS, where [FONT=MathJax_Math-italic]λ[FONT=MathJax_Main]([FONT=MathJax_Math-italic]p[FONT=MathJax_Main])[FONT=MathJax_Main]≠[/FONT][FONT=MathJax_Main]0[/FONT][/FONT][/FONT][/FONT][/FONT]λ(p)≠0 is a number which depends only on [FONT=MathJax_Math-italic]p[/FONT], S is a minimal surface.[/FONT][/FONT][/FONT]
[FONT=MathJax_Math-italic]
[/FONT]


Thank you!

 

[tkhunny]

What say you? Your views and impressions are required.