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I heard a proof of comparison principle a while ago and figured it out today. In GT, they define w=u-v which is quite misleading. In nonlinear PDE, we should better not to substract two solutions. The translation (or other perturbation) is the right way of thinking.

http://dl.dropbox.com/u/23145891/Blog/Comparison%20principle.pdf

Last week, a student in the Modern Geometry class showed me a brilliant construction of an orthogonal frame E_i such that \nabla_{E_i}E_j(p)=0. We take a normal coordinates at p and diagonalize coordinate vectors by Gram-Schimidt. We get E_i = a_ij \frac{\partial}{\partial xj} where a_ij only involved g_{ij}. When we compute \nabla_{E_i}E_j(p), each term contains some \partial_k g_{ij}(p)=0.