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Let $(M,g)$ be a surface that can be immersed into $\mathbb{R}^3$. Denote by $\nabla$ the associated Levi Civita connection. Further, let $X_1,X_2$ be the directions of principal curvature which are orthonormal with respect to the metric $g$.

What can one say about the following expressions?

$$\nabla_{X_1}X_2 \qquad \nabla_{X_2}X_1$$

Are they zero and can one regard the Levi Civita connection as nothing else but a directional derivative?

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  • $\begingroup$ your expressions depend on both the length and the direction of the vectors. Knowing the direction is not enough. $\endgroup$ – Xipan Xiao Jul 28 '14 at 17:17
  • $\begingroup$ They are of unit length. I only want to consider direction for now, but if there is a quick of answering on how it depends on length, I would also be interested. $\endgroup$ – madison54 Jul 29 '14 at 7:04
  • $\begingroup$ $\nabla_{X_1}X_2 - \nabla_{X_2}X_1 = [X_1, X_2]$ $\endgroup$ – Xipan Xiao Jul 29 '14 at 13:43
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Taking the unit sphere $(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$ as an example, let $X_1=\frac{\partial}{\partial \theta}=(\cos\theta \cos\phi, \cos\theta \sin\phi, -\sin\theta)$ and $X_2=\frac{1}{\sin\theta}\frac{\partial}{\partial \phi}=(-\sin\phi,\cos\phi,0)$, then $$\nabla_{X_1}{X_2}=0$$ $$\nabla_{X_2}{X_1}=\frac{1}{\sin\theta}(-\cos\theta\sin\phi, \cos\theta\cos\phi,0)=\cos\theta X_2 \ne 0$$ It is the orthogonal projection of the directional derivative of the ambient space.

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