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I am an undergraduate studying basic differential geometry.

My question is:

Let $X\colon U\to\mathbb{R}^{3}$ be a regular surface and $(e_{1},e_{2},e_{3})$ a moving frame associated with it.

What is the point in writing the coefficients of the first and second fundamental forms in terms of the co-referential and connection forms of the moving frame?

Is it just a matter of simplifying notation? Thanks.

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Your question is a bit vague. First of all, assuming you mean an orthonormal moving frame, the first fundamental form will just be $(\omega_1)^2+(\omega_2)^2$. But expressing $\omega_1$ and $\omega_2$ in terms of your parameters $u,v$ will allow you then to compute the connection form $\omega_{12}$ (from which parallel translation can be computed) and the Gaussian curvature. The extrinsic geometry is entirely contained in the second fundamental form (knowing what $\omega_{13}$ and $\omega_{23}$ are), and you can get Gaussian curvature that way as well.

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