Let $U=\{ (\theta,\phi,r):\theta \in \mathbb{R}, \phi \in ]0,\pi[,r\gt 0\}$, how I can find the moving frame. I thougt: Consider the parametization for $U$ $$(\theta,\phi,r)\mapsto (r\cos(\theta)\sin(\phi),r\sin(\theta)\sin(\phi),r\cos(\phi)) $$

So we have a basis for $U$:

$$\{\frac{\partial}{\partial \theta},\frac{\partial}{\partial \phi},\frac{\partial}{\partial r}\}$$

Where: $$\frac{\partial}{\partial \theta}= (-r\sin(\theta)\sin(\phi),r\cos(\theta)\sin(\phi),0))$$

$$\frac{\partial}{\partial \phi}= (r\cos(\theta)\cos(\phi),r\sin(\theta)\cos(\phi),-r\sin(\phi))$$

$$\frac{\partial}{\partial \theta}= (-\cos(\theta)\sin(\phi),\sin(\theta)\sin(\phi),\cos(\phi))$$

We can see that basis is orhtogonal, therefore, we only need to divide each vector for its norm.Therefore we have:

$$\{\frac{1}{r\sin(\phi)}\frac{\partial}{\partial \theta},\frac{1}{r}\frac{\partial}{\partial \phi},\frac{\partial}{\partial r}\}$$

It's a orthonormal frame for $U$. But I don't how to find the dual frame, neither the conecctions forms. And I need to use that in the unity sphere, how to restrict that referencial in the sphere? Could someone help me? Thank you!


Hint: Start by showing $\theta^1 = dr$, $\theta^2=r\,d\phi$, $\theta^3=r\sin\phi\,d\theta$ is the dual basis (if you switch your first and third tangent vectors).

Next, compute $d\theta^i$ and find $\omega^i_j=-\omega^j_i$ satisfying $d\theta^i=\sum\theta^j\wedge\omega^i_j$. (I do not know which way you're being taught this material.)


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