Consider $M := \mathbb R^3$ as a smooth manifold with a Riemannian metric $g := \sum_{i=1}^3 dx^i\otimes dx^i$, where $(x^1, x^2, x^3)$ is the standard coordinate of $M$. Let $N\subset M$ be a submanifold defined as $S^2 \cap H$, where $H = \{(x^1, x^2, x^3) \mid x^3 > 0\}$. Then $N$ has the coordinate $(x^1,x^2)$.
The problem is to represent the restriction of $g$ to the tangent bundle $TN$ w.r.t. the (global) frame $(dx^1, dx^2)$.
I tried to solve it in the following way: I introduced a coordinate neighborhood $(M \cap H , \psi)$ to $M$, where $\psi:M\cap H \rightarrow(0,\infty)\times(0,\pi)\times(0,\pi)$ is a diffeomorphism defined by $\psi^{-1}(r, \phi, \theta) = (r\cos\phi, r\sin\phi\cos\theta, r\sin\phi\sin\theta)$. Then I write $g$ in terms of $dr$, $d\phi$ and $d\theta$. Since I get $g|_{TN^{\otimes 2}}$ by dropping the terms that involve $dr$ from $g$, I rewrite this in terms of $dx^1, dx^2$ by applying inverse mapping theorem to obtain the result.
While this seems to work in theory, it is a bit cumbersome. I would be most grateful if you could help me solve this problem more easily.