Consider the Riemannian manifold $(\mathrm S^3_+, i^*g)$, where $i : \rm S^3 \rightarrow\Bbb R^4$ is the inclusion, $g = g_{\Bbb R^4}$ is the euclidean metric and $\rm S^3_+ = \{ (x,y,z,t)^T : t >0\}$. We denote $\tilde g = i^*g$, analytically determine the coefficients $\tilde g_{i,j}$, $i,j = 1,2,3$.
I was thinking that since $\rm S^3 \subset \Bbb R^4$ the inclusion act as the identity and pullback of identity is also identity, so could I say that the coefficients are exactly the same of the standard euclidean metric? Otherwise I'm not really sure how to proceed. Thank you very much.