computing the components of $f^*g_N$

Question

Let $M$ and $N$ are to compact complex manifolds of dimensions $m$ and $k$ respectively, and $f:M\to N$ is a holomorphic map then how can we compute the components of $f^*g_N$

Answer 1

The pullback of a Riemannian metric is treated like the general pullback for tensor fields. In other words, $f^*\mathrm{g}(X_q,Y_q)=\mathrm{g}(f_*X_q,f_*Y_q)$. If we are working in a specific chart, we can write out the metric as $$\mathrm{g}=\sum_{i,j}g_{ij}\mathrm{d}x^i\otimes\mathrm{d}x^j.$$ For the ease of calculation, we often just write this out as a matrix, where the $(i,j)^\text{th}$ entry is $g_{ij}$. In this case, you can apply $\mathrm{g}$ to two tangent (column) vectors $v$ and $w$ by left-multiplying this matrix by the transpose of $v$ and right-multiplying by $w$. Since we are working within a local coordinate system, we can represent the pushforward by the Jacobian.

Thus, the new matrix for $f^*\mathrm{g}_N$ would be given by left-multiplying the matrix for $\mathrm{g}_N$ by the transpose of the Jacobian of $f$ in the given coordinates and right-multiplying by the Jacobian of $f$. You can get the new components $g'_{ij}$ from this matrix.