Questions tagged [pullback]

Here $\phi^*(f_I)$ is defined by $$\phi^*(f_I)=f_I\circ\phi,$$ the composition of $\phi$ and $f_I$. This means $\phi^*(f_I)(\mathbf x)=f_I(\phi(\mathbf x))$; in other words, $\phi^*(f_I)$ is the function resulting from $f_I$ by substituting…

Consider the mapping $f:\mathbb R^3\to\mathbb R^4$ between two Euclidean spaces: $$f(\mathbf x)=(x_1,x_2,x_3,1/\|\mathbf x\|)^T$$ How to calculate the induced metric by $f$ on $\mathbb R^3$ (i.e. the pullback metric)?