how to calculate the induced metric

Question

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)?

Answer 1

Assuming $f$ is defined by

$$f(\mathbf x) = f(x_1,x_2,x_3) = (x_1,x_2,x_3, \frac{1}{\sqrt{x_1^2+x_2^2+x_3^2}})$$

and the metric on $\mathbb R^4$ is defined by the Euclidean norm, it looks like the distance between $(x_1, x_2, x_3)$ and $(y_1, y_2, y_3)$ according to the pullback metric is

$\sqrt{ (x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2 + ( \frac{1}{\sqrt{x_1^2+x_2^2+x_3^2} }-\frac{1}{\sqrt{y_1^2+y_2^2+y_3^2}})^2}.$