I'm a beginner in differential geometry. I got stuck with this exercise:
Let $V$ be real $n$-dimensional vector space. Let $S(V)$ denotes the unit sphere in $V$. Consider the map $ f: \mathbb {R}^+ \times S(V) \longrightarrow V$, $(z,v)\longmapsto \sqrt{z} v$.
If $d \rho$ and $d \sigma$ denote the volume forms in $V$ and $S(V)$ respectively, show that $f^* (d \rho )= \frac{1}{2} z^{\frac{n-2}{2}}dz \wedge d \sigma$.
I greatly appreciate your help!
We have $d\rho(zv) = z^{n-1} \, dz \wedge d\sigma(v)$ where $zv \in V$ with $z \in \mathbb{R}^+$ and $v \in S(V).$ This gives the pullback $$ f^*(d\rho)(z,v) = (d\rho\circ f)(z,v) = d\rho(f(z,v)) = d\rho(\sqrt{z}v) \\ = (\sqrt{z})^{n-1} d(\sqrt{z}) \wedge d\sigma(v) = (\sqrt{z})^{n-1} \frac{dz}{2\sqrt{z}} \wedge d\sigma(v) \\ = \frac{1}{2} z^{\frac{n-2}{2}} dz \wedge d\sigma(v) . $$