If $M,N$ are smooth manifolds with Riemann metrics $g,h$ and $f:M\to (0,+\infty)$ is a smooth function, the warped product $M\times_f N$ is defined as the smooth manifold $M\times N$ with the Riemann metric $g\times_f h$, defined as follows:
if $(p,q)\in M\times N$ and $v_1,v_2\in T_p M,\ w_1,w_2\in T_q N$, then $$\left(g\times_f h\right)_{(p,q)}\left((v_1,w_1),(v_2,w_2)\right)=g_p(v_1,v_2)+f(p)^2 h_q(w_1,w_2).$$ (Recall that the tangent space of $M\times N$ at $(p,q)$ is identified with $T_p M\times T_q N$.)
I want to prove that $\mathbb{R}^n$ with the Euclidean Riemann metric is locally isometric to the warped product $\mathbb{R_+}\times_{\lVert \cdot \rVert} S^{n-1}$, where $\lVert\cdot \rVert$ denotes the Euclidean metric, $\mathbb{R_+}=(0,\infty)$, and $S^{n-1}\subseteq \mathbb{R}^n$ is the $(n-1)-$dimensional sphere.
Do you have any ideas? Thanks in advance.
