Warped Product Metric: assumption on the metric I am studying a book of differential geometry where the author says: Let $(M,g)$ be a Riemannian manifold of dimension $2$ with coordinates $(t,x)$ endowed with the following warped product metric:
\begin{align*}
M=\mathbb{R} \times \mathbb{S}^1, \quad g(t,x)=dt^2+f^2(t) dx^2,
\end{align*}
where $f: \mathbb{R} \rightarrow (0,\infty)$ is smooth, odd with $f^{\prime}(0)=1$. Could you please explain to me these assumptions on $f$? For example, I speculate that $f$ must be smooth and odd so that one can extend it to all of $\mathbb{R}$, right? But why do we need $f^{\prime}(0)=1$? Is this without loss of generality?
 A: ADDED: More detailed explanation:
This is a metric on a circular cylinder of varying radius. $t$ is a parameterization of the central axis, and for each $t$, $f(t)$ is the radius of the circle at that point on the axis.
If $f$ is odd, then $f(0)= 0$ and the cylinder "pinches" down to a point as $t \rightarrow 0$. In general this results in a cone with a singular point at $t = 0$. A natural question is when is the surface in fact smooth with a smooth metric for $t$ near $0$.
The assumptions that $f'(0) = 1$ and $f$ is odd imply that the surface and metric are smooth at the origin. To see this a little more easily, it's better to change the variable names from $t$ and $x$ to $r$ and $\theta$.
If $x = r\cos\theta$ and $y = r\sin\theta$, then
$$ dx = dr\,\cos\theta - d\theta\,r\sin\theta\text{ and }dy = dr\,\sin\theta + d\theta\,r\cos\theta $$
Solving for $dr$ and $d\theta$, we get
\begin{align*}
dr &= \frac{x\,dx + y\,dy}{r}\\
d\theta &= \frac{-y\,dx + x\,dy}{r^2}.
\end{align*}
Now assume that $f$ can be written as $f(r) = r\phi(r)$. Then using the formulas above and doing some algebra, you get
\begin{align*}
g &= dr^2 + f^2\,d\theta^2\\
&= \frac{(x^2 + y^2\phi^2)\,dx^2 + 2xy(1-\phi)\,dx\,dy + (y^2 + x^2\phi^2)\,dy^2}{r^2}
\end{align*}
If $\phi$ is even and $\phi(0) = 1$, then it is easy to check that this metric is smooth with respect to $(x,y)$ including at the origin. These conditions on $\phi$ are equivalent to $f'(0) = 1$ and $f$ odd.
The standard examples are flat Euclidean space, where $f = r$, the sphere of radius $R$, where
$$ f = R\sin\frac{r}{R}, $$
and hyperbolic space, where
$$ f = \sinh r $$
