 Solution

Writing the upper cone as $(r\cos \theta ,r\sin \theta ,r)$ gives metricOn any polar branch, the mappulls this back to $d{s}^2+s^{2}d{\phi }^2$, so it is a local isometry. Strictly, because $\sqrt{2}\phi$ is not $2\pi$-periodic, this formula is globally single-valued on the universal polar cover and locally on ${\mathbb{R}}^2\setminus \{0\}$; that is the developing map used here.

An isometry fixing $p$ lifts locally to a Euclidean isometry fixing a lift of $p$. Compatibility with the cone's angular identification leaves only the identity and reflection across a radial line. Descending to the cone, these are the identity and reflection in the plane through the cone axis and $p$.