 Solution

Let $g(x,y,z)=x^2+y^2-z^2$. The upper cone away from the apex is the part of $g^{-1}(0)$ with $z>0$. ThereThe regular level set theorem says that a level set of a smooth scalar function with nonzero gradient is an embedded submanifold of codimension one. Intersecting it with the open half-space $z>0$ preserves this local property. Thus $M$ is a two-dimensional submanifold of ${\mathbb{R}}^3$.