I am not very familiar to differentiable manifolds, so I would appreciate some hints or reasonings about why the cone $$ M = \{(x,y,z)\in\mathbb R^3:x^2+y^2-z^2 = 0, z\geq0\} $$ is not a regular submanifold of $\mathbb R^3$.
Also, is it possible to give $M$ structure of differentiable manifold of dim. 2 with the usual topology of $\mathbb R^3$?
If M is a submanifold and p is a point of V, the set of all first derivatives of all curves through p is a vector subspaces of the ambient space.
If you do not know this, then you should prove it.
Once you have it, show that this does not hold for the apex of the cone