# Example of non regular surface

I was reading definition of surface in differential geometry book which defined as follows

A subset $$S\subset \mathbb R^3$$ is regular surface if $$\forall p\in S$$ there is open set in S such that $$p\in V$$ and $$\exists \phi :U\to V\in S$$ where U is open set in $$\mathbb R^2$$ such that map is surjective , smooth and homeomorphism with image and also $$\forall q\in U, dX_q:\mathbb R^2\to \mathbb R^3$$ is injective

TO better understand cocept I wanted to know counterexample of regular surface.

Any Help will be appreciated

• Another easy example is the cone $z^2=x^2+y^2$ or the half-cone $z=\sqrt{x^2+y^2}$. – Ted Shifrin Jun 4 at 15:53

If $$f:\Bbb R\to\Bbb R^2$$ is defined by $$f(x)=(x^2,x^3)$$ then the subset $$S:=f(\Bbb R)\times\Bbb R$$ is a counterexample.
One reason for this is that any tangent vector at $$0_{\Bbb R^3}\in S$$ must have the first two coordinates equal to zero. This is because if $$c:\Bbb R\to \Bbb R^3$$ is a smooth curve with value in $$S$$ such that $$c(0)=0_{\Bbb R^3}$$ then $$c_1^\prime(0)=0$$ because $$c_1$$ is positive, and because $$c_2=(c_1)^{3/2}$$ you have $$c_2^\prime(0)=0$$. Hence $$v=c^\prime(0)=(0,0,c_3^\prime(0))$$
But for a surface the set of tangent vectors at a point is a vector space of dimension $$2$$.
Note that if we define $$\phi:\Bbb R^2\to \Bbb R^3$$ by $$\phi(x,y)=(f(x),y)$$ then $$\phi$$ is smooth and is a homeomorphism onto its image $$S$$, with inverse $$\phi^{-1}:(x,y,z)\mapsto (y^{1/3},z$$). So $$\phi$$ has all properties you are looking for, except the last one: $$d_0\phi$$ is not one-to-one. This shows you why this property is important, which is not clear at first in my opinion.