The graph of $f:\mathbb R^2\to \mathbb R$ can be embbeded in $\mathbb R^2$ or only in $\mathbb R^3$?

Question

Let $f:\mathbb R^2\to \mathbb R$ a continuous function. I'm a bit confuse with something :

The manifold is $$M=\{(x,y,z)\mid z=f(x,y)\},$$ so it's a subset of $\mathbb R^3$. This mean that $M$ is a submanifold of $\mathbb R^3$.

A parametrization of $M$ is given by $\varphi (u,v)=(u,v,f(u,v))$, and thanks to this, we can define a measure on $\mathbb R^2$ (by $\mu(A\times B)=\int_{A\times B}\sqrt{\det g(u,v)}dudv$ where $(g(u,v))_{ij}=\left<\gamma _i,\varphi _j\right>$ for $i,j\in \{u,v\}$). What the meaning of that ? Does it mean that $M$ can be embedded in $\mathbb R^2$ ?

Answer 1

If we have a $\mathcal{C}^\infty$ function $f:\mathbb{R}^2\to\mathbb{R}$, and we study its graph $M=\{(x,y,z):z=f(x,y)\}$ then indeed the space is a smooth manifold with a (global) parametrization $(x,y)\mapsto (x,y,f(x,y))$ from $\mathbb{R}^2\to M$. $M\subseteq \mathbb{R}^3$, and as such it is a submanifold of $\mathbb{R}^3$. Not only that: because we have a globally defined coordinate chart $\mathbb{R}^2\to M$, we get that $M$ is diffeomorphic to $\mathbb{R}^2$ and hence embeds in $\mathbb{R}^2$. If we replace $f$ smooth with $f$ continuous, this works except that we only get a homeomorphism, and the resulting surface might not be smooth.

As for the last bit involving the measure, this is saying that because we have a map $f:\mathbb{R}^2\to M$ which is a diffeomorphism, we can define a measure on $A\times B$ by integrating over the region $f(A\times B)$. The meaning of the notation you gave isn't exactly clear to me (what are $\gamma$ and $\phi$?), so I'll give an explanation in more elementary terms. Recall from multivariable calculus that for $A,B\subseteq \mathbb{R}$ $$ \text{Area}(f(A\times B))=\int_{f(A\times B)}1\cdot dA=\int_{A\times B}1\cdot \lVert f_x\times f_y\rVert \cdot dA$$ here $$f_x=(\partial_x (x),\partial_x(y),\partial_x(f(x,y))=(1,0,\partial_xf(x,y))$$ and $$f_y=(\partial_y(x),\partial_y(y),\partial_y(f(x,y))=(0,1,\partial_yf(x,y))$$ Then $$ f_x\times f_y=\det \begin{bmatrix} i&j&k\\ 1&0&\partial_xf(x,y)\\ 0&1&\partial_yf(x,y) \end{bmatrix}=i(-\partial_xf(x,y))-j(\partial_yf(x,y))+k.$$ $$ \lVert f_x\times f_y\rVert=\sqrt{(\partial_x f)^2+(\partial_yf)^2+1}.$$ So, at long last we have defined a measure $$ \mu_f(A\times B)=\int_{A\times B} \sqrt{(\partial_x f)^2+(\partial_yf)^2+1}dxdy.$$ To check that this is a measure in the sense of measure theory is not too hard.

Answer 2

$M$ is homeomorphic to $\mathbb{R}^2$. A canonical homeomorphism is given by $h : M \to \mathbb{R}^2, h(x,y,z) = (x,y)$. In particular $M$ can be embedded into $\mathbb{R}^2$. This does not mean that there exists an embedding which preserves the measure of sets.

In fact $M$ is a (topological) submanifold of $\mathbb{R}^3$. Define $\phi : \mathbb{R}^3 \to \mathbb{R}^3, \phi(x,y,z) = (x,y,z-f(x,y))$. This is a homeomorphism, its inverse being $\psi(x,y,z) = (x,y,z+f(x,y))$. We have $\phi(M) = \mathbb{R}^2 \times \{ 0\}$ which proves our claim.