Calculate the Euler Characteristic of M Let $M$ be the following subsets of $\mathbb R^4$:$$M= \{(x,y,z,w), 2x^2+2=z^2+w^2, 3x^2+y^2=z^2+w^2 \}$$
we know $M$ is a submanifold of $\mathbb R^4$,what is the Euler Characteristic of M?
 A: Let $F:\mathbb{R}^4\rightarrow\mathbb{R}^2$ be defined by $F(x,y,z,w)=(x^2+y^2,z^2+w^2-2x^2)$. Then $M=F^{-1}(2,2)$. I'll assume that you know how to check that $(2,2)$ is a regular value of $F$ (hint: on $M$, $x,y$ do not vanish simultaneously and the same happens for $z,w$). Then $M$ is a surface embedded in $\mathbb{R}^4$. I claim that $f:M\rightarrow\mathbb{R}$ defined by $f(x,y,z,w)=w$ is a Morse function. Then, counting the indices of its critical points we will get the Euler characteristic of $M$. Let's do that:
Step 1: recall that $T_{(x,y,z,w)}M=\ker d_{(x,y,z,w)}F$. I consider two (non-exclusive) cases (you can compute yourself the kernel of $dF$ and check that the spaces I'm giving are correct).
1) $z\neq 0$: $T_{(x,y,z,w)}M=\langle yz\partial_x-xz\partial_y+2xy\partial_z,w\partial_z-z\partial_w\rangle$.
2) $w\neq 0$: $T_{(x,y,z,w)}M=\langle yw\partial_x-xw\partial_y+2xy\partial_w,w\partial_z-z\partial_w\rangle$.
Step 2: Since $df=dw$, none of the points of case 1) is critical, while in case 2) the critical points are those satisying $xy=0$ and $z=0$. You get 8 of these points: $(0,\pm\sqrt{2},0,\pm\sqrt{2}),(\pm\sqrt{2},0,0,\pm\sqrt{6})$.
Step 3: The Hessian of $f$ at these critical points is
$$
\begin{pmatrix}
2w(y^2-x^2) & 0\\
0 & -w
\end{pmatrix}.
$$
If you plug in this matrix the coordinates of the eight points above you'll see that there are 4 points of index 1, 2 of index 0 and 2 of index 2, so the Euler characteristic of $M$ is 2-4+2=0.
A: I'll give another answer without making use of Morse theory. Since we suspect $M$ is a torus we may try to find a diffeomorphism between $M$ and $S^1\times S^1$. The map
$$
\begin{array}{ccc}
M & \longrightarrow & S^1\times S^1\\
(x,y,z,w) &\longmapsto & \left(\left(\frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}}\right),\left(\frac{z}{\sqrt{2x^2+2}},\frac{w}{\sqrt{2x^2+2}}\right)\right)
\end{array}
$$
should do the trick.
