Show the regular submanifold

Question

Please help me how sdo I show such a problem? I Will be happy to teach me. Thank you

Answer 1

Suppose $\dim M_1= m_1,\dim S_1=s_1\leq m_1$.From the defnition of the regular submanifold, there exists a chart $(U_1, \phi_1)$ of $M_1$ such that $$S_1\cap U_1=\{x\in M_1:x_{{s_1}+1}=x_{{s_1}+2}=\dots=x_{{{m_1}}}=0\}$$ Do the same for $S_2,M_2$:

Suppose $\dim M_2= m_2,\dim S_2=s_2\leq m_2$.From the defnition of the regular submanifold, there exists a chart $(U_2, \phi_2)$ of $M_2$ such that $$S_2\cap U_2=\{y\in M_2:y_{{s_2}+1}=y_{{s_2}+2}=\dots=y_{{{m_2}}}=0\}$$

Now, take the chart $U_1\times U_2$ of the product manifold $M_1\times M_2$

For $S_1\times S_2\subset M_1\times M_2$ it holds :

$$(S_1\times S_2)\cap (U_1 \times U_2)=(S_1\cap U_1)\times (S_2\cap U_2)=$$ $$\{(x,y)\in S_1\times S_2 :x_{{s_1}+1}=x_{{s_2}+2}=\dots=x_{{{m_1}}}=y_{{s_2}+1}=y_{{s_2}+2}=\dots=y_{{{m_2}}}=0 \}$$

Therefore, $S_1\times S_2 $ is a regular submanifold of $M_1\times M_2$ and $\dim (S_1\times S_2)=s_1+s_2$