Manifolds Part 3

Problem 1.78 Let $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ be given by $f(x, y, z)=x^{2}+y^{2}-1$. (i) Prove that $C=f^{-1}(0)$ is an embedded 2-submanifold of $\mathbb{R}^{3}$. (ii) Prove that a vector $$ v=\left(a \frac{\partial}{\partial x}+b \frac{\partial}{\partial y}+c \frac{\partial}{\partial z}\right)_{(0,1,1)} $$ is tangent to $C$ if and only if $b=0$. (iii) If $j: S^{1} \rightarrow \mathbb{R}^{2}$ is the inclusion map, prove that $j \times \mathrm{id}_{\mathbb{R}}: S^{1} \times \mathbb{R} \rightarrow \mathbb{R}^{3}$ induces a diffeomorphism from $S^{1} \times \mathbb{R}$ to $C$.

Problem 1.79 Let $\varphi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ be the map given by $$ u=x^{2}+y^{2}+z^{2}-1, \quad v=a x+b y+c z, \quad a, b, c \in \mathbb{R}, a^{2}+b^{2}+c^{2}=1 $$ (i) Find the points at which $\varphi$ is a submersion. (ii) Find $\varphi^{-1}(0)$. (iii) Find the points where $\varphi$ is not a submersion, and its image.

Problem 1.80 Consider the differentiable map $\varphi: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2}$ given by $$ u=x^{2}+y^{2}+z^{2}+t^{2}-1, \quad v=x^{2}+y^{2}+z^{2}+t^{2}-2 y-2 z+5 . $$ (i) Find the set of points of $\mathbb{R}^{4}$ where $\varphi$ is not a submersion, and its image. (ii) Calculate a basis of $\operatorname{ker} \varphi_{*(0,1,2,0)}$. (iii) Calculate the image by $\varphi_{*}$ of $(1,0,2,1) \in T_{(1,2,0,1)} \mathbb{R}^{2}$ and the image by $\varphi^{*}$ of $(\mathrm{d} u+2 \mathrm{~d} v)_{(-1,5)} \in T_{(-1,5)}^{*} \mathbb{R}^{2}$, choosing the point $(0,0,0,0)$ in $\varphi^{-1}((-1,5))$.