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\title{Smooth Manifolds Fall 2023 - Homework 4.5}

\begin{document}

\maketitle

\begin{problem}
Consider spheres with centers $x_1,x_2 \in \R^3$ and radii $r_1,r_2 \in \R_+$, respectively. Find the quadruples $(x_1,x_2,r_1,r_2)$ for which the intersection is transverse, and show that your answer is correct. 
\end{problem}

\begin{problem}
Let $f : \R^2 \to \R$ be a $C^\infty$ function. Find a diffeomorphism between $\R^2$ and $\Gamma_f$, the graph of $f$. Compute the tangent bundle to the graph of $f$ in two ways:

\begin{enumerate}
\item at each point $z \in \Gamma_f$, find a pair of linearly independent vectors in $T_z\Gamma_f$
\item at each point $z \in \Gamma_f$, find a functional $\varphi_z$ such that $T_z\Gamma_f = \ker \varphi_z$
\end{enumerate}

Finally, consider two $C^\infty$ functions $f$ and $g$ on $\R^2$. Find a necessary and sufficient condition on the functions $f$ and $g$ so that there exists $c \in \R$ such that $\Gamma_f$ and $\Gamma_{g+c}$ intersect nontrivially and transversally.
\end{problem}

\begin{problem}
Show that for almost every matrix $A \in M_2(\R) = \R^4$, the function $F_A : S^1 \to \R^2$ is transverse to $S^1$, where $F_A(v) = Av$. For which matrices $A$ does the image of $F_A$ intersect $S^1$ nontrivially? For which values is $F_A$ an immersion? An embedding? 
\end{problem}

\begin{problem}
Let $X = M_2(\R) = \R^2$, and $F : X \to X$ denote the function $F(A) = A^2$. Compute the differential of the function $F$ as a $4 \times 4$ matrix.
\end{problem}


\end{document}




