1. • Let $f: ℝ^n → ℝ$ be a continuously differentiable function. State a sufficient condition for $$ H_f=f^{-1}(0) ⊂ ℝ^n $$ to be a submanifold of $ℝ^n$.
   • If $H_f ⊂ ℝ^n$ is a submanifold, describe its tangent space at a point $p ∈ H_f$. Prove that the tangent space of $H_f$ at $p$ is contained in a space defined by the gradient of the function $f$ at $p$. State a precise characterisation of the tangent space of $H_f$ at $p$ in terms of this gradient.
   • Suppose that $f, g: ℝ^n → ℝ$ are continuously differentiable functions such that both $H_f=f^{-1}(0)$ and $H_g=g^{-1}(0)$ are submanifolds of $ℝ^n$, with $H_f$ and $H_g$ disjoint. Assume further that $p ∈ H_f$ and $q ∈ H_g$ are such that $$ {\|p-q\|} ⩽{\|x-y\|}   \text { for all }   x ∈ H_f, y ∈ H_g, $$ where $\|⋅\|$ denotes Euclidean length in $ℝ^n$. Prove that the vector $p-q ∈ ℝ^n$ is orthogonal to the tangent space of $H_f$ at $p$, as well as to the tangent space of $H_g$ at $q$.
   • Let $n=2$, and the functions $f, g$ be given by $$ f\left(x_1, x_2\right)=x_1^2+x_2^2-1 $$ and $$ g\left(x_1, x_2\right)=x_1-\frac{1}{2} x_2^2+2 $$Show that $H_f, H_g$ are submanifolds of $ℝ^2$. Find the smallest possible distance between points $p, q$ with $p ∈ H_f$ and $q ∈ H_g$. Sketch graphs of $H_f, H_g ⊂ ℝ^2$, indicating where nearest pairs of points lie.

1. • $H_f⊂ℝ^n$ is a submanifold if for every $p∈H_f$, $∇{f|}_p≠0$ [or $\operatorname{rank}∇{f|}_p=1$, or $\operatorname{rank}D{f|}_p=1$].
   • A tangent vector to $H_f$ at $p$ is $γ'(0)∈ℝ^n$ where $γ:(-ϵ,ϵ)→H_f$ is a $C^1$-function with $γ(0)=p$. The space of all tangent vectors is the tangent space $T_pH_f$ we have $T_pH_f=\ker(∇{f|}_p)⊂ℝ^n$
     Claim $T_pH_f⊂\ker(∇f_p)$
     Proof As $γ(t)⊂H_f$, we have $f(γ(t))=0$, by Chain Rule, $0=D(f∘γ)(t)=Df(γ(t))γ'(t)$
     So for $t=0,γ'(0)⊂\ker(∇{f|}_p)$
   • As before, let $γ:(-ϵ,ϵ)→H_f$ be a path, with $γ(0)=p$. Then the function $\left\|γ(t)-q\right\|^2$ has extremum at $t=0$, by the stated inequality. So\[0=\frac{d}{dt}\left\|γ(t)-q\right|^2\Big|_{t=0}=2(γ(0)-q)⋅γ'(0)=2(p-q)⋅γ'(0)\]Hence $p-q$ is orthogonal to every tangent vector at $p$ and thus to the whole tangent space. An identical argument show it is also orthogonal to $T_qH_g$.
   • $∇H_{(x_1,x_2)}=(2x_1,2x_2)≠(0,0)$ unless $(x_1,x_2)=(0,0)$, not in $H_f$. So indeed $H_f⊂ℝ^2$ is a submanifold. $∇{g|}_{(x_1,x_2)}=(1,-x_2)≠(0,0)$, so indeed $H_g∈ℝ^2$ is also a submanifold.
     To apply (b), we need to check $H_f∩H_g=∅$. But\[\left.\begin{aligned}x_1^2+x_2^2&=1\\x_1-\frac{x_2^2}2&=-2\end{aligned}\right\}⇒\frac{x_2^4}4-x_2^2+3=0\text{ no real solution.}\]so $H_f∩H_g=∅$