Problem.
(1) Suppose $F:S^1 \to \mathbb{R}^3$ is a smooth immersion (i.e. $dF$ is nowhere zero). Prove that there is a unit vector $\mathbf v$ in $\mathbb{R}^3$ such that $\pi_{\mathbf v} \circ F : S^1 \to \mathbb{R}^3$ is also a smooth immersion, where $\pi_{\mathbf v}$ is the orthogonal projection to the vector subspace which is orthogonal to $\mathbf v$.
(2) Show by example that there is such an $F$ that is infective but for all $\mathbf v$ , $\pi_{\mathbf v} \circ F : S^1 \to \mathbb{R}^3$ fails to be injective.
*Remark:*Personally, I think this problem must be related to the following lemma (cf.GTM 218 Introduction to Smooth manifold), which is for Whitney Embedding Theorem. But, here the dimension condition in the lemma $N >2n+1$ doesn't hold.
As in the lemma, to show there is a $\pi_{\mathbf v}$ is an immersion, it suffices to show the following map $$ \tau : TM \smallsetminus M_0 \to RP^2 $$ is not surjective, where $M=F(S^1)$. However, I cannot continue.
