For a $k$-manifold $X$ in $\mathbb{R}^M$, define its tangent bundle $T(X) \to X$ and the normal bundle $N(X) \to X$. Let $B$ denote the open punctured unit ball in $\mathbb{R}^3$, i.e., $B = \{y \in \mathbb{R}^3 \mid 0 < ||y|| < 1\}$. Prove or disprove the following.
There exist $X$, $k$, $M$ such that $T(X)$ is diffeomorphic to $B$.
There exist $X$, $k$, $M$ such that $N(X)$ is diffeomorphic to $B$.
for (1), tangent bundle is always even dimensional manifold. Here open punctured unit ball is a 3 dimensional manifold so there doesn't exist any manifold which has tangent bundle diffeomorphic to open punctured unit ball.
for (2) I guess $\mathbb{S}^2$ is the correct manifold , but the ambient space should be $\mathbb{R}^6$ to make the normal bundle 3 dimensional manifold. Am I correct?