Using the definition of normal bundle from this question:
Suppose $M\subseteq \Bbb R^n$ is an embedded $m$-dimensional submanifold. For each $x\in M$, we define the normal space to $\pmb > M$ at $\pmb x$ to be the $(n-m)$-dimensional subspace $N_xM\subseteq T_x\Bbb R^n$ consisting of all vectors that are orthogonal to $T_xM$ with respect to the Euclidean dot product. The normal bundle of $\pmb{M}$, denoted by $NM$, is the subset of $T\Bbb R^n\approx \Bbb R^n\times\Bbb R^n$ consisting of vectors that are normal to $M$: $NM = \big\{(x,v) \in \Bbb R^n\times\Bbb R^n : x\in M,\ v\in N_xM \big\}.$
Observing that $N_xM$ is a vector space of dimension $\mathbb{R}^{n-m}$, I would like to know if there are conditions when there exists a diffeomorphism from NM to $M\times\mathbb{R}^{n-m}$.