Let $M$ be a Riemannian manifold (possibly non-compact, possibly non-complete) and $N\subseteq M$ a smooth submanifold (possibly non-compact).
Does there exist a continuous $\mu\colon M\rightarrow (0,\infty)$, such that exponential map of $W$ $$exp\colon TW\supseteq U\rightarrow W$$ is defined on $\{(m,v)\in \vartheta(M\subseteq W)\colon||v||<\mu(m)\}$ and restricts to an embedding $$exp\colon\{(m,v)\in \vartheta(M\subseteq W)\colon||v||<\mu(m)\}\rightarrow W,$$ which provides a tubular neighbourhood of $M$ in $W$, where I denote by $\vartheta(M\subseteq W)$ the normal bundle of $M$ in $W$ thought as a subbundle of $TW$.
In other words, does the elementary construction of tubular neighbourhoods of submanifolds in euclidian space via the map $(x,y)\mapsto x+y$ generalize?
I suppose this is true. Does anyone know a reference for this?