The Soul Theorem states that in every complete, connected riemannian manifold $M$ with $\mathrm{sec}(M)\geq 0$, there exists compact, totally convex, totally geodesic submanifold $S$ such that $M$ is diffeomorphic to the normal bundle of $S$.
I don't know if I'm being too formal on my approach but I don't totally get how a manifold could be diffeomorphic to a bundle given that they're in different classes of objects. What is the precise meaning of diffeomorphic here? In general the normal bundle of $S$ (any submanifold) will be of dimension les than $\mathrm{dim}(M)$ right? So how is it possible that they're diffeomorphic?