Suppose we define a manifold with boundary, using the locally ringed space definition, with the local model being either open subsets of Euclidean spaces, or open subsets of the half-spaces in $\mathbb R^n$, together with the sheaf of differentiable functions on this.
Is there any danger in proceeding with this definition? Will everything go through as with the usual definition? Or does one need to be careful that problems might crop up somehwere? As an example, does the correspondence between locally free sheaves and vector bundles hold, as in the case of manifolds without boundaries?