can one construct a solid Klein bottle - a 3 manifold whose boundary is a Klein bottle as follows. - Start with a solid cylinder and identify the two bounding disks by a reflection. - The boundary becomes a Klein bottle but is this a smooth manifold whose boundary is this Klein bottle? - If so does this manifold deform onto its central circle just as a solid torus would? - Since reflection is an isometry of the disk, can one give this manifold a flat metric? In general if the boundaries of two Riemannian manifolds are identified by an isometry do their metrics extend?