- #1

lavinia

Science Advisor

Gold Member

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## Main Question or Discussion Point

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?

- 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?