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I wanted to show that this property of geodesics doesn't even characterise the topology of the surface, and I think this is an example.alt text

The cut off edges should be asymptotic to parallel planes, and the wormhole parts can be

This is a bunch of cylinders of the kind Rbega mentions, deformed slightly so as to be joined smoothly together. The cut off edges should be asymptotic to parallel planes.
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I wanted to show that this property of geodesics doesn't even characterise the topology of the surface, and I think this is an example.alt text

The cut off edges should be asymptotic to parallel planes, and the wormhole parts can be cylinders of the kind Rbega mentions, deformed slightly so as to be joined smoothly together.