As indicated in the title,
suppose $S$ is a surface with genus $g$, then
what is the definition of "two parallel copies of S"?
2026-03-29 14:06:41.1774793201
what is the definition of "two parallel copies of a surface S"
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If $S$ is embedded in an orientable manifold, then there is a well-defined normal direction. So you can push off a copy of $S$ along this normal direction to get two parallel copies. It is the same as embedding $S\times [0,1]$ in the ambient manifold and considering the restriction $S\times\{0,1\}$.