- The man lives in the same house with a woman but never have sex or desire to have sex.
- Hey is comfortable with being nude with her and so is she, but not sexual touch or for lust.
- He cuddles and does things socially together.
- He sleeps in the same bed with her but again not sexual.
- He supports her and provides but is not married or joined together.
- He loves deeply but cannot marry the woman.
closed as too localized by Jasper Loy, simchona♦, kiamlaluno, Marthaª, RiMMER Dec 4 '11 at 23:53
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As the comments indicate, such men are sufficiently uncommon that we probably don't have a special noun for them, but the generic term is
Note that asexual is normally an adjective, but it is used as a noun too.