under heaven—
1. on earth; in the world:
- You travel for twenty miles over one of the most unfortunate, desolate countries under heaven.
2. is used to emphasize an interrogative sentence:
- And why under heaven should one write a history of anything—even a tramp dog!—without a sense of humor!
under the sky—
1. (also: under the open sky) in the open air; not inside a house or building:
- I will sleep under the sky rather than stay in the city for another night.
2. = under heaven 1:
- Am I deceived, I who have known more liars than any man under the sky?
Note: Neither expression correlates in meaning with the phrase under a cloud—suspected of smth. reprehensible:
- He leaves the service, possibly under a cloud…. The inference in the club think tank was that the cloud was sexual.