How would gravity affect a structure that completely encircled the earth

If a circular structure could be built that would wrap around the entire earth, but was of a larger radius, would it 'float' above the earths surface or would it somehow fall onto the surface at some points and be pushed further away at others? In particular, could the ring be stable without moving, or would it need to orbit at the same speed as if it were a satellite.

This construct would be similar to a dyson sphere but only a band rather than a completely enclosing sphere. However, differences and similarities between the structures would be great to know about!

I'm more interested in how it would theoretically act than its feasibility, so feel free to explain materials/stresses etc but "it's the future" can be used to explain away those kind of issues.

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How would gravity affect a structure that completely encircled the earth

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