The Manhattan Bridge is the steel suspension bridge just east of the Brooklyn Bridge, linking DUMBO and Chinatown. It is less polished as a visitor experience than its famous neighbour, but the pedestrian and bike paths give excellent views of the East River, the skyline, and the Brooklyn Bridge itself.
Use the signed access from the Brooklyn side near Jay Street, or cross from the Manhattan side around Bowery and Canal. It is noisy when trains pass, which is part of the character rather than a flaw.