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What are manifolds?

What are manifolds? Here are some definitions.

Noun
  1. plural of manifold
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Included in the intermediate chapters are introductions to differentiable manifolds and Lie groups.
The most important result in this paper is that two real algebraic manifolds are equivalent if and only if they are analytically homeomorphic.
The exhaust manifolds are dual-skin, with stainless steel inners.
One stir plate is pretty much like another, ditto the separatory funnels flasks, vacuum manifolds, etc.
The rules require all cylinder blocks, heads and intake manifolds to have factory part numbers.
Topology is the mathematical discipline concerned with surfaces or manifolds in higher dimensions.

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