1. In Mathematics:
* A topological space that locally resembles Euclidean space. This means that at every point on the manifold, you can find a small region that looks like a flat space. Think of a curved surface like a sphere: at any given point, you can zoom in enough to see a tiny flat patch.
* A smooth manifold: This is a special type of manifold where the "local resemblance" to Euclidean space is even stronger. The transition between these local patches is smooth, meaning there are no abrupt changes or "corners".
2. In General Usage:
* Many and varied. This is the more common, everyday meaning. For example, "The restaurant offers a manifold of choices for appetizers."
* A device that creates a copy of something. This is often used in the context of machinery, for example, "A manifold is used to distribute fuel to the engine cylinders."
Here are some examples of how "manifold" is used in different contexts:
* "The theory of manifolds is a cornerstone of modern geometry." (Mathematics)
* "The artist explored the manifold emotions of love and loss in her paintings." (General Usage)
* "The intake manifold on this engine is designed to improve airflow." (Mechanical Engineering)
To understand which meaning is intended, you need to pay attention to the context of the sentence or phrase.