Manifold

Intrinsic and extrinsic view

Every real manifold can be embedded in some Euclidean space. That has been proven by Hassler Whitney in the 1930s. Whitney even gave accurate bounds on dimensions — a manifold of dimension n can be embedded in Euclidean space of dimension 2n. This is the extrinsic view. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space it is always clear whether a vector at some point is tangential or normal to some surface through that point.

Related Topics:
Embedded - Hassler Whitney - Tangential - Normal

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When we view a manifold simply as a topological space without any embedding, then it is much harder to imagine what a tangent vector might be. This is the intrinsic view. An ant on a 2-dimensional manifold, say the surface of Earth, has the intrinsic view. A space ship seeing the Earth's surface has the extrinsic view.

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