Defining a Point on a Manifold: Intrinsic vs Embedded Space

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Discussion Overview

The discussion revolves around defining a point on a manifold, particularly in the context of differentiable manifolds and their embeddings in higher-dimensional spaces. Participants explore the distinctions between intrinsic and extrinsic definitions, as well as the implications of using coordinates or tangent spaces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants inquire about the definition of the manifold itself, suggesting that the nature of the manifold is crucial for discussing how to define a point on it.
  • There is a suggestion that defining a point on a manifold may be analogous to defining a vector in a vector space, raising questions about the context of the vector space.
  • One participant emphasizes the need for a coordinate frame, origin, and directions when discussing points on a manifold, noting that these are only defined locally.
  • Another participant points out that in an arbitrary manifold, no point is uniquely suited to be an origin, implying that all points can serve as potential origins.
  • There is mention of using a comoving coordinate system and the importance of understanding the manifold's atlas to define points effectively.

Areas of Agreement / Disagreement

Participants express differing views on how to approach the definition of points on manifolds, with no consensus reached on a singular method or framework. The discussion remains unresolved regarding the best approach to defining points.

Contextual Notes

Participants highlight limitations in defining points based solely on the term "differentiable manifold," indicating that further specification of the manifold's atlas is necessary for clarity.

dsaun777
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Say you have some n dimensional manifold embedded in a higher space. what is the best way to describe or define a point on a manifold with or without coordinates. How could I do this either intrinsically or using the embedded space. Would you use the tangent space somehow using basis vectors?
 
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I would primarily ask how your manifold is defined?
 
fresh_42 said:
I would primarily ask how your manifold is defined?
Differentiable manifold
 
This is not a description, this is an arbitrary object in a category.

You basically asked something like "what is the best way to describe or define a vector in a vector space with or without coordinates?" and answered to "Which vector space?" by "Finite dimensional vector space." How would you answer this question?
 
fresh_42 said:
This is not a description, this is an arbitrary object in a category.

You basically asked something like "what is the best way to describe or define a vector in a vector space with or without coordinates?" and answered to "Which vector space?" by "Finite dimensional vector space." How would you answer this question?
Well, in euclidean space a point is simply coordinates or a position vector. Is there an analog to differential manifolds?
 
You have embedded it in a higher and I assume Euclidean space, so this embedding provides naturally coordinates. If we only have the manifold itself, then the question is how it is defined. We need a frame for coordinates, an origin and directions. On an arbitrary manifold we have those only locally, i.e. a different frame at every point, and no point is naturally suited to be an origin, or better: all points are. We often have paths within a manifold, so we could use a comoving coordinate system. Whatever you want to do, the first question is always: what do you have?

If "differentiable manifold" is your only answer, then its atlas is mine. Show me the atlas and I show you your points.
 
fresh_42 said:
You have embedded it in a higher and I assume Euclidean space, so this embedding provides naturally coordinates. If we only have the manifold itself, then the question is how it is defined. We need a frame for coordinates, an origin and directions. On an arbitrary manifold we have those only locally, i.e. a different frame at every point, and no point is naturally suited to be an origin, or better: all points are. We often have paths within a manifold, so we could use a comoving coordinate system. Whatever you want to do, the first question is always: what do you have?

If "differentiable manifold" is your only answer, then its atlas is mine. Show me the atlas and I show you your points.
"different frame at every point, and no point is naturally suited to be the origin" was kinda the answer was looking for
 

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