Defining a Point on a Manifold: Intrinsic vs Embedded Space

[dsaun777]

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?

 

[fresh_42]

I would primarily ask how your manifold is defined?

 

[dsaun777]

I would primarily ask how your manifold is defined?

Differentiable manifold

 

[fresh_42]

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?

 

[dsaun777]

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?

 

[fresh_42]

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.

 

[dsaun777]

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