Differentiable manifoldfresh_42 said:I would primarily ask how your manifold is defined?
Well, in euclidean space a point is simply coordinates or a position vector. Is there an analog to differential manifolds?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?
"different frame at every point, and no point is naturally suited to be the origin" was kinda the answer was looking forfresh_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.
A manifold is a mathematical concept that describes a space that locally resembles Euclidean space, meaning that it can be mapped onto a flat plane without any distortion. It can have any number of dimensions, but the most commonly studied manifolds are three-dimensional.
A point on a manifold is a specific location within the space that is being studied. It is usually described by a set of coordinates that correspond to the dimensions of the manifold. For example, a point on a two-dimensional manifold would be described by two coordinates, such as (x,y).
Intrinsic space refers to the space that is defined by the properties of the manifold itself, without any external reference or embedding. It is the natural space of the manifold and is independent of any coordinate system or embedding into a higher dimensional space.
Embedded space refers to the space in which the manifold is embedded or placed. This is usually a higher dimensional space that is used to visualize or study the manifold. The embedding is not inherent to the manifold and can vary depending on the chosen coordinate system.
The main difference between intrinsic and embedded space is that intrinsic space is defined by the properties of the manifold itself, while embedded space is defined by the chosen coordinate system or embedding into a higher dimensional space. Intrinsic space is independent of any external reference, while embedded space is dependent on it. Additionally, intrinsic space is usually lower dimensional than embedded space.