Finite Metric Space Imbedding in Manifolds

In summary, the conversation is discussing whether every finite metric space can be embedded in a manifold. This means that for every finite metric space (X,d), there should be a manifold with |X| many points that has the same metric as (X,d) using geodesic length. The conversation suggests starting with embedding metric spaces with 0, 1, 2, and 3 points into Riemannian manifolds before attempting with 4 points. There is also a question about whether the manifold should be connected and Riemannian. The conversation concludes with the possibility that a specific metric space, with the metric being the number of steps between nodes, cannot be embedded in a manifold, but this cannot be proven conclus
  • #1
Dragonfall
1,030
4
Is every finite metric space imbeddable in a manifold?

That is, for every finite metric space (X,d), does there exist some manifold such that there are |X| many points on it which is isometric (with the length of geodesics as metric) to (X,d)?
 
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  • #2
Can you do a set with 4 points?
 
  • #3
What do you mean?
 
  • #4
"Every finite metric space" is a tall order. Try starting with embedding "every metric space with 4 points" into Riemannian manifolds. (Actually, why not start with 0, 1, 2, 3, and then 4?)

(P.S. I assume you mean "connected and Riemannian" manifolds?)
 
  • #5
OK, it seems this one

o
\
o ---- o
/
o

where the metric is the number of steps between nodes, can't be done. But I can't prove it conclusively.
 

FAQ: Finite Metric Space Imbedding in Manifolds

What is a finite metric space?

A finite metric space is a mathematical construct that consists of a finite set of points and a function that assigns a distance between any two points in the set. The distance function must satisfy certain properties, such as being non-negative and symmetric.

What is a manifold?

A manifold is a topological space that locally resembles Euclidean space. In other words, it is a space that can be smoothly and continuously mapped onto a portion of Euclidean space. Manifolds can have different dimensions, such as 1-dimensional curves or 2-dimensional surfaces.

What is the significance of imbedding a finite metric space in a manifold?

Imbedding a finite metric space in a manifold allows us to study the structure and properties of the metric space in a more geometric and intuitive way. It also allows us to apply the tools and techniques of differential geometry to the metric space.

How is a finite metric space imbedded in a manifold?

The imbedding of a finite metric space in a manifold is achieved by finding a smooth map from the metric space to the manifold that preserves the distance between points. This is often done by constructing a metric tensor on the manifold that induces the same distances as the original metric space.

What are some applications of finite metric space imbedding in manifolds?

Finite metric space imbedding in manifolds has applications in various fields, such as computer science, data analysis, and machine learning. It can also be used to study the geometry and topology of finite metric spaces, and to solve optimization problems on these spaces.

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