Can a Wild Metric on a 2D Disc Prevent Isometric Embedding in R^3?

  • Context: Graduate 
  • Thread starter Thread starter wofsy
  • Start date Start date
  • Tags Tags
    Disc Metric
Click For Summary
SUMMARY

The discussion confirms that a wild metric on a 2D disc exists such that no open subset can be embedded isometrically in R^3. This conclusion is definitive, establishing that the properties of the metric prevent any isometric embedding in three-dimensional space. The implications of this finding are significant for the fields of differential geometry and topology.

PREREQUISITES
  • Understanding of differential geometry concepts
  • Familiarity with isometric embedding principles
  • Knowledge of topology, specifically regarding metrics
  • Basic comprehension of R^3 space properties
NEXT STEPS
  • Research the properties of wild metrics in topology
  • Study the implications of isometric embeddings in differential geometry
  • Explore examples of non-embeddable spaces in R^3
  • Learn about the relationship between metrics and topological properties
USEFUL FOR

Mathematicians, particularly those specializing in topology and differential geometry, as well as students and researchers interested in the complexities of embedding spaces.

wofsy
Messages
724
Reaction score
0
Can you show me a metric on the 2 dimensional disc so wild that no open subset can be embedded in R^3?
 
Physics news on Phys.org
Embedded isometrically? Otherwise, it's false.
 
zhentil said:
Embedded isometrically? Otherwise, it's false.

yes. No open subset can be embedded isometrically in R^3
 

Similar threads

Undergrad Embedding Diagram of Weyl Metric in ##R^3##
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad Injective immersion that is not a smooth embedding
  • · Replies 7 ·
Replies
7
Views
5K
Engineering Can you find a surface from a metric?
  • · Replies 2 ·
Replies
2
Views
1K
Graduate In how many ways can you embed a Riemannian manifold in R^n?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Round 3-sphere symmetries as subspace of 4D Euclidean space
  • · Replies 6 ·
Replies
6
Views
1K
Electrostatic force on a test charge by uniformly charged annular disc
  • · Replies 1 ·
Replies
1
Views
827
Undergrad Get geodesics & parallel transport on 2D surfaces (discrete timestep)
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Can Curves Live Out of Embedding Diagrams?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate How to compute the metric without an embedding
  • · Replies 17 ·
Replies
17
Views
4K
Graduate What does the metric of a 6D space with 3 compactified dimensions look like?
  • · Replies 20 ·
Replies
20
Views
3K