Increasing the dimensions of a manifold

Click For Summary
SUMMARY

The discussion centers on the theoretical manipulation of a R^3 manifold with curvature, aiming to create a flat R^3 chart by introducing a fourth dimension. The participant proposes extracting curvature information from the original manifold and depositing it into this new dimension, which would require a suitable topology or mapping to describe the curvature. However, it is established that altering the number of dimensions fundamentally changes the manifold itself, indicating that the proposed transformation may not be feasible within the constraints of manifold theory.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with manifold theory
  • Knowledge of curvature and its implications in topology
  • Basic concepts of dimensionality in mathematical structures
NEXT STEPS
  • Research the properties of Riemannian manifolds
  • Study the implications of dimensionality changes in topology
  • Explore methods for extracting and representing curvature information
  • Learn about the relationship between topology and manifold dimensions
USEFUL FOR

Mathematicians, theoretical physicists, and students of advanced geometry interested in manifold theory and curvature analysis.

sqljunkey
Messages
183
Reaction score
8
Suppose I have a R^3 manifold that goes into R^3 charts, if that is possible. The manifold has curvature and is Riemannian and has a metric. I want to eliminate all curvature in R^3 charts, so I want to add another dimension to the manifold, I would extract all the curvature information from the manifold and deposit it into this new 4th dimension. I would probably also have to add some kind of topology or map to the new dimension describing the curvature. But I would have flat R^3 chart. Is this possible at all?
 
Physics news on Phys.org
I'm not sure what your question is exactly, but the number of dimensions is a fundamental topological property of a manifold, so if we change the number of dimensions we have a different manifold.
 

Similar threads

Undergrad Differential structure on topological manifolds of dimension <= 3
  • · Replies 11 ·
Replies
11
Views
4K
Undergrad Maximal atlas of topological manifold
  • · Replies 37 ·
2
Replies
37
Views
5K
Undergrad 2-sphere manifold intrinsic definition
  • · Replies 44 ·
2
Replies
44
Views
7K
Undergrad Definition of manifolds with boundary
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Pseudo-Riemaniann isometries
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Grassmannian as smooth manifold
  • · Replies 6 ·
Replies
6
Views
3K
High School Ways to abstract curves or surfaces
  • · Replies 19 ·
Replies
19
Views
2K
Undergrad What is the difference between Gaussian and sectional curvature?
  • · Replies 1 ·
Replies
1
Views
3K
High School Can Curvature and Geodesics be Generalized in Higher Dimensions?
  • · Replies 10 ·
Replies
10
Views
3K
Graduate A claim about smooth maps between smooth manifolds
  • · Replies 20 ·
Replies
20
Views
6K