Differential geometry vs differential topology

Click For Summary
SUMMARY

Differential topology and differential geometry are distinct fields within mathematics, focusing on different aspects of smooth manifolds. Differential topology studies properties that remain invariant under diffeomorphisms, emphasizing topological invariants and cobordism classes. In contrast, differential geometry examines properties reliant on a metric, particularly the smooth metric tensor on bundles over manifolds. These definitions, while simplified, effectively delineate the core differences between the two disciplines.

PREREQUISITES
  • Understanding of smooth manifolds
  • Familiarity with diffeomorphisms
  • Knowledge of metric tensors
  • Basic concepts of topological invariants
NEXT STEPS
  • Research the cobordism classes of smooth manifolds
  • Explore topological invariants in differential topology
  • Study the role of smooth metric tensors in differential geometry
  • Investigate the conditions under which combinatorial manifolds admit smooth structures
USEFUL FOR

Mathematicians, students of topology and geometry, and researchers interested in the properties of smooth manifolds will benefit from this discussion.

sabw1992
Messages
1
Reaction score
0
in a nutshell, what is the difference between those two fields?
 
Physics news on Phys.org
I think of differential topology as the study of the properties of smooth manifolds that do not depend upon a metric. One might say that it is the study of invariants of smooth manifolds under diffeomorphsms. Differential geometry on the other hand studies the properties of smooth manifolds that do depend upon a metric, particuarly a smooth metric tensor on a smooth bundle over the manifold.

These two definition are overly restricitve but capture the basic idea of the difference.

Classical subjects in Differential topology are

- What are the cobordism classes of smooth manifolds?

- what are the topological invariants of smooth manifolds?

- when does a combinatorial manifolds admit a smooth structure?

- What are the smooth structures on a manifold?Differential geometry covers a vast array of subjects
 

Similar threads

Undergrad Differential Geometry with GNU/Linux
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Differential structures over a topological manifold
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Definition of manifolds with boundary
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad 2-sphere with any topology can't be homeomorphic to the plane
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad How can a sphere be transformed using differential geometry?
  • · Replies 17 ·
Replies
17
Views
4K
Undergrad Differential structure on topological manifolds of dimension <= 3
  • · Replies 11 ·
Replies
11
Views
4K
Graduate What Is the Gauss Embedding Theorem Mentioned in the Video Talk?
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Can we always rewrite a Tensor as a differential form?
  • · Replies 8 ·
Replies
8
Views
4K
Graduate Differential Forms or Tensors for Theoretical Physics Today
  • · Replies 70 ·
3
Replies
70
Views
17K
Undergrad Differential Geometry: Comparing Metric Tensors
  • · Replies 37 ·
2
Replies
37
Views
7K