Inside Sphere: Does It Have Negative Curvature?

  • Context: Graduate 
  • Thread starter Thread starter feynmanisbest
  • Start date Start date
  • Tags Tags
    Sphere
Click For Summary
SUMMARY

The inside surface of a hollow sphere does not have negative curvature; it has positive curvature. Regardless of perspective, a sphere maintains consistent curvature in all directions. In contrast, a saddle shape exemplifies negative curvature, curving differently along different axes. This fundamental distinction highlights the geometric properties of spheres versus saddle shapes.

PREREQUISITES
  • Understanding of basic geometric concepts
  • Familiarity with curvature types: positive and negative
  • Knowledge of surface geometry
  • Concept of closed versus open surfaces
NEXT STEPS
  • Research the properties of positive curvature in spherical geometry
  • Explore the characteristics of negative curvature using saddle shapes
  • Study the implications of curvature in differential geometry
  • Learn about the applications of curvature in physics and cosmology
USEFUL FOR

Mathematicians, physicists, geometry enthusiasts, and anyone interested in the properties of surfaces and curvature in mathematical contexts.

feynmanisbest
Messages
6
Reaction score
0
Does the inside of a sphere have negative curvature?
 
Physics news on Phys.org
I mean does the inside surface of a hollow sphere have negative curvature?
 
feynmanisbest said:
I mean does the inside surface of a hollow sphere have negative curvature?

the sphere is infinitely thin so it really has no inside or outside. The curvature is the same no matter how you look at it.
 
The answer is no.

A sphere (inside or outside) has positive curvature. Any direction in which you choose to measure, it will curve in the same direction. If the surface is extended far enough it will be closed.

An example of a shape that has a negative curvature is a saddle. If you go front-to-back it curves one way; but side-to-side it curves in a different direction. A saddle shape, if extended indefinitely, is open.
saddle_point.png
 
Last edited:

Similar threads

Undergrad 2-sphere manifold intrinsic definition
  • · Replies 44 ·
2
Replies
44
Views
6K
Undergrad 2-sphere with any topology can't be homeomorphic to the plane
  • · Replies 5 ·
Replies
5
Views
2K
High School Hopf fibration, Bloch sphere and quantum rotations - interactive site
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad How can a sphere be transformed using differential geometry?
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Triangles on Spheres: Isosceles and Shortest Distance Inferences
  • · Replies 8 ·
Replies
8
Views
3K
Graduate Torus and sphere with positive or negative curvature
  • · Replies 7 ·
Replies
7
Views
7K
Undergrad Round 3-sphere symmetries as subspace of 4D Euclidean space
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad Global coordinate chart on a 2-sphere
  • · Replies 37 ·
2
Replies
37
Views
6K
High School Electric Field Inside a Hollow Sphere
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Approximating smooth curved manifolds with "local bits" of curvature?
  • · Replies 5 ·
Replies
5
Views
3K