Derived metrics on surfaces of positive curvature

In summary: I don't understand what that means.There is no change of metric everywhere, but there is a change of metric at an umbilic point where the principal curvatures are equal.Does this metric also have positive Gauss curvature? If so repeat the process and calculate the Gauss curvature again. In case one gets an infinite sequence of surfaces of positive Gauss curvature what is the limiting metric?Yes, the metric has positive Gauss curvature.
  • #1
lavinia
Science Advisor
Gold Member
3,317
708
TL;DR Summary
Using the principal curvatures of an embedded surface of positive Gauss curvature, what is the curvature of the derived metric?
Start with a closed surface of positive Gauss curvature embedded smoothly in ##R^3##. At each point, choose two independent eigenvectors of the shape operator whose lengths are the corresponding principal curvatures. By declaring them to be orthonormal one gets - I think - a new metric on the surface.

- How does one compute the Gauss curvature of this new metric?
- Can the surface also be embedded in ##R^3##?
- Does this metric also have positive Gauss curvature? If so repeat the process and calculate the Gauss curvature again. In case one gets an infinite sequence of surfaces of positive Gauss curvature what is the limiting metric?
- Does one always get a limiting metric?
 
Last edited:
Physics news on Phys.org
  • #2
I'm a little confused by the question. You can rescale eigenvectors to have unit length, so you can always pick an orthonormal basis of eigenvectors for the shape operator.

Do you have a way of picking 'favored' eigenvectors that you want to declare as having unit length for your new metric?
 
  • #3
Infrared said:
I'm a little confused by the question. You can rescale eigenvectors to have unit length, so you can always pick an orthonormal basis of eigenvectors for the shape operator.

Do you have a way of picking 'favored' eigenvectors that you want to declare as having unit length for your new metric?

Take the vectors that are of length the principal curvatures and redefine the metric so that these vectors are length 1.
 
  • #4
"Start with a closed surface of positive Gauss curvature embedded smoothly in R3. The eigenvectors of the shape operator are an orthogonal basis for the tangent plane at every point but they may not be of unit length. By declaring them to be orthonormal one gets - I think - a new metric on the surface."

How does this work? Eigenvectors of a linear operator don't usually come with a natural length, do they? And of course at an umbilic point — where the principal curvatures are equal — the choice of orthogonal directions would be arbitrary (not that this is important — it isn't).

(
But the idea of a closed Coo surface in R3 of positive Gaussian curvature everywhere is a beautiful thing. If we also consider its interior, this will be a convex body in R3 with a smooth boundary. The Minkowski sum of two smoth convex bodies is also a smooth convex body: By definition, the Minkowski sum is

B1 + B 2 = {v + w ∈ R3 | v ∈ B1 and w ∈ B2}.

So the set of such convex bodies forms a commutative monoid https://en.wikipedia.org/wiki/Monoid#Commutative_monoid.

Each such convex body B has a unique center of gravity or centroid. And at least 4 double normals. (A double normal is a chord, a line segment whose endpoints lie on the boundary ∂B of B, such that it is perpendicular to the tangent planes of ∂B at each of its endpoints.)
)
 
  • #5
zinq said:
"Start with a closed surface of positive Gauss curvature embedded smoothly in R3. The eigenvectors of the shape operator are an orthogonal basis for the tangent plane at every point but they may not be of unit length. By declaring them to be orthonormal one gets - I think - a new metric on the surface."

How does this work? Eigenvectors of a linear operator don't usually come with a natural length, do they? And of course at an umbilic point — where the principal curvatures are equal — the choice of orthogonal directions would be arbitrary (not that this is important — it isn't).

(
But the idea of a closed Coo surface in R3 of positive Gaussian curvature everywhere is a beautiful thing. If we also consider its interior, this will be a convex body in R3 with a smooth boundary. The Minkowski sum of two smoth convex bodies is also a smooth convex body: By definition, the Minkowski sum is

B1 + B 2 = {v + w ∈ R3 | v ∈ B1 and w ∈ B2}.

So the set of such convex bodies forms a commutative monoid https://en.wikipedia.org/wiki/Monoid#Commutative_monoid.

Each such convex body B has a unique center of gravity or centroid. And at least 4 double normals. (A double normal is a chord, a line segment whose endpoints lie on the boundary ∂B of B, such that it is perpendicular to the tangent planes of ∂B at each of its endpoints.)
)

The principal curvatures are the eigenvalues and the principal directions are the eigen vectors. One gets two orthogonal vectors by multiplying orthonormal eigenvectors by their corresponding principal curvatures. Declare these two vectors to be orthonormal and this gives a new metric at that point. In the case of an umbilic there is no change of metric.

For instance for the sphere there is no change of metric anywhere. For an ellipsoid there is no change of metric at its four umbilics. Across the whole surface one gets a new Riemannian metric. It is clear intuitively I think that if the surface starts out to be close enough to a sphere then the new metric is still of positive Gauss curvature.
 
Last edited:
  • #6
This seems like an fruitful thread. One thought might be to write out the equations for the Gauss curvature of the derived metric using principal coordinates on surfaces with isolated umbilics. As an experiment one might try the simple case of a spheroid - an ellipsoid of revolution - since the curvature will be constant in one of the principal directions. There might be some approximation theorems here which say that simple examples like this can be used to approximate the general case.
 

Related to Derived metrics on surfaces of positive curvature

1. What are derived metrics on surfaces of positive curvature?

Derived metrics on surfaces of positive curvature are mathematical quantities that are calculated based on the intrinsic geometry of a surface with positive curvature. These metrics are used to describe the shape and properties of the surface, and can be derived from the curvature tensor.

2. How are derived metrics on surfaces of positive curvature used in science?

Derived metrics on surfaces of positive curvature are used in various fields of science, such as physics, mathematics, and engineering. They are used to study and understand the properties of curved surfaces, which are common in nature and in man-made structures.

3. What is the difference between intrinsic and extrinsic curvature?

Intrinsic curvature refers to the curvature of a surface that is measured from within the surface itself, without any reference to an external space. Extrinsic curvature, on the other hand, is measured from the outside of the surface, in relation to an external space. Derived metrics on surfaces of positive curvature are based on intrinsic curvature.

4. How do derived metrics on surfaces of positive curvature relate to the Gauss-Bonnet theorem?

The Gauss-Bonnet theorem states that the integral of the Gaussian curvature over a surface is equal to 2π times the Euler characteristic of the surface. Derived metrics on surfaces of positive curvature, such as the Gaussian curvature and the mean curvature, are used in the calculation of the Euler characteristic, making them essential in proving the Gauss-Bonnet theorem.

5. Can derived metrics on surfaces of positive curvature be applied to real-world problems?

Yes, derived metrics on surfaces of positive curvature have practical applications in various fields, such as computer graphics, material science, and geodesy. They are used to analyze and optimize the shape and properties of curved surfaces in real-world scenarios, such as designing efficient structures or modeling natural phenomena.

Similar threads

  • Differential Geometry
Replies
13
Views
2K
  • Differential Geometry
Replies
2
Views
2K
  • Differential Geometry
Replies
6
Views
2K
  • Differential Geometry
Replies
1
Views
3K
  • Special and General Relativity
Replies
24
Views
645
  • Cosmology
Replies
24
Views
3K
  • Engineering and Comp Sci Homework Help
Replies
2
Views
810
Replies
4
Views
2K
  • Differential Geometry
Replies
3
Views
2K
Replies
1
Views
2K
Back
Top