Geodesic Transport of Small 2D Surface on 3D Manifold

In summary, if you transport points on a small flat surface in a vertical direction above a manifold, the surface area and shape will change. This is a property of the Weyl tensor. This is true for 4D manifolds, but not for any lower dimension.
  • #1
dismachaerus
9
0
Hello,
I've just read and I think I have understood the following result :
If we were to geodesically transport all points of a small 2D surface, so small that it would be flat for all purposes, in a direction vertically above it, and if this surface belongs in an arbitrary 3D manifold, then in general both its area and its shape will change. It is impossible to achieve a change in shape without its area to dwindle or expand simultaneously. And this is a property of the so-called Weyl tensor.
I can somehow visualize how this is happening without recourse to advanced maths, but what a great result this is!
In a 4D manifold, it is also true that the object can retain its volume even if the tidal forces (curvature) change its shape, but not in any lower dimension than 4.
I have nothing more to ask, just to verify if this is true. What a great result! If it is true, it is a moment of revelation and enlightening for me.
 
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  • #2
dismachaerus said:
Hello,
I've just read and I think I have understood the following result :
If we were to geodesically transport all points of a small 2D surface, so small that it would be flat for all purposes, in a direction vertically above it, and if this surface belongs in an arbitrary 3D manifold, then in general both its area and its shape will change. It is impossible to achieve a change in shape without its area to dwindle or expand simultaneously. And this is a property of the so-called Weyl tensor.
I can somehow visualize how this is happening without recourse to advanced maths, but what a great result this is!
In a 4D manifold, it is also true that the object can retain its volume even if the tidal forces (curvature) change its shape, but not in any lower dimension than 4.
I have nothing more to ask, just to verify if this is true. What a great result! If it is true, it is a moment of revelation and enlightening for me.
Yes, but where did you read it ?
 
  • #3

Related to Geodesic Transport of Small 2D Surface on 3D Manifold

1. What is geodesic transport of small 2D surface on 3D manifold?

Geodesic transport of small 2D surface on 3D manifold refers to the movement of a small 2D surface on a 3D curved surface, following the shortest path (geodesic) between two points. This concept is important in understanding the curvature and geometry of a 3D manifold.

2. Why is geodesic transport of small 2D surface on 3D manifold significant?

Geodesic transport of small 2D surface on 3D manifold is significant because it helps us understand the intrinsic geometry and curvature of a 3D manifold. It also has practical applications in fields such as computer graphics and computer vision.

3. How is geodesic transport of small 2D surface on 3D manifold calculated?

Geodesic transport of small 2D surface on 3D manifold is calculated using differential geometry and calculus. It involves finding the shortest path between two points on a 3D manifold and determining the change in position and orientation of the small 2D surface along this path.

4. What are some real-world examples of geodesic transport of small 2D surface on 3D manifold?

One example is the movement of a 2D map on the curved surface of a globe. The map follows the shortest path between two points on the globe's surface, which is a geodesic. Another example is the motion of a small object on a curved surface, such as a ball rolling on the surface of a hill.

5. How does geodesic transport of small 2D surface on 3D manifold relate to general relativity?

Geodesic transport of small 2D surface on 3D manifold is a concept that arises from the principles of general relativity. In this theory, gravity is explained as the curvature of 3D spacetime. The geodesic path, which is the shortest path between two points, is the path that a freely moving object would follow in this curved spacetime.

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