Metric Tensors for 2-Dimensional Spheres and Hyperbolas

In summary, the metric tensor for a 2 dimensional sphere is coordinate dependent and can be expressed as ds^2 = R^2 (dtheta^2 + sin^2theta dphi^2). The metric tensor is also different for the surface of a hyperbola and it is currently unknown. Help in finding the metric tensor for a hyperbola would be greatly appreciated.
  • #1
thehangedman
69
2
Does anyone know what the metric tensor looks like for a 2 dimensional sphere (surface of the sphere)?

I know that it's coordinate dependent, so suppose you have two coordinates: with one being like "latitude", 0 at the bottom pole, and 2R at the northern pole, and the other being like longitude, 0 on 1 meridian and Pi * R on the opposite side (here, 2 Pi R gives you the same location as 0).

I've searched online and can't find a simple example of this basic metric tensor... :-(

The other one I'm curious about is the surface of a hyperbola (again, think 2-D surface of a shape in 3 dimensions). What is the metric on THAT surface?

Any type of help is greatly appreciated...
 
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  • #2
thehangedman said:
Does anyone know what the metric tensor looks like for a 2 dimensional sphere (surface of the sphere)?

The standard metric is
[tex]ds^2 = R^2 \left( d\theta^2 + sin^2\theta d\phi^2 \right).[/tex]
 
  • #3
Note: George Jones is using the physics notation which takes [itex]\phi[/itex] as the "longitude" and [itex]\theta[/itex] as "co-latitude", the opposite of mathematics notation.
 

1. What is a metric tensor for a sphere?

A metric tensor for a sphere is a mathematical tool used to describe the geometry and distances on a sphere in a precise and consistent way. It is a symmetric matrix that relates the local coordinates on a sphere to the distances between points on the sphere.

2. How is a metric tensor calculated for a sphere?

The metric tensor for a sphere can be calculated using the spherical coordinate system, which uses two angles (θ and φ) to describe a point on the sphere. The metric tensor is then derived from the relationship between these angles and the distances between points on the sphere.

3. What does the metric tensor tell us about a sphere?

The metric tensor provides important information about the geometry of a sphere, such as the curvature and distances between points. It also allows us to calculate important quantities like the surface area and volume of a sphere.

4. How does the metric tensor change on a curved surface like a sphere?

Unlike on a flat surface, the metric tensor on a curved surface like a sphere is not constant. It changes based on the position and orientation on the sphere, reflecting the non-Euclidean geometry of the surface.

5. Can the metric tensor be used for other shapes besides a sphere?

Yes, the metric tensor can be used to describe the geometry of any curved surface in three-dimensional space. It is a fundamental tool in the field of differential geometry and is used to study a wide range of shapes and surfaces, including spheres, ellipsoids, and more complex shapes.

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