Length of a curve on a manifold

In summary, the conversation is about finding the length of a circle on the unit sphere with a given latitude of 90 degrees. The formula for calculating the length is related to the integral of the square root of the metric tensor g_ij with respect to the parameter T. The formula can be found on the Wikipedia page for Riemannian manifold. The conversation also discusses alternative methods for finding the length, such as parametrization and using geometry.
  • #1
whattttt
18
0
Can anyone help with finding the length of a circle (theta) =pi/2 (latitude 90') on the unit sphere. I know it is related to the equation
L=
integral from 0 to T of
Sqrt(g_ij (c't,c't))
The formula is on the wikipedia page called Riemannian manifold so you can get a better idea what it looks like. Any help greatly appreciated. Thanks
 
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  • #2
whattttt said:
Can anyone help with finding the length of a circle (theta) =pi/2 (latitude 90') on the unit sphere. I know it is related to the equation
L=
integral from 0 to T of
Sqrt(g_ij (c't,c't))
The formula is on the wikipedia page called Riemannian manifold so you can get a better idea what it looks like. Any help greatly appreciated. Thanks

parametrize the sphere and write down the circle with the parameters. The do the integral.
easier might be to do a little geometry and figure out the radius of the circle.
 
  • #3
I assume the sphere is
x= cos(theta)sin(alpha)
Y= sin(theta)sin(alpha)
Z= cos(alpha)


For a circle theta = pi/2 can you please point me in the right direction how to implement the formula.
I guess g_ij is worked out from the sphere but am not sure how to do the rest. Thanks for any help

Is the final answer pi, as if it is I think I know how it works
 
Last edited:
  • #4
whattttt said:
I assume the sphere is
x= cos(theta)sin(alpha)
Y= sin(theta)sin(alpha)
Z= cos(alpha)For a circle theta = pi/2 can you please point me in the right direction how to implement the formula.
I guess g_ij is worked out from the sphere but am not sure how to do the rest. Thanks for any help

Is the final answer pi, as if it is I think I know how it works

Write down the equation for the circle in your trigonometric coordinates. What is it?
 
  • #5
It works out to be (sin(alpha))^2. do I just put this into the formula
 
  • #6
whattttt said:
It works out to be (sin(alpha))^2. do I just put this into the formula

I don't think that equation is right.

You need to figure out what x,y, and z are as functions of alpha.
 

Related to Length of a curve on a manifold

1. What is a curve on a manifold?

A curve on a manifold refers to a continuous path or trajectory that is defined on a manifold, which is a mathematical space that is curved or non-Euclidean. Manifolds can have any number of dimensions, and a curve on a manifold can be described by a set of coordinates or parametric equations.

2. How is the length of a curve on a manifold measured?

The length of a curve on a manifold is measured using a mathematical concept called the arc length. The arc length is calculated by integrating the square root of the sum of the squares of the derivatives of the coordinates of the curve with respect to the parameter. This calculation can become complex for higher-dimensional manifolds.

3. Why is the length of a curve on a manifold important?

The length of a curve on a manifold is important because it helps us understand the geometry and topology of the manifold. It can also be used to calculate other properties of the curve, such as its curvature and torsion, which have implications in various fields such as physics and engineering.

4. Can the length of a curve on a manifold be infinite?

Yes, the length of a curve on a manifold can be infinite. This can occur when the curve is defined on an unbounded or infinite manifold, such as a line or a plane. It can also happen when the curve has points of infinite curvature, such as a cusp or a point of infinite slope.

5. How is the length of a curve on a manifold affected by the choice of coordinates?

The length of a curve on a manifold is independent of the choice of coordinates. This is because the arc length is calculated using the derivatives of the coordinates, and these derivatives are invariant under coordinate transformations. However, the parametric equations of the curve may change with different coordinate systems, resulting in a different representation of the curve.

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