What's the definition of angle in a curved space embedded in a higher Eucledian space?

In summary, angle in a cuved space embedded in a higher eucledian space is the angle between vectors tangent to the surface and pointing in the same direction as the lines where they meet.
  • #1
Ahmed1029
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I don't want to post this in a math forum because it's very basic and I just want a straightforward answer, not something math heavy . What's the definition of angle in a cuved space embedded in a higher eucledian space? Like when I have a spherical surface in 3d eucledian space and want to work out the interior angles of a triangle on this 2 diemnsional sphere, what does "angle" mean here?
 
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  • #2
If you have a Euclidean space in which your lower dimensional space is embedded, then it's the angle between vectors tangent to the surface and pointing in the same direction as the lines where they meet.

You have to work a bit harder if you don't have a nice space to embed your curved space in, which is the case in GR.
 
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  • #3
Ahmed1029 said:
Like when I have a spherical surface in 3d eucledian space and want to work out the interior angles of a triangle on this 2 diemnsional sphere, what does "angle" mean here?
This is why we need the concept of “tangent space”
 
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  • #4
Ahmed1029 said:
I don't want to post this in a math forum because it's very basic and I just want a straightforward answer, not something math heavy . What's the definition of angle in a cuved space embedded in a higher eucledian space? Like when I have a spherical surface in 3d eucledian space and want to work out the interior angles of a triangle on this 2 diemnsional sphere, what does "angle" mean here?

Let's take your example of angles on a 2-sphere. At any point on the 2-sphere, we can generate a flat tangent plane to the sphere at that point. An illustration from a random webpage might be helpful here.

Then you can measure the angle in the flat tangent space.
maxresdefault.jpg

Mathematically, if you have two vectors, and an inner product operator, the relation between angle and inner product is given by the well known relation
$$\vec{A} \cdot \vec{B} = ||A|| ||B|| \sin theta$$

Useful wiki links.
Tangent Space: https://en.wikipedia.org/wiki/Tangent_space
Inner Product Space: https://en.wikipedia.org/wiki/Inner_product_space

You'll note from the wiki the general observation
wiki said:
The inner product of two vectors in the space is a scalar, often denoted with angle brackets . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors.

An important mathematical property of vectors is that they add commutatively
$$\vec{A} + \vec{B} = \vec{B} + \vec{A}$$
Note that finite displacements on the sphere are not vectors. As Misner remarked, if you go 500 miles north and 500 miles east on the surface of the Earth (modeled for the sake of the example as a sphere), you do not necessarily wind up at the same location as if you go 500 miles east then 500 miles north. But this difficulty doesn't arise in the tangent plane to the sphere, in fact vectors on a sphere are commonly regarded as existing in the tangent space, with every point on the sphere having it's own tangent space.

If you need more detail, the topic you need to read about is called "differential geometry".

((note: not sure what's up with the LATEX not working, I don't see my error)).
 
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  • #5
pervect said:
$$\vec{A} \cdot \vec{B} = ||A|| ||B|| \sin theta$$
Typo:$$\vec{A} \cdot \vec{B} = \|A\| \|B\| \cos \theta$$
 
  • #6
pervect said:
$$\vec{A} \cdot \vec{B} = ||A|| ||B|| \sin theta$$
DrGreg said:
Typo:$$\vec{A} \cdot \vec{B} = \|A\| \|B\| \cos \theta$$
Apart from the cos vs sin and ##\theta## vs ##theta##, do note the subtle difference in the typesetting of the ##\|##. It is one of those things you might miss if you are not aware of it, even more subtle than the use of ##\ll## over ##<<## or ##|\psi\rangle## over ##|\psi >##. Once you are aware it is however difficult to unsee.
 
  • #7
For some reason, the latex wasn't formatting for me at all, earlier. The error with cos instead of sin was however a mental error, my bad. I do hope that with this error corrected, the post is useful in explaining how the inner product of vectors gives a definition of angle between vectors, though it was rather short and ommited quite a lot of detail.
 
  • #8
pervect said:
For some reason, the latex wasn't formatting for me at all, earlier.
You were the first user or LaTeX on this page, and there's a bug where LaTeX won't render in preview or in new posts unless there is already LaTeX on the page. You can either make your best effort, post, and then refresh the page, or with sone LaTeX in the editor go into preview mode and refresh the page then (it's worth copying your post to clipboard before refreshing if you do that). Either way, the refresh wakes MathJax up and makes it start rendering.
 
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  • #9
Nugatory said:
This is why we need the concept of “tangent space”
Isn't this like the only case when we don't need the concept of the tangent space? If our space is embedded in R^n, then the angle between e.g. two curves that go through a point is just the angle in R^n.

We need tangent spaces, when our space is not embedded anywhere.
 
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  • #10
Pony said:
Isn't this like the only case when we don't need the concept of the tangent space? If our space is embedded in R^n, then the angle between e.g. two curves that go through a point is just the angle in R^n.

We need tangent spaces, when our space is not embedded anywhere.
You need the tangent space but you also need a metric. When you have an embedding of your space into a metric space then a metric on your space may be defined through the pullback of the metric in the embedding space. This is general and not restricted to ##\mathbb R^n## as the embedding space.
 
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Related to What's the definition of angle in a curved space embedded in a higher Eucledian space?

1. What is an angle in a curved space embedded in a higher Euclidean space?

An angle in a curved space embedded in a higher Euclidean space is a measure of the amount of rotation between two intersecting curves or surfaces. It is determined by the amount of curvature in the space and the distance between the curves or surfaces.

2. How is an angle in a curved space different from an angle in a flat space?

An angle in a curved space is different from an angle in a flat space because in a curved space, the distance between two intersecting curves or surfaces is not constant. This means that the amount of rotation between the curves or surfaces will vary depending on the amount of curvature in the space.

3. Can an angle in a curved space be measured using the same methods as in a flat space?

No, an angle in a curved space cannot be measured using the same methods as in a flat space. In a flat space, angles can be measured using a protractor or by using trigonometric functions. However, in a curved space, the non-constant distance between the intersecting curves or surfaces makes these methods inaccurate.

4. How is an angle in a curved space calculated?

An angle in a curved space is calculated using mathematical formulas that take into account the amount of curvature in the space and the non-constant distance between the intersecting curves or surfaces. These formulas are more complex than those used for calculating angles in a flat space.

5. Why is it important to understand angles in a curved space?

Understanding angles in a curved space is important in fields such as physics, astronomy, and geometry. It allows us to accurately measure and describe the relationships between objects in a curved space, which is essential for understanding the laws of nature and the structure of the universe.

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