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

[Ahmed1029]

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?

 

[Ibix]

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.

 

[Nugatory]

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”

 

[pervect]

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.

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

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)).

 

[DrGreg]

$$\vec{A} \cdot \vec{B} = ||A|| ||B|| \sin theta$$

Typo:$$\vec{A} \cdot \vec{B} = \|A\| \|B\| \cos \theta$$

 

[Orodruin]

$$\vec{A} \cdot \vec{B} = ||A|| ||B|| \sin theta$$

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.

 

[pervect]

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.

 

[Ibix]

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.

 

[Pony]

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.

 

[Orodruin]

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.