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.
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)).