Rotations in differential geometry

[meteo student]

Simple and basic question(maybe not). How are rotations performed in differential geometry ?

What does the rotation matrix look like in differential geometry? Let us assume we have orthogonal set of basis vectors initially.

I am looking to calculate the angle between two geodesics. Can this local angle be related to the tangent vectors ?

 

[WWGD]

Simple and basic question(maybe not). How are rotations performed in differential geometry ?

What does the rotation matrix look like in differential geometry? Let us assume we have orthogonal set of basis vectors initially.

I am looking to calculate the angle between two geodesics. Can this local angle be related to the tangent vectors ?

If I understood you correctly, the angle between (differentiable) curves at a given point is defined as the angle between their tangents at the common point.

 

[meteo student]

Excellent. So if the two curves are geodesics then the "local angle" could be defined in terms of latitude and longitude. Then the tangent vectors could be defined in terms of latitude and longitude. Am I correct ?

 

[WWGD]

Excellent. So if the two curves are geodesics then the "local angle" could be defined in terms of latitude and longitude. Then the tangent vectors could be defined in terms of latitude and longitude. Am I correct ?

Sorry, I don't get the reference to latitude and longitude outside of spheres. Would you elaborate?

 

[meteo student]

I am assuming that two curves are arcs of two great circles on the surface of the sphere with unit radius. That is assuming the Earth is a perfect sphere and not an ellipsoid.

 

[Fredrik]

The angle $\theta$ between two vectors x and y in a real inner product space is defined by $\cos\theta =\frac{\langle x,y\rangle}{\|x\|\|y\|}$.

If your two geodesics intersect at a point p in the manifold M, then their tangents (velocity vectors) are elements of $T_pM$, i.e. the tangent space at p. If M is a Riemannian manifold, then $T_pM$ is a real inner product space.

 

[meteo student]

Thanks for that answer. But I am not a mathematician and I am dealing with Euclidean space in three dimensions - latitude, longitude and r (radius of uniform sphere). I am trying to correlate the mathematical definitions to definitions in cartography- differential geodesy. So if I have a geodesic pointing to true north and I have another arc which is east of that geodesic we call that angle between two arcs as the azimuth. So the tangent vectors of the two geodesics would give me the angle that I require ?

 

[Narasoma]

Rotation

Simple and basic question(maybe not). How are rotations performed in differential geometry ?

What does the rotation matrix look like in differential geometry? Let us assume we have orthogonal set of basis vectors initially.

I am looking to calculate the angle between two geodesics. Can this local angle be related to the tangent vectors ?

Rotation seems to be a special concepts for three dimensional Euclidean space. But in a more general manifold orientation using n-form is usually used...

 

[zinq]

(No: n-dimensional differential forms, and rotations, are two different concepts. )

A rotation of n-dimensional Euclidean space, for any n ≥ 1, is a mapping

f: ℝⁿ → ℝⁿ ​

of the space to itself such that

a) f(0) = 0;

b) all distances are preserved:

||f(x) - f(y)|| = ||x - y||​

for all points x, y of ℝⁿ;

and

c) there is a continuous family {f_t | 0 <= t <= 1} of mappings

f_t: ℝⁿ → ℝⁿ ​

satisfying a) and b) such that f₀ = f and f₁ = the identity mapping (i.e., taking each point of ℝⁿ to itself).

* * *

This definition given by a), b), and c) above is a conceptual definition.

But it turns out to be very useful that it is equivalent to the following computational definition:

f: ℝⁿ → ℝⁿ ​

is a rotation according to a), b), and c) if and only if f is an invertible linear mapping, given by an n × n matrix L satisfying

i) L^-1 = L^t

(i.e., the inverse of L equals the transpose of L) and

ii) det(L) > 0.

(In fact when i) and ii) hold, then det(L) = +1, necessarily.)

 

[meteo student]

I do not understand why it should be restricted to Euclidean space.
I am trying to use derivatives to describe rotations so that it can be linked to curvature eventually. If you have a conformal map then the transformation can be described in terms of a scaling term times the Jacobian matrix. Is that correct ? Let us assume I have two curves(geodesics) defined in terms of latitude and longitude on the surface of a sphere(or an ellipsoid). The first curve is defined with respect to one set of axes and the second curve is defined with respect to a rotated set of axes.

Assuming I have these two functions
λ' = f(λ,Φ)
and Φ' = f (λ,Φ)

Then I can define these basis vectors $e' = ∑ P_i e_n$
between the unrotated and rotated system.
I am going to call δ the local angle formed between two meridians at the point P.

 

[zinq]

There are a few things I don't understand.

You write: "If you have a conformal map then the transformation can be described in terms of a scaling term times the Jacobian matrix. Is that correct ?"

1) I have to ask: What transformation do you mean by "the" transformation ?

1a) What do you mean by "the" Jacobian matrix? Which Jacobian matrix?

2) Where you write:

"Assuming I have these two functions λ' = f(λ,Φ) and Φ' = f (λ,Φ)",

aren't you making λ' and Φ' equal to each other, since you have set each of them equal to the same expression, f(λ,Φ) ?

3) What do λ, λ', Φ, Φ' have to do with the two curves that you mention? Or with the conformal map you started with?

4) Where you write: "Then I can define these basis vectors . . . between the unrotated and rotated system," this is the first time you mention any rotation. Which rotation?

 

[meteo student]

1) Regarding my usage of the word transformation - I have a curve defined with respect to a (λ,Φ) system. Then I rotate the axes of this system and define another curve in the rotated system with respect to (λ',Φ') system. Would that mapping be conformal ?

2) Sorry there was an error in the transformation functions

λ' = f(λ,Φ)

and Φ' = g(λ,Φ)

3) and 4) follow from (1). Please let me know if you need any clarification.

 

[meteo student]

Rotation

Rotation seems to be a special concepts for three dimensional Euclidean space. But in a more general manifold orientation using n-form is usually used...

If I had an object rotating around the earth(an imperfect ellipsoid) in a latitude circle or a great circle what sort of motion would that be ?

 

[meteo student]

1) Regarding my usage of the word transformation - I have a curve defined with respect to a (λ,Φ) system. Then I rotate the axes of this system and define another curve in the rotated system with respect to (λ',Φ') system. Would that mapping be conformal ?

2) Sorry there was an error in the transformation functions

λ' = f(λ,Φ)

and Φ' = g(λ,Φ)

3) and 4) follow from (1). Please let me know if you need any clarification.

I presume f and g would have invertible mappings from the definition that you outlined.

 

[WWGD]

Thanks for that answer. But I am not a mathematician and I am dealing with Euclidean space in three dimensions - latitude, longitude and r (radius of uniform sphere). I am trying to correlate the mathematical definitions to definitions in cartography- differential geodesy. So if I have a geodesic pointing to true north and I have another arc which is east of that geodesic we call that angle between two arcs as the azimuth. So the tangent vectors of the two geodesics would give me the angle that I require ?

Sorry for getting back so late: If the two geodesics intersect at a point p, then the angle between the curves is defined as the angle between the tangent spaces.

An issue with your reference to conformal maps is that these maps are defined on a sphere, instead of on Euclidean n-space ( or complex plane) , so you need to use some additional machinery (pre-and post- compose with charts, chart-maps) to define conformal maps on a sphere ( or in a general manifold) that you do not need to use in Euclidean space.

 

[meteo student]

Thanks for that note. Things are becoming clear now.
I realized later I was not on a Euclidean space.
So the tangent vectors can also be called the basis vectors of the transformation. Am I right ?

So if each basis vector is defined in the following way

$e_i' = ∑ P_i e_i$
and given two transformation functions f and g - does the matrix of partial derivatives give me the matrix P ? I forgot to mention that f and g are conformal maps.

 

[zinq]

"Sorry for getting back so late: If the two geodesics intersect at a point p, then the angle between the curves is defined as the angle between the tangent spaces."

This is almost true. Actually, the angle between two parametrized curves that intersect at a point p where neither of them has zero velocity is the angle between their tangent vectors. (And for the sign of the angle to be well-defined, there should be an order assigned to the curves: a first curve and a second curve.)

For instance: If in the plane, curve

c(s) = (s,0)​

for s ≥ 0, and curve

d(t) = (1+2t,t)​

for t ≥ 0, then they intersect at the point p = (1,0) with tangent vectors c'(1) = (1,0) and d'(0) = (2,1).

Hence the angle θ between curve c and curve d at point p satisfies

cos(θ) = (1,0)/1 ⋅ (2,1)/√5 = 2/√5 = √(4/5).

And, θ assumed taken counterclockwise from curve c to curve d will be a positive angle with 0 < θ < π/2.

(If we had just used the tangent spaces, there would be an ambiguity not just with the sign of θ but worse, whether we meant θ or its supplement π-θ.)

 

[WWGD]

"Sorry for getting back so late: If the two geodesics intersect at a point p, then the angle between the curves is defined as the angle between the tangent spaces."

This is almost true. Actually, the angle between two parametrized curves that intersect at a point p where neither of them has zero velocity is the angle between their tangent vectors. (And for the sign of the angle to be well-defined, there should be an order assigned to the curves: a first curve and a second curve.)

For instance: If in the plane, curve

c(s) = (s,0)​

for s ≥ 0, and curve

d(t) = (1+2t,t)​

for t ≥ 0, then they intersect at the point p = (1,0) with tangent vectors c'(1) = (1,0) and d'(0) = (2,1).

Hence the angle θ between curve c and curve d at point p satisfies

cos(θ) = (1,0)/1 ⋅ (2,1)/√5 = 2/√5 = √(4/5).

And, θ assumed taken counterclockwise from curve c to curve d will be a positive angle with 0 < θ < π/2.

(If we had just used the tangent spaces, there would be an ambiguity not just with the sign of θ but worse, whether we meant θ or its supplement π-θ.)

So you specify ahead of time which of the angles you are interested in.

 

[WWGD]

1) Regarding my usage of the word transformation - I have a curve defined with respect to a (λ,Φ) system. Then I rotate the axes of this system and define another curve in the rotated system with respect to (λ',Φ') system. Would that mapping be conformal ?

2) Sorry there was an error in the transformation functions

λ' = f(λ,Φ)

and Φ' = g(λ,Φ)

3) and 4) follow from (1). Please let me know if you need any clarification.

If your transformation is only a rotation, then yes, this is a conformal map. And I think you are considering Mobius maps , i.e., maps
of the type:

$\frac {az+b} {cz+d}$ (with $ad \neq bc$) , then, yes, you can use standard polynomial division to express them as a composition of inversions, rotations and scalings (maybe some other transformation I cannot remember now) And all of these are bijective conformal maps from the sphere $\mathbb CP^1 =S^2$ to itself, where = is loosely used to mean manifold-isomorphic.

 

[meteo student]

I am just considering pure rotations at the moment. So I just calculate the matrix of partial derivatives right ?

 

[WWGD]

I am just considering pure rotations at the moment. So I just calculate the matrix of partial derivatives right ?

Are these rotations defined on the circle? And, sorry, what are you seeking to do by doing the Jacobian? If your transfromation is a Mobius transformation that is a rotation, and $ab \neq cd$ then your map is conformal (which you can show by showing each of the component transformatins is itself conformal). EDIT: all Mobius transformations are the bijective conformal maps of the Riemann sphere to itself:

https://en.wikipedia.org/wiki/Möbius_transformation

 

[meteo student]

Yes they are defined on a circle. I want to get the slope of the two meridians at a point.

 

[zinq]

I know that for me, it would be easier to respond to your question if you included more information.

Let's consider the question "So I just calculate the matrix of partial derivatives right ?"

First of all, "the matrix of partial derivatives" would be the matrix of partial derivatives of some function. Which function is that?

1) Functions come in all shapes and sizes; in particular a function has to be from some space and to some space (which could be the same as the one it's from). Which spaces are these? (They are called the domain and codomain, respectively.) Earlier you referred to latitude and longitude. Maybe the function is both from and to a sphere:

f: S² → S²​

?

2) For the function to have partial derivatives, the function would need to be differentiable. Also, partial derivatives have to be with respect to some particular variables, also known as coordinates. Since you don't specify the coordinates, you may be referring to the latitude and longitude as earlier?. Or maybe the standard coordinates of a Euclidean space (like the (x,y) of a plane)?

3) Especially puzzling for me are the repeated references to a) conformal maps and b) rotations in what seems to be the same context.

A conformal map is one that preserves angles, and sense (clockwise or counterclockwise). A rotation is a map that preserves distances (and orientation). It is easy to see that a rotation preserves angles, so that all rotations are conformal.

But the converse is very far from true: In the plane, many conformal maps from some part of the plane to another are not much like rotations at all.

 

[WWGD]

Yes they are defined on a circle. I want to get the slope of the two meridians at a point.

You may have to adapt the concept of slope that is usually used for lines in the plane ro this setting of maps betwwen 2-spheres.

 

[meteo student]

@zinq - This is the paper that I am reading right now. The only difference is that I am in the latitude,longitude, sigma space.

http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922

 

[meteo student]

You may have to adapt the concept of slope that is usually used for lines in the plane ro this setting of maps betwwen 2-spheres.

Thank you. That is very useful for me.

As I mentioned I have two functions(yes they are differentiable)

λ' = f (λ,Φ)

and Φ' = g(λ,Φ) . Both f and g do have inverse mappings.

These functions represent rotations to and from a 2-sphere.

 

[WWGD]

Thank you. That is very useful for me.

As I mentioned I have two functions(yes they are differentiable)

λ' = f (λ,Φ)

and Φ' = g(λ,Φ) . Both f and g do have inverse mappings.

These functions represent rotations to and from a 2-sphere.

But there is also the additional issue that slope is usually associated to/with functions of one variable, while yours is a function of two variables. The closest analog of slope then I guess would be a directional derivative. But then : at what point do you want to find this derivative and in which direction?

 

[meteo student]

Great question. I raised the same point sometime ago - what does the Jacobian matrix of partial derivatives mean to me ? You talked about directional derivatives. I presume the Jacobian 2x2 matrix represents the full set of directional derivatives. Am I correct ?

I have ∂f/ ∂λ ∂f/Φ
∂g/∂λ ∂g/ ∂Φ

 

[WWGD]

O.K, to simplify things, you can express your map as a function from $\mathbb R^2$ to itself. Then the Jacobian gives you a local expression for the differential/pushforward https://en.wikipedia.org/wiki/Pushforward_(differential) which is the meaning , in the context of a manifold, of the derivative.

 

[meteo student]

Very nice. Learned something new. Pushforward and pullback.

I will calculate the Jacobian using the transformation functions and attempt to reproduce the published expressions.

 

[WWGD]

Very nice. Learned something new. Pushforward and pullback.

I will calculate the Jacobian using the transformation functions and attempt to reproduce the published expressions.

Good luck, sorry I could not give you something more direct/clear.