Curved surfaces that do not preserve area

[diegzumillo]

Hi all

Making this title was harder than I thought. It certainly makes the topic look more advanced than it actually is.

I studied differential geometry during my masters but never went much in depth, just enough so I could apply basic concepts to my specific problems at the time. Now I'm trying to answer a simple question: do parallel transport on closed loops on curved spaces conserve scale? scale as in volume. Also, the question is specific to 2D spaces, or surfaces, so scale would be the area of a polygon. So bear with me because my knowledge on the subject is very limited in solid math and too reliant on my intuition.

I spent a long time dismissing this question in my head because my intuition was making some assumptions that I now question. Clearly the area of an object can change, (with respect to a projection on an euclidean surface) but if I take this object for a walk around space and come back to the initial position, its orientation might change but its scale doesn't seem like it would. Unless there is some kind of discontinuity, or holes, on this space or I impose specific boundary conditions that connect edges of different sizes. Which makes me wonder if this question is more suitable for the topology forum than differential geometry, because holes and connections scream topology to me.

Does this make sense?

 

[lavinia]

Hi all

Making this title was harder than I thought. It certainly makes the topic look more advanced than it actually is.

I studied differential geometry during my masters but never went much in depth, just enough so I could apply basic concepts to my specific problems at the time. Now I'm trying to answer a simple question: do parallel transport on closed loops on curved spaces conserve scale? scale as in volume. Also, the question is specific to 2D spaces, or surfaces, so scale would be the area of a polygon. So bear with me because my knowledge on the subject is very limited in solid math and too reliant on my intuition.

I spent a long time dismissing this question in my head because my intuition was making some assumptions that I now question. Clearly the area of an object can change, (with respect to a projection on an euclidean surface) but if I take this object for a walk around space and come back to the initial position, its orientation might change but its scale doesn't seem like it would. Unless there is some kind of discontinuity, or holes, on this space or I impose specific boundary conditions that connect edges of different sizes. Which makes me wonder if this question is more suitable for the topology forum than differential geometry, because holes and connections scream topology to me.

Does this make sense?

Have you tried differentiating inner products of parallel translated factors?

 

[diegzumillo]

Not really. Why? I thought area was related to cross product. But in any case I am convinced the angle between parallel transported vectors doesn't change. And I know the length is also preserved throughout any path (not necessarily closed). But I don't really know how that relates to area.

What about my intuition about area not returning to its original value when there's a discontinuity? are these kinds of spaces violating some assumption on the above statements? if the closed path cannot be continuously shrink to a point, is that a problem?

The reason I ask this is because I have a very specific example in my head but it's hard to explain with words, I could draw it though. But in it an object does change scale but the path necessarily engulfs a hole on the surface.

edit: clarity.

 

[lavinia]

Not really. Why? I thought area was related to cross product. But in any case I am convinced the angle between parallel transported vectors doesn't change. And I know the length is also preserved throughout any path (not necessarily closed). But I don't really know how that relates to area.

What about my intuition about area not returning to its original value when there's a discontinuity? are these kinds of spaces violating some assumption on the above statements? if the closed path cannot be continuously shrink to a point, is that a problem?

The reason I ask this is because I have a very specific example in my head but it's hard to explain with words, I could draw it though. But in it an object does change scale but the path necessarily engulfs a hole on the surface.

edit: clarity.

I am a little confused. So bear with me.

Given two vectors, the area of the parallelogram that they span will not change if the angle between them and their lengths do not change. In 3 space this is measured by the cross product. So if their lengths and the angle between them is preserved during parallel translation then so is the area of the parallelogram that they span. This is true on any surface with a Levi-Civita connection.

I am not sure what discontinuities you have in mind.

While it is true that parallel translation might not preserve orientation of the two vectors, on an oriented surface it will. On the Klein bottle - which is unorientable - it will not.

 

[diegzumillo]

I think I know where my intuition is failing. I'm confusing the space itself with a projection. So ok, I'm now convinced the area does not change. At least not on the usual smooth surface. But what if I define some periodic boundary conditions where one side is smaller than the other? This is the example I had in mind: http://dosketch.com/view.php?k=ZlLxIB2eZ6uNGjEeXlZB , and what I think would be a topologically equivalent surface.

Sorry for being confusing, by the way And thanks for trying to make sense of my babbling

While it is true that parallel translation might not preserve orientation of the two vectors, on an oriented surface it will. On the Klein bottle - which is unoriented, it will not.

Sorry, I meant direction! I understand orientation has a very specific meaning in differential geometry but I forgot.

edit: the sketch disappeared. What a great web tool.

 

[WWGD]

Maybe if you can express the transformation in coordinates, you can see what happens to the Jacobian (must equal 1).

 

[diegzumillo]

What happens if it's not equal to one? I made a quick example here: square of side length 1, right edge is associated with half of left edge (from y=0 to 0.5). The transformation is is x' = x+1 and y' = y/2. So that's a jacobian (determinant) of 1/2.

 

[WWGD]

Then, if I understood you correctly, the area was halved into 1/2, equal to the determinant. The determinant of the Jacobian gives you a general measure of the infinitesimal change in the n-dimensional volume (area here ). This is from geometric algebra. Look at an example where, e.g., you double one side and halve the other, i.e., $x \rightarrow 2x , y \rightarrow y/2$, preserving the product. Then notice the determinat is 1. Sure this is not a proof, but an illustrating case.

 

[diegzumillo]

Wow, this is magical.

OK, so that makes sense. And for all purposes my questions have been answered but I'm still curious. Usually there is a manifold that is topologically equivalent to this one constructed from the square (like the plane, torus, Mobius strip etc). Something tells me this particular topology will require some holes (even matching the remaining sides in the most sensible way).

edit: The problem is not just holes. But how can objects shrink? as we know areas cannot change.

 

[lavinia]

Sorry, I meant direction! I understand orientation has a very specific meaning in differential geometry but I forgot.

Parallel translation of a vector in along a closed loop can alter its direction. If this happens along a sufficiently small loop, changed direction is evidence of non-zero curvature.

 

[diegzumillo]

Yep. That much I know. :)

 

[zinq]

Lots of differential geometry questions involve some concepts from topology. But if the question (on a differentiable curve, surface, or manifold) mentions length, area, volume, angle (among other things), then it is usually in the area of differential geometry.

Also, although orientatability and orientation may be expressed in terms of differential geometry concepts, these are fundamentally topological concepts.