• Moliere
    I'm mostly just "checking in" here to say I am reading along, but couldn't come up with much last Saturday.
  • fdrake

    It's really hard going! Spending hours on paragraphs means you know it's hard.
  • Moliere
    Yes! :D Reading him reminds me of an old math prof I had, actually -- his mind was so intuitively mathematical that what seemed like not worth mentioning to him was something that was crucially important for me to follow his reasoning.

    I'm not giving up or anything. It's just taking time.
  • fdrake

    I'm finding the same thing, what I'm benefitting most from I think is trying to integrate the imaginative background from the first section with the mechanical mathsy bits in the second. I imagine philosophically the first section and the final section would do, so if others feel like this isn't progressing quick enough to the philosophical juicy bits I could summarise the maths so far and then we could move on to §3.
  • Moliere
    Personally I don't mind. I think at least getting a gist of the math is important for understanding the broader themes.
  • unenlightened
    Doing my best to remain silent so's not to remove all doubt, but thought y'all might like some dimensional confusion. Why read, when you can play?

  • Moliere
    To find the simplest cases, I shall seek first an expression for manifoldnesses of n - 1 dimensions which are everywhere equidistant from the origin of the linear element; that is, I shall seek a continuous function of position whose values distinguish them from one another. In going outwards from the origin, this must either increase in all directions or decrease in all directions; I assume that it increases in all directions, and therefore has a minimum at that point. If, then, the first and second differential coefficients of this function are finite, its first differential must vanish, and the second differential cannot become negative; I assume that it is always positive. This differential expression, of the second order remains constant when ds remains constant, and increases in the duplicate ratio when the dx, and therefore also ds, increase in the same ratio; it must therefore be ds2 multiplied by a constant, and consequently ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx, in which the coefficients are continuous functions of the quantities x.

    Alright, I have a question about this. I thought I was following until the end here.

    As I understand it what Reimann is saying is that the displacement of a line does not alter the length of the line -- it's not like the coordinates themselves follow some kind of progression where the space between 1 and 2 is smaller than the space between 3 and 4. The space is equidistant.

    In the quoted bit it seemed to me that he was considering a manifold which is a straight line (segment?), and he is trying to establish the rate of change of x with respect to s (or vice versa?). But then I get lost when he is using the 2nd derivative, because by my figuring that would be equal to zero since he was considering a straight line?
  • Moliere
    I actually own that game. :D I just got to watch the video. It's a great game.
  • JimRoo
    My understanding is when Riemann is talking about the 2nd derivative, he is talking about the 2nd derivative of the metric function, the function for ds:

    ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx

    He uses the example of this function in Space:

  • Moliere
    Okay, so it's actually more general than just a straight line. It's just arclength of whatever manifold we're interested in.
  • fdrake
    This isn't from the paper, it uses mostly external ideas to what's been presented in the paper so far.

    If I can make an analogy, imagine yourself as a point rolling down a hill. When the hill changes shape, so does your acceleration. A 'straight line' on the surface of the hill isn't a 'straight line' in the embedding space. To be sure, if we have that the manifold is locally flat, 'straight lines' of tiny extent on the manifold will look like straight lines in a Euclidean embedding space. But they don't actually have to be straight (in the sense of the embedding space) because of the possibility of curvature.

    Though, I think some of your intuition about the derivative is correct. Imagine if we place two points A, B on the hill really close together and draw a smooth path between them, like a piece of string bound tightly to the surface.


    Imagine that we parametrise this path so that s=0 at point A and s=k at point B - this uniquely specifies every point on the path. The average change in the function (per unit length on the path) that this path corresponds to would be:

    so the second derivative would be 0, since the first would be a constant (except if the points A and B coincided). But what does this calculation actually mean? All the function f does is take the arclength along the curve between A and B and spit it back out. IE f(s)=s, with f(A)=0 and f(B)=k. The rate of change of the arclength with respect to itself is always 1. More generally, the rate of change of any function with respect to itself is always 1.

    Another thing to note is that the tangent vector to a manifold at a point - a line that is visualised in the embedding space - agrees with the first derivative of the manifold in its direction, but the second derivative of the tangent vector is 0 - whereas the manifold itself 'curves away' from it, showing the presence of a nonzero second derivative of the manifold (with respect to the coordinate system we're using) - curvature.

    The situation that we usually have on the manifold is more like the form:

    where there is more than one coordinate required in the specification of the path, if it's a one dimensional curve we also have that y(s) = f(x(s)). This means that we can consider how the curve bends over the coordinate system x(s), y(s). The curve itself, like the length of wiring, can be straightened out, so the curvature it has isn't intrinsic to its shape. This ability to straighten out something precisely means that there is a smooth transformation from distances within the shape to distances like they behave in a Euclidean (flat) space. You can also visualise this as the curvature of spaces rendering the linear approximation to their surface (like the tangent plane or vector) worse and worse when you go away from the point of approximation.

    So it might be that we can bend the wire, but we don't introduce any irremovable/intrinsic curvature. What intrinsic curvature actually measures is how movements of oriented objects constrained within the surface change the orientation of those objects when moving around closed paths - paths with the same start and end point.

    This even applies to forming a circle out of it, circle boundaries don't have intrinsic curvature whereas the surfaces of spheres do! A tangent vector to a circle at one point, transported around the circle while remaining tangent, is still in the same direction as it was when you started! You can check this by rotating, say, your phone around the top of a coffee mug. So let's talk about the surface of a sphere.

    Another task we might imagine is taking a perfectly taut bit of copper wiring and trying to wrap it onto the surface of a sphere. You can press it onto a point, and it'll touch the point. But, if the sphere is absolutely huge relative to the size of the perfectly taut bit of string, the taut bit of wire might resemble the surface of the sphere very well, it wraps away slowly from the wire with respect to spatial changes (points flowing away from the wire). If the sphere is tiny compared to the bit of string, it just glances off it and the sphere quickly wraps away from it. The acceleration with which the sphere wraps away from the taut bit of string is its curvature (no acceleration = flat!) - in the small sphere case you'd have to bend the wire a lot to fit the surface, in the large sphere case you'd have to bend the wire a tiny amount to fit the surface. The curvature of the surface of a sphere is not something that can be removed by these smooth transformations.

    The presence of curvature, like on the surface of a sphere, does something pretty strange to straight lines from the embedding space. If you take the iron wire, press it onto a point on the surface of the sphere, and you hold it onto the sphere, what happens when you take the iron wire, still taut and straight, from around a closed path on the sphere? This requirement that it's still tight equates to that the iron wire must be tangent to the sphere. If you move it along a path, where the start and end points of the path are the same (say carving out a quarter of the sphere). By the time it's back to its start, the direction you're holding the iron wire in actually changes. The presence of these changes signals intrinsic curvature.

    You can check this with your fist and your mobile phone. Clench your fist with your thumb toward you, take your phone and press it hard onto the knuckle of your index finger. Push the phone away from your body along the line of your knuckles. When it gets past your final knuckle, still keeping it pressed onto your hand, force the phone towards you from the first finger joint on your pinky to the first finger joint on your index finger. Finally, bring it back from the first joint on your index finger to the base of your thumb, and back up again to the first knuckle. You should see that the phone has inverted. Contrast this to the mug and phone thing from earlier.
  • SapereAude
    I have been a bit busy lately and have just now taken a look at the essay. Is there someone who can give me some background on the work: Why is it important? Who wrote it? Historical background? Philosophical background?
  • fdrake

    Have you read the thread so far? We've discussed things relevant to your questions.
  • StreetlightX
    @fdrake”: "Aw man, behind on this and §3 (of section I!) is really killing me, even with your exposition. I get that the overall aim here is to show how to ‘decompose’ a manifoldness, but I don’t quite understand the demonstration he sets up to explain it. Two questions to begin with:

    (1) “Let us suppose a variable piece of a manifoldness of one dimension” - I’m not sure what work the word ‘variable’ does here in ‘variable piece of a manifoldness’. Can one take an invariable piece of a manifold? And what would this distinction mean?

    (2) “Let us take a continuous function of position within the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness.” - Here, I’m not sure what work ‘not constant’ is doing. Is it the variation of position on the manifoldness we are asked to think of is ‘not constant’?

    I don’t think I can work my way through the rest of the paragraph without getting these fundamentals down.
  • fdrake
    (1) “Let us suppose a variable piece of a manifoldness of one dimension” - I’m not sure what work the word ‘variable’ does here in ‘variable piece of a manifoldness’. Can one take an invariable piece of a manifold? And what would this distinction mean?StreetlightX

    I read that as saying variable piece of a manifoldness might be a connected chunk of a manifoldness. But when we take 'a' variable piece of a manifoldness Riemann intends us to be discussing an arbitrary one. The other bit 'of one dimension', connotes that the manifoldness he's considering is just a 1 dimensional curve.

    So you might imagine cutting a cylinder down the middle really finely to produce a circle.

    (2) “Let us take a continuous function of position within the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness.” - Here, I’m not sure what work ‘not constant’ is doing. Is it the variation of position on the manifoldness we are asked to think of is ‘not constant’?StreetlightX

    This continuous function describes what point you are on on the previously considered curve. If the function was something like f(x)=x for x<1 and f(x)=1 for all x>=1, this makes the entire region [1,infinity) map to 1, so it can't be used to uniquely specify the position. More specifically, in this example, if you know the function is f(x) = x, you can take an output of this function f(x) and directly map it to an input x, allowing you to translate between the position described using the function and the position on the manifold. However, when this x becomes greater than 1, all this function tells you is that it's equal to 1. Which means the input which caused the function to be 1 could be anywhere between 1 and infinity - so we can't invert the function to uniquely specify the point on the curve.
  • fdrake
    Sorry for not continuing with the exegesis, I've been busy IRL and sitting down to concentrate for the length of time required to understand what's going on in the next paragraph has been difficult. I have tomorrow off and intend to give it a try.
  • fdrake
    Right, so I finally have some purchase on the construction in section 2 §2. This post will be heavily edited to include diagrams later. Anyone more experienced with math in this field please correct me.

    Imagine we're on the surface of a wibbly wobbly sphere, and we pick a point O and call it the origin. We're going to look at the distance from O to nearby points.

    For this purpose let us imagine that from any given point the system of shortest limes going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin.

    From O, we move out in every direction along the shortest possible path. We can imagine this as inscribing a wibbly wobbly circle on the wibbly wobbly sphere and drawing lines on it which hug the surface and are as straight as possible.

    We then pick another point on the wibbly wobbly sphere within the wibbly wobbly circle. Since we drew the 'system of shortest lines (geodesics)', and drew all of them, this point will lay on one of the geodesics. Therefore, we can relate the position of this point to its position on the geodesic. In order to do this, we need to look at how the geodesic hugs the sphere - which means we need to look at how the geodesic changes over the wibbly wobbly circle within the wibbly wobbly sphere. That is, we need to relate the new point to the old point using the geodesic line and the coordinate system the wibbly wobbly sphere is in (the embedding space).

    Riemann's recipe for this is:

    It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. Let us introduce now instead of the dx0 linear functions dx of them, such that the initial value of the square of the line-element shall equal the sum of the squares of these expressions, so that the independent varaibles are now the length s and the ratios of the quantities dx. Lastly, take instead of the dx quantities x1, x2, x3,..., xn proportional to them, but such that the sum of their squares = s2

    The quantities 'in' the geodesic are the embedding coordinate system quantities which vary with it - like the angle when moving on the boundary of a circle. The ratios are the rates of change of each coordinate with respect to every other. EG, moving around the boundary of a circle, we never change the distance from the origin-we have a fixed radius, so the rate of change with respect to the distance from the origin is 0, whereas the rate of change with respect to the angle from the horizontal axis is 1. IE we are looking at and , the being the distance of the point in the embedding space from the origin and the angle of rotation from the positive x-axis , rotating clockwise around the boundary of the circle by moves with respect to changes and with respect to . So the distance between two points on the boundary of a circle only increases with respect to the angle (sweeping out an interpoint distance of infinitesimally, and does not increase with respect to the radius since the distance from the origin does not change.

    Riemann wants to generalise from this notion, instead of necessarily having two independent coordinates, the system of points going out from the origin might (and in general will) be functions of multiple dimensions from the embedding space - a general interpoint distance on the wibbly wobbly surface depends on changes in all the in the embedding space coordinates. This means instead of just looking at independent sums where , he wants it to be . What this looks like for a curve with inputs from the embedding space that outputs a position on the surface is.

    where is a matrix that stores all the at the chosen origin. This interfaces with the y variables through the chain rule:
    and stores the results in the vector . This vector can be thought of as a linear displacement from 0 - (edit: like the gradient operator combined with the chain rule), and Riemann then insists that when the displacement is infinitesimally small - when we increment along the curve by , this gives us the square of the line element:

    the intuition here is that we need to increment along 'all the quantities in the geodesic', which are the , so that the infinitesimal increment in position becomes:

    and the norm of this increment is then

    through Pythagoras (and the flat space stuff from before). Which Riemann states as if it is incredibly obvious:

    When we introduce these quantities, the square of the line-element is \sum dx^2 for infinitesimal values of the x,

    . It should be noted here that the embedding space coordinates are kind of extraneous so long as we are considering increments confined to the surface - IE, whenever we write a y or a dy it's also secretly an x or a dx, and all this chain rule stuff does is express how the changes on the surface coordinates (system of geodesics) work with respect to the embedding coordinates which we initially used to express them. This local linear approximation of changes on the manifold with respect to the embedding space using the variables on the manifold is what Riemann achieves through the proportion construction

    . Let us introduce now instead of the dx0 linear functions dx of them, such that the initial value of the square of the line-element shall equal the sum of the squares of these expressions, so that the independent varaibles are now the length s and the ratios of the quantities dx.

    by fixing the 'initial value' to be the sum of the squares of the increment's norm. The increment's norm gives us the local linearity, the remaining discussion refines the approximation of to include quadratic terms that express the curvature.
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