Intuitive explanation of parallel transport and geodesics

[Dom Claude]

"It seems completely false that you MUST approximate the line by geodesics. Can you prove this?"

Axiom: A line on a plane is the extremal distance between 2 points. ( Euclid )

Axiom: A line on a surface is the extremal distance between 2 points on the surface. ( Karl Friedrich Gauss)

Axiom: ( of differentiation ) Any curve shall be approximated by segments of lines. ( Gottfried Wilhelm Leibnitz ? )

I shall leave you to complete the proof.


***Please note***

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[mathwonk]

Another nice elementary discussion of angle sum and curvature for polygons occurs in chapter 7 of Experiencing Geometry by David Henderson. The extension to compact surfaces via dissections and coverings is in chapter 17 of that book. No calculus is used there.

 

[lavinia]

"It seems completely false that you MUST approximate the line by geodesics. Can you prove this?"

Axiom: A line on a plane is the extremal distance between 2 points. ( Euclid )

Axiom: A line on a surface is the extremal distance between 2 points on the surface. ( Karl Friedrich Gauss)

Axiom: ( of differentiation ) Any curve shall be approximated by segments of lines. ( Gottfried Wilhelm Leibnitz ? )

I shall leave you to complete the proof.


***Please note***

On more than one occasion I have notified the administrator of this forum to ban this account on grounds of personal reasons. I do not mind if my posts are left indefinitely in this forum. If this account has not been banned within one week, I shall publicly surrender the credentials to this account.

i am not sure what you are saying here. I wrote that approximation by geodesic segments should work.

And I do not think that these are axioms but rather theorems as you have said elsewhere.

In any case thanks for your comments. I like the idea of visualizing parallel translation through piece wise geodesic approximation.

I was just objecting to the MUST part of what you were saying.

BTW: Parallel translation along closed geodesics can also have non-trivial holonomy. this is true even when the curvature tensor is identically zero. I think that the examples given so far here have assumed that the geodesic polygons are small.

 

[mathwonk]

Dom, I cannot find your axiom in my translation, by Heath, of Euclid. Mine says:

2."a line is breadthless length",

3. " the extremities of a line are points", and then

4. "a straight line is a line which lies evenly with the points on itself".Are you re - translating #3. so that the word "extremities", given here as endpoints, means instead something minimal or maximal in distance? That would seem to restructure the form of the sentence.

Are you reading a Greek version?

Here is Heiberg's version of #3:

γʹ. Γραμμῆς δὲ πέρατα σημεῖα. translated by Richard Fizpatrick as "And the extremities of a line are points."

(The first word is line, the last is points, and the next to last is extremities or ends, but I do not know the grammar.)

If you have determined not to post more, I shall not be offended.

 

[mathwonk]

for this point of view of curvature lavinia, take a look at page 4.A - 14 of volume II of spivak's differential geometry book, his translation of riemann's "on the hypotheses which lie at the foundation of geometry", section II.3, I believe.

Here is another nice article giving Gauss' definition of curvature (as well as revealing that Euler's formula was found much earlier by Descartes. Descartes wrote his treatise about 1620, some 87 years prior to Euler's birth):

http://www.ams.org/samplings/feature-column/fcarc-descartes6

 

[lavinia]

Here is another nice article giving Gauss' definition of curvature (as well as revealing that Euler's formula was found much earlier by Descartes. Descartes wrote his treatise about 1620, some 87 years prior to Euler's birth):

http://www.ams.org/samplings/feature-column/fcarc-descartes6

thanks for the link.

I read in one book that Gauss curvature was thought of as the area of a small region on a surface divided into the area on the sphere swept out by the unit normal - take the limit as the region shrinks to a point. I wonder if this definition came from astronomical observations.

The formula above, is the Gauss-Bonnet formula but I never thought of it before as defining curvature. Very cool.

In geodesic polar coordinates one gets the Gauss curvature from the area of a geodesic circular disk and its geodesic radius. This I think is the method used in General Relativity.

I learned Gauss curvature as the coefficient of the exterior derivative of the connection 1 form written in terms of the volume element of the surface. This definition leads to the theorem that the total Gauss curvature is the sum of the indices of a vector field. This also follows from the geodesic triangle formula. It would be interesting to develop of thread where all of these relationships are worked out.

 

[mathwonk]

Lavinia, for this version of gauss's curvature, read the abstract by gauss at the end if his 1827 paper, general investigations on curved surfaces. he both gives the ratio of areas definition and the connection with angle excess for a triangle there in plain language.

 

[wisvuze]

An interesting interpretation of the gaussian curvature along these lines can also be found in Do Carmo's Differential Geometry of Curves and Surfaces on page 292. The exponential map is used, along with the taylor expansion of a component of the first fundamental form of a geodesic coordinate patch

 

[teodron]

First of all, I apologize for not replying during all this time.

I started this post in order to code a simple application that can show how a vector is modified while its support point, which lies on a polyhedral surface (since computers work only with discrete representations of mathematical abstractions such as numbers and geometric objects). If you want to see an example of this, please look at the already classic applet: http://torus.math.uiuc.edu/jms/java/dragsphere/

What I wanted to express in my utterly unconvincing and confusing statement (the initial post), is that when moving along a geodesic, the particle should not exert any force/pressure onto a manifold. lavinia tried to explain this as either zero geodesic curvature, which I now understand as being a measurable amount that gives an idea of how much "pressure" an "object" moving along such a path would exert in a direction that's not normal to the surface (coming from Kg + Kn = K decomposition - as vectors!). So, in this respect, one can use a virtual knife to cut a surface along a geodesic.. as long as that knife contains the normal to the surface and does not exert any side-forces on the surface.

Although the non-parallizable attribute of the sphere makes the notion of "keeping a vector in a certain direction while moving it along the surface" a bogus explanation, it is the most commonly found in so many textbooks that I can't really renounce it until this aspect is clarified. Then, how would one move a vector from point A to point B, that are indefinitely close on a manifold, and perceive a notion of direction? there was an example of a roman soldier holding a javelin facing forwards when he started marching from the north pole towards the south pole. When he'd reached the equator, he had to move along this great circle a quarter of its circumference and then return to the north pole, all of this while trying to keep the javelin in the same direction (parallel transport). Then when he got back from where he started, the javelin was.. rotated (by 90 degrees). So, these notions of direction _must_ be coupled with what happens in the tangent space. Although the sphere, the cone and cylinder are trivial objects, how would you deal with understanding how this soldier should perceive a notion of direction if he had to move on a more "distorted" surface (with hills, valley, saddles)?

Thank you all for your discussions, sorry for seeing a pseudo-flame-war taking place. I appreciate all of your interventions and corrections.

 

[lavinia]

Yes, I was trying to answer to lavinia's suggestion of actually cutting a manifold (e.g. sphere) with a knife that contains the normal vector and this way describing a geodesic. I exemplified by cutting along the 45 deg N lat circle. What I did was to squeeze a cone that contained this circle and had its vertex in the center of the sphere. If you take small Δl elements of that circle and join them with the cone's vertex, you can get lavinia's knife sticking deep inside the sphere. If you move along the circle, you can cut the sphere and see the knife always "containing" the normal.
I understood what you said: the cone's tangent planes coincide with the sphere's tangent planes, and hence the parallel transported vectors lie on a cone. If developed in a plane, the vectors shift in orientation by exactly the angle deficit, as you specified. Although very important as an example, this cannot be generalized, whereas Lavinia's "method" of "knifing" the manifold should work regardless of the surface (we cannot fit a developable surface to contain those vectors for any kind of surface, unfortunately, otherwise I'd have been satisfied with putting "cone hats" on manifolds and detect how much off a curve is from being a true geodesic).

Regards!
P.S. I am continuing this debate to provide students like I have been with a starting point before they dive into the Del operator, Christoffel symbols and other wonders that efficiently hide any geometric elegance from their users. I am extremely angry at most math professors for writing the same information in their lecture notes, books, etc. and providing students with stupid pictures of obvious things. When they start discussing tensors, curvature, bundles, geodesics, they again conjure up a sphere or a cylinder and draw the obvious. None of them tries to actually digest these things, as to explain and track their origins. I would bet my life that Riemann and Levi Civita didn't submit to swallowing up definitions and formulae, then using them to develop ground breaking mathematical devices. It simply cannot be. Have we advanced so much that we cannot stop any longer and analyze the intuitive meaning of the concepts we even develop our PhD theses on? If so, I am very much disappointed and sad. I don't have the time to re-read the whole theory, but I do believe one can understand some concepts without maneuvering all those general aspects (resuming to 3d curvature, torsion of 2d manifolds and curves). I am willing to swear that most of my University Professors were nothing more than skilled users of calculus, many of them lacking more fundamental insights on the objects they were invoking in their papers. Can it be that a TA can't provide you with answers to some questions derived from what your professor writes on the blackboard? I sincerely want people to understand such things without wasting many hours that most do not have.. also, not all students are as bright as Einstein to grasp the notions without much detailing.
Kind regards to all of you interested people.

I have been trying to learn this stuff on my own for a long time. As usual I strongly recommend the book by Singer and Thorpe, Lecture Notes on Elementary Topology and Geometry which is elementary yet completely modern. I also read Struik's book on classical differential geometry which has a wealth of examples. the two books work well together.

I do not know but I imagine that classically, geodesics were thought of as curves of least constraint, that is they are unconstrained except that they must lie on a surface. If one imagines wrapping a perfectly stretchable elastic along the surface and letting go of it, it would slide along the surface until its tension is minimized. If you assume that the tension is proportional to the amount of stretching - not a bad assumption if the elastic is not stretched to the breaking point - then its minimum tension curve will also be of minimal length - at least with respect to near by curves. So a minimal tension curve, a geodesic, locally minimizes length since it locally minimizes tension in the elastic.

In this case, the remaining forces on the elastic are perpendicular to the surface. They are the only forces that the elastic can not eliminate by sliding along the surface. They are the forces arising from the constraint that the elastic must lie on the surface. This is why the perpendicular knife slice defines a geodesic. The curve of intersection of the surface and the plane of the knife will have its acceleration vector normal to the surface - if it has unit speed of course - so the only forces it feels are normal to the surface. But this is a curve of least constraint i.e. a geodesic.

I think differential geometry of surfaces was well developed by the time parallel translation was first defined. Parallel translation seems to be an idea of intrinsic geometry which was a later development. While I like the idea suggested here of intuitively thinking of parallel translation along an arbitrary curve as being well approximated by fixed angle sliding along geodesics - I still feel that there should be a more profound intuition coming from the analysis of what it means to compare measuring rods at different points of space.

I think in modern pedagogy, it is considered more important to quickly learn basic ideas then start to use them. One really learns what they mean by solving problems. One learns from practice. Then as time goes by, one learns origins on an as needed basis. No math book or math professor will teach you mathematics. You learn by thinking about it, solving problems, asking questions and answering them for yourself, making up problems for yourself, always asking "what does this really mean?" I do not believe there are any bad math books, only students who don't think enough.