Non Euclidean Geometry: examples of Riemannian and Lobachevsky - Bolyai geometries
The main reason that I chose to include models of non Euclidean geometry is that it is so abstract. When you think about making a picture of an axiomatic set you are thinking about a second degree of abstraction, and that is really fun for me. Many people will not be interested in this model for that reason. And it is OK to skip this model because in the final analysis I do not, at this time, think that it will affect a final determination regarding elements of consistency among the various models. On the other hand, this model arises from Differential Geometry which is the Mathematical Foundation for General Relativity and is useful in Quantum Mechanics; and perhaps the reader might wish to gain an elementary understanding of how this works.
History:
The history of these non Euclidean models goes back to the Pythagoreans circa 475B.C. Originally, the Pythagoreans tried to describe all things in terms of whole numbers. Eventually they came up with the Pythagorean Theorem (a^2 + b^2 = c^2 for right triangles). The problem was that the hypotenuse of a right triangle, with each side equal to 1, was equal to the square root of 2. The square root of 2 was discovered by the Pythagoreans to be irrational (that is it could not be of the form p/q where p and q were whole numbers).
It is said that one of the Pythagoreans leaked this fact and, later on a sea voyage, he was thrown over board for his heresy.
Euclid (c325B.C. - 265B.C.) wanted to make a complete system of mathematics which would not fail like that of the Pythagoreans. The main problem was that the Pythagorean Theorem was so successful that he needed his geometry to come up with the same theorem. The only way to accomplish this was to include the fifth postulate. (1)
The current statement of the fifth postulate is:
Given a line and a point not lying on that line, there is one and only one line passing through that point which does not intersect the original line.
Many Mathematicians have tried, unsuccessfully, to prove that postulate could be derived form the other original postulates, and finally some Mathematicians tried to show that there could be alternative geometries which assumed (a) that there were no lines that satisfied the fifth postulate or (b) the there were an infinite number of lines that satisfied the fifth postulate.
The solution for (a) is credited to Riemann and (b) is jointly credited to Lobachevsky & Bolyai.
Text: (skip to The Answers if you are not technical or curious):
A surface S is described by the set {(x,y,z) such that x = f1(u,v), y = f2(u,v), and z = f3(u,v) where fi are infinitely differentiable Real functions and u and v are independent Real numbers such that u[size=75]0[/size][size=100]<=[/size] u<= u[size=75]1[/size] [size=100]and[/size] v[size=75]0[/size][size=100]<= [/size]v<= v[size=75]1[/size]}
An [size=100]arbitrary[/size] line L on surface S is given by the set {(x,y,z) such that x = f1(u(t),v(t)), y = f2(u(t),v(t)), z = f3(u(t),v(t)) where fi is defined as above and t0<=t<=t1 where t is also Real}. The variable t is said to be a parameter for the line L.
Because the point (x,y,z) corresponds to a vector, Differential Geometry is generally expressed in the form of vector analysis.
The arc length is given by s(t) = integral t[size=75]0[/size] to t [size=100]square[/size] root ((df1(t)/dt)^2 + (df2(t)/dt)^2 + (df3(t)/dt)^2). By taking the derivative of s with respect to t we get ds/dt = square root ((df1(t)/dt)^2 + (df2(t)/dt)^2 + (df3(t)/dt)^2) and squaring ds/dt we get (ds/dt)^2 = (df1(t)/dt)^2 + (df2(t)/dt)^2 + (df3(t)/dt)^2 which equals dx*dx.
Because x is a composite of u and v we can write dx = (dx/du)du + (dx/dv)dv. By convention dx/du is the normalized partial derivative of x with respect to u given by (df1(u,v)/du, f2(u,v)/du, df3(u,v)/du) / |(df1(u,v)/du, df2(u,v)/du,df3(u,v)/du)| where |(df1(u,v)/du, df2(u,v)/du,df3(u,v)/du)| is the magnitude of that vector and dx/dv is defined analogously.
To get a more intuitive feel consider the following figure:
We shall denote dx/du as x[size=75]u[/size] [size=100]and[/size] dx/dv as x[size=75]v[/size] . [size=100]Then[/size] x[size=75]u[/size] [size=100]is[/size] a unit tangent vector originating at the center of the intersection of the blue lines and x[size=75]v[/size] is [size=100]also[/size] a unit vector originating at the intersection of the blue lines going along the other blue line.
Now we can write ds^2 = ((dx/du)du +(dx/dv)dv)((dx/du)du +(dx/dv)dv) = (x[size=75]u[/size]du [size=100]+[/size] x[size=75]v[/size][size=100]dv[/size])(x[size=75]u[/size]du [size=100]+[/size] x[size=75]v[/size]dv) [size=100]=[/size] (x[size=75]u[/size][size=100][/size]x[size=75]u[/size])[size=100]du[/size]^2 + 2(x[size=75]u[/size][size=100][/size]x[size=75]v[/size])[size=100]dudv[/size] + (x[size=75]v[/size][size=100]*[/size]x[size=75]v[/size])[size=100]dv^2[/size].
The equation ds^2 = Edu^2 +2Fdudv + Gdv^2 is called the first fundamental form.
It turns out that this form is invariant with respect to the choice of parameters. Additionally, any line can be described in terms of s, it’s own arc length.
The second fundamental form is the result of the fact that the normal vector N is perpendicular to the plane defined by xu and xv. (N is defined by N = x[size=75]u[/size] [size=100]x[/size] x[size=75]v[/size]/[size=100]|[/size]x[size=75]u[/size] [size=100]x[/size] x[size=75]v[/size]| where in general if c = a x b then c = (a2b3 -b2a3, a3b1 - b3a1, a1b2 - b1a2). Therefore t*N = 0 where t is a tangent vector dx/ds.
Now we can write:
d(tN)/ds = 0 and thus (dt/ds)N + t(dN/ds) = 0. Therefore (dt/ds)N = -t*(dN)/ds
Substituting t = dx/ds we get :
(dt/ds)N = -(dx/ds)(dN/ds) = - (dx)*(dN)/ds^2 = -(dx)*dN)/dxdx
Here (dt/ds)*N is the [size=100]normal[/size] scalar curvature kn (the geometric interpretation of which I will discuss shortly) and dxdx is the second fundamental form so we can write:
kn = -(dx)(dN)/Edu^2 + 2Fdudv + Gdv^2
But dN = N[size=75]u[/size][size=100]du[/size] + N[size=75]v[/size]dv [size=100]and[/size] since dx = x[size=75]u[/size]du [size=100]+[/size] x[size=75]v[/size]dv [size=100]we [/size]can write:
kn = (edu^2 + 2fdudv + gdv^2)/(Edu^2 + 2Fdudv + Gdv^2) where
e = -x[size=75]u[/size][size=100][/size]N[size=75]u[/size], 2f [size=100]=[/size] -(x[size=75]u[/size][size=100][/size]N[size=75]v[/size] [size=100]+[/size] x[size=75]v[/size][size=100][/size]N[size=75]u[/size]), [size=100]g[/size] = -x[size=75]v[/size][size=100][/size]N[size=75]v[/size].
[size=100]The[/size] second fundamental form is defined by the equation -dx*dN = edu^2 +2fdudv + gdv^2
The fundamental theorem of surface theory is:
If E, F, G and e, f, and g are functions of u and v and they are sufficiently differentiable and EG -F^2 <>0, E > 0, and G > 0; and EFG and efg satisfy the Gauss - Codazzi equations then there exists an unique (excepting its placement in space) Real surface in which EFG are the coefficients of the first fundamental form and efg are the coefficients of the second fundamental form.
Going into the Gauss - Codazzi equations is beyond the scope of this paper.
One of the most significant quantities in all of differential geometry is the term K = eg - f^2 / (EG-F^2). K is called the Gaussian Curvature. The geometry of a surface is determined by this constant. If K = 0 then the surface at that point is Euclidean, if K is Positive then the surface is Riemannian, and if K is negative then the surface has the Lobachevsky - Bolyai geometry. Again the proof of this statement is well beyond the scope of this paper.
At this point, I am going to introduce the concept of the oscillating sphere.
Consider the following drawing.
The blue circle is intended to represent a three dimensional sphere that lies tangent to an arbitrary point on a line on a 2 dimensional surface.
I shall not go into a proof that such a sphere exists and how we determine it’s radius other than to simply say that it does exist (except on a flat plane, where the radius is said to be infinite) and that it’s radius is |1/kn| where kn = dt/ds*N. (The reader should be aware the dt/ds is the second derivative of x with respect to s). This is why kn is called the normal scalar curvature.
Now we consider the above figure. Here t = dx/ds (the red vector), N = x[size=75]u[/size] [size=100]x[/size] x[size=75]v[/size][size=100]/[/size]|x[size=75]u[/size] [size=100]x[/size] x[size=75]v[/size]| ([size=100]the[/size] black vector), u = t x N/|t x N| (the green vector). The reader should notice the two blue vectors x[size=75]u[/size] [size=100]and[/size] x[size=75]v[/size] are in the tangent plane, as well as the tangent vectors t and u.
To relate dt/ds = kn = k to kn (the black vector) and kg (the green vector) we simply define kg such that kn + kg = k. (The reader should note that kn is not equal to kn).
kg (the length of kg) is called the geodesic curvature and one of the important theorems of Differential Geometry is that if L is any line on a surface between two points A and B on that surface then L is the shortest distance between those to points only if kg = 0.
The Answers:
Riemannian Sphere:
This is a surface of revolution given by r = acos(u/a), z = asin(u/a) + constant, where r^2 = x^2 + y^2. The lines on this surface are great arcs and they satisfy the Riemannian geometry. (Technically, diametrically opposite points need to be defined as identical as is shown by the blue lines). Any line on this sphere will also intersect any other line and is therefore model a, in which there were no lines that satisfied the fifth postulate.
Pseudosphere (my favorite):
This is a surface of revolution given by r = be^(u/b), z = integral of (1 - e^(2u/b))^½, where r^2 = x^2 + y^2. The lines on this surface correspond to latitudinal lines and curves that sweep away from the “equator” which asymptotically approach the appropriate pole. Since there are an infinite number of lines (see the vertical blue and green lines) that lie off a latitudinal line that pass through a point not on that line, this geometry corresponds to model b the Lobachevsky - Bolyai geometry.(2)
A final comment:
In the Riemannian geometry a triangle has more than 90 degrees and in the Lobachevsky - Bolyai geometry a triangle has less than 90 degrees. In neither case does the Pythagorean Theorem hold. It appears that Euclid, to meet his goal of including the Pythagorean Theorem, was right to include the Fifth Postulate.
(1) The history is found in “God Created the Integers” by Stephen Hawking
(2) Most of the formal relationships come from “Differential Geometry” by Dirk Struik
I think that much of organization of this work is the result of class work taken from Dr. Green, while I was a student at the University of Minnesota Department of Technology.
Model Comparisons:
Model - Newtonian Gravity
Model - Non Euclidian Geometry
At this point we can start comparing models. One of the points that strikes me, is that critical analysis in both models requires differentiable functions. This is a very stringent requirement as is seen in comparison with the Weierstrass functions (continuous but no where differentiable) and constructed Fractal Lines (continuous but who’s inverse is not). The other fact that seems obvious is fact the underlying entities x and t in Newtonian Physics and u and v in non Euclidean geometry are continuous. Please add any observations that you might have.