TheDestroyer said:
Hello guys,
I keep hearing that Euclidean parallel postulate was broken through differential geometry, can someone please explain how that happens, and in what sense?
I do understand differential geometry notations and tensors, so explanation with them is OK.
Thank you :)
I am responding a second time because I believe there is a misconception in this thread about how differential geometry replaced Euclidean geometry historically.
Originally,people thought that space is intrinsically Euclidean, that Euclidean geometry is embedded in the very idea of space itself. Geometry was not thought of in terms of calculus or curvature but in terms of axioms that people believed uniquely defined the structure of space.These axioms were simple and described the way lines separate planes and how lines intersect with each other. One axiom stated that a line separates a plane into two half planes. Another says that two lines can intersect in at most one point. Another says that two points determine a unique line.
People felt that the Parallel Postulate which says that in a plane, there is a unique parallel to a given line passing through any point, should not be a postulate but was actually a necessary intrinsic feature of space. This meant that it should be provable from the other axioms. People tried to find this proof for two thousand years and failed repeatedly. The thing that got them was uniqueness. They could not show that there was only one parallel through a point rather than many. They were able to show that there is at least one parallel and I will describe how they did this.
18'th century mathematicians knew that if two lines are both perpendicular to a third line then they must be parallel. Why? Because if they were not then by symmetry they would have to intersect in both half planes and thus would intersect in two points, a violation of the axioms for lines. So they knew that parallels always exist. What they could not prove was that these parallels were unique. This made them question what straightness really meant and led them to accept the possibility that uniqueness wasn't true.
Gauss finally discovered an example of a plane geometry in which all of the axioms of space were still true but in which the parallel postulate failed. In his geometry there were infinitely many parallels through a point. This ended the search for a proof and wiped out two thousand years of belief in the Euclidean universe.
Gauss was able to arrive at this new axiomatic geometry because people were already questioning what the idea of straight really was and this allowed him to find a model of the axioms where lines were actually curves. But he did not use differential geometry nor the idea of Gauss curvature in these thoughts. He just realized that the axioms could be modeled using curves in a plane as lines and still hold, except for the parallel postulate itself.
After Gauss's new geometry was discovered, people thought that there were two possible intrinsic geometries for space, Euclidean and Gaussian, and Gauss went out and measured large triangles on the Earth to try to find out which one of the two was true for the universe. If the universe was Euclidean, the sum of the angles of a triangle would be 180 degrees, if it was Gaussian the sum would be less than 180 degrees. This was an Earth shattering event in the history of thought. People suddenly saw that the geometry of space must be determined empirically, that there is no way of knowing it a priori!
Gauss did not use differential geometry or the notion of Gaussian curvature in these early thoughts. It was only later that he realized that the deviation from 180 degrees in a geodesic triangle could be determined from the Gauss curvature and that knowing this, his new geometry could be modeled as a surface of constant negative curvature.
After these key initial discoveries mathematicians still continued to wonder what the idea of straightness really was. How the idea of a geodesic was adopted I am not sure but the notion of a path of least constraint, as Gauss called it, was already known from the calculus of variations, and differential geometry was already a well developed subject largely through the researches of Monge and his school in France. I am sure that the connection of differential geometry to non-Euclidean geometries was not made until Gauss discovered that Gaussian curvature is an intrinsic property of a surface. The whole idea in the search for the true geometry of space was to find its intrinsic properties. Until Gauss curvature was known to be intrinsic I can not see how it would have helped research on non-Euclidean geometry.
By the way, the sphere is not a model for a non-Euclidean plane geometry and was not considered as a candidate in early research. Lines on a sphere intersect in two points or conversely two points sometimes determine more than one line. That violates the axioms of space.
One might try to pass to projective space by modding out all of the opposite poles and letting the lines be the projections of great circles onto the quotient. Then two lines will intersect in exactly one point and there will be no parallel lines. Sadly, another axiom is then violated, the axiom that says that a line separates a plane into two half planes. It is hopeless. This is why the sphere did not play a role in the fall of the Parallel Postulate.
