Riemann
On the 10th of June 1854 Riemann gave his famous "Habilitationsvortrag" in the Colloquium of the Philosophical Faculty at Gottingen. His talk with the title " Uber die Hypothesen, welche der Geometrie zu Grunde liegen" is often said to be the most important in the history of differential geometry. Gauss, at the age of 76, was in the
audience and is said to have been very impressed by his former student. Riemann’s revolutionary ideas generalized the geometry of surfaces which had been studied earlier by Gauss, Bolyai and Lobachevsky. Later this lead to an exact definition of the modern concept of an abstract Riemannian manifold.
audience and is said to have been very impressed by his former student. Riemann’s revolutionary ideas generalized the geometry of surfaces which had been studied earlier by Gauss, Bolyai and Lobachevsky. Later this lead to an exact definition of the modern concept of an abstract Riemannian manifold.
P/S: The Riemannian metrics, which define the structure of a manifold, are what is difficult in the modern mechanics problem. The "manifold" is where the "physical laws" come into place in many physical problems. It describes the "behavior" of a system. Mechanics is a very classical problem, and they are "geometrical" in nature. "Geometry" is what always confuse people – they assume sin(th), cos(th), … are what they mean by "geometry", and abuse their notation (or wordings) in many literature – this is what I don’t like in most cases. They are not only confusing people, but also confusing themselves. "Geometry", in Riemannian setting, means the topology of a set. They are rules (or axioms) that describe the "structure" (structure looks like geometry, that’s why people are confused) of the set in space. Riemannian metrics itself have some rules (or axioms) built upon it as well, of course I’m not going to elaborate here. Truly, the classical mechanics have been well-advancing since these mathematicians come out with these amazing tools. We, mechanical engineer, should take advantage of these tools to design better (or improve) mechanical device so that they improve our life.
Hilbert, the 6th question out of his famous 23 "Hilbert problems" presented in Second International Congress in Paris on August 8, 1900, asked whether or not physics can be axiomized, and Smale (1998) asked in his 18th problem that "what are the limits of intelligence, both artificial and human?" – so my question is: any metric can capture physics and intelligence in the first place?