Monday, 8 February 2010
what is geometry?
For most of my life I answered this question somewhat like: "Geometry is the study of shapes; of measurement of lengths, angles, curvatures, volume and so on." It did not occur to me that there are differences between Euclidean and e.g. spherical geometry other than the basis of the vector space. But as I had to learn there are a lot of differences between those geometries. Certainly I was not the only one caught in that assumption. For more than 2000 years the prefix Euclidean in Euclidean geometry was not needed since there were no other geometries known. Euclid was ~300 BC the first with an axiomatic attempt to geometry. His Elements had such an impact, that for the next 2000 years there were no answer to the question what is geometry other than Euclid's axioms. But then, even without asking, other geometries emerged. Spherical geometry was the most obvious but not alone. Others like hyperbolic-, projective-, Möbius- and Lie-geometry among others came into being and it became interesting again to ask: What is geometry? Finally in 1872 Felix Klein (the one with the bottle) came to a new viewpoint of geometry. In his Erlangen program he stated that there is a group of symmetries associated with every geometry. The hierarchy of the group resembles the hierarchy of the different geometries for example this means, that any theorem proven in projective geometry also holds in euclidean geometry because its associated group O(n) is a subgroup of GL(n) which is a subgroup of PGL(n+1). One major flaw of Klein's approach was that he was unable to include Riemannian geometry. Later this dilemma was fixed by Cartan and his connections which revolutionized differential geometry like Klein did with geometry.
All that I have learned about this brought me slightly to a better answer on the above question but not to fulfillment. So I conclude that I have to know more about group theory, Cartan's work
and algebraic geometry.