What is geometry?

It is sometimes rash to try to give precise definitions of entire areas of mathematics. I would not attempt to define combinatorics, or analysis, and if I gave a reasonable sounding definition of number theory it would probably exclude about seventy percent of research in the subject.

The situation with geometry is a little different, however, in that there is a way of thinking about the subject, introduced by Klein, which, although it falls short of a complete definition, certainly clarifies what is meant by the word "geometry". This approach to geometry is known as Klein's Erlanger Programm ("Erlanger" doesn't mean anything - it just refers to where Klein lived when he thought of it) or the transformational approach.

What Klein suggested was that it is fruitful to regard geometry as follows. Given a group G of transformations, say of R^2, the geometry associated with that group is the study of properties of subsets of R^2 that do not alter when the subsets are subjected to transformations from G. For example, if G is the group of all rigid motions of the plane, then the resulting geometry is normal Euclidean geometry, while if instead you take G to be the group of Möbius transformations (of the extended plane), then you get Möbius geometry instead. If you take all invertible affine transformations (that is, transformations of the form x goes to Ax+b where A is an invertible linear map) then you get affine geometry. If you take all homeomorphisms (continuous maps with continuous inverses) then you get topology. If you take the surface of a sphere and use O(3) as your group of transformations then you get spherical geometry. In the course you will see what the transformations are for hyperbolic geometry.

The sorts of properties that one might consider are, for example, that of being a circle, or a triangle. The notion of a circle makes sense in Euclidean geometry, and also does in Möbius geometry once one adapts it to allow for circles containing the point at infinity. It does not make sense for affine geometry, since an affine map can transform a circle into an ellipse, but the notion of an ellipse does make sense here. In topology, one can no longer speak of a circle or an ellipse, but one can speak of a simple closed path. A triangle makes sense in Euclidean and affine geometry, but isosceles triangles and right-angled triangles belong only to Euclidean geometry.

If you bear this in mind, then you will have a clearer picture of what is going on in the course, and why it is appropriate to speak of spherical, Euclidean, Möbius and hyperbolic "geometry".

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