I heard that there's several kind of geometries for instance projective geometry and non euclidean geometry besides the euclidean geometry. So the question is what do you mean by a geometry, do you need truly many geometries and if yes what kind of results we can find in one geometry and not in the others. Thanks a lot.
Different geometries denote different sets of axioms, which in turn result in different sets of conclusions. I'll concentrate on the planar cases.
Projective geometry is pure incidence geometry. The basic relation expresses whether or not a point lies on a line or not. One of its axioms requires that two different lines will always have a point of intersection, which in the case of parallel lines is usually interpreted as being infinitely far away in the direction of those parallel lines. Projective geometry does not usually come with any metric to measure lengths or angles, but using concepts by Cayley and Klein, many different geometries can be embedded into the projective plane by distinguishing a specific conic as the fundamental object of that geometry. This includes Euclidean and hyperbolic geometry as well as pseudo-Euclidean geometry and relativistic space-time geometry, among others.
Non-Euclidean geometries would in the literal sense be any geometry which doesn't exactly follow Euclid's set of axioms. More specifically, though, it is usually used for geometries which satisfy all of his postulates except for the parallel postulate. This will always include hyperbolic geometry and, depending on how you interpret the other axioms, usually includes elliptic geometry as well. One important difference between these and Euclidean geometry is the way lengths and angles are measured. It turns out that hyperbolic geometry describes the geometry on an infinite surface of constant negative curvature, whereas elliptic geometry is the geometry on a positively curved surface and therefore closely related to spherical geometry.