What Connects Different Types of Geometry in Mathematics?

[tgt]

On the one hand there are Differential Geometry, Algebraic Geometry

On the other there are Euclidean geometry, Hyperbolic geometry and elliptical geometry

On the other there are Affine geometry, projective geometry.

How do they all link up? Or are they all a bit different.

 

[micromass]

This is a very exciting question. It turns out that Euclidean, hyperbolic, elliptical, affine and projective geometry are all versions of the same thing, called a Cayley-Klein geometry. The Klein program says that it's the allowed transformations that characterize the geometry. So Euclidean geometry, or hyperbolic geometry are the same thing, except for the allowed transformations. The Cayley-Klein model generalized this situation and derives very general results which hold for all these types of geometries (for example, a general law of sines). Furthermore, it exhibits all these geometries as subsets of projective geometry.

Differential and algebraic geometry don't fit as well here. In my opnion, they're not an actual geometry, but they describe methods for studying geometry. So algebraic geometry will study geometry through algebraic methods, while differential geometry will study it through smooth, calculus methods. The two have very parallel results though.

 

[tgt]

This is a very exciting question. It turns out that Euclidean, hyperbolic, elliptical, affine and projective geometry are all versions of the same thing, called a Cayley-Klein geometry. The Klein program says that it's the allowed transformations that characterize the geometry. So Euclidean geometry, or hyperbolic geometry are the same thing, except for the allowed transformations. The Cayley-Klein model generalized this situation and derives very general results which hold for all these types of geometries (for example, a general law of sines). Furthermore, it exhibits all these geometries as subsets of projective geometry.

So is projective geometry inside Cayley-Klein geometry or vice versa as you seem to suggest at the end?

Differential and algebraic geometry don't fit as well here. In my opnion, they're not an actual geometry, but they describe methods for studying geometry. So algebraic geometry will study geometry through algebraic methods, while differential geometry will study it through smooth, calculus methods. The two have very parallel results though.

Can you derive geometrical results not get-able from the results above (such as in projective geometry) because of their extra structure?

Also are the geometrical results gotten from either differential or algebraic geometry results about either euclidean geometry, hyperbolic geometry or elliptical geometry?

 

[micromass]

So is projective geometry inside Cayley-Klein geometry or vice versa as you seem to suggest at the end?

Every type of geometry can be found inside projective geometry. This is the Cayley-Klein formalism. So a Cayley-Klein geometry is not a type of geometry, but rather a formalism to study many different types of geometries.

Can you derive geometrical results not get-able from the results above (such as in projective geometry) because of their extra structure?

Depends on what you mean with a "geometrical result". The Klein Erlanger program describes a geometric result exactly as those results which can be obtained from the Cayley-Klein formalism.

Also are the geometrical results gotten from either differential or algebraic geometry results about either euclidean geometry, hyperbolic geometry or elliptical geometry?

You can study projective geometry in differential or algebraic setting perfectly. You can study Euclidean and affine geometry in differential and algebraic setting. Hyperbolic and elliptical geometry however seem more suited for differential geometry.

 

[tgt]

Every type of geometry can be found inside projective geometry.

Ok but does that also mean any geometrical facts (in any geometries) can be derived as theorems in projective geometry?

So geometrical facts derived using either differential geometry or algebraic geometry can in theory be derived in projective geometry?

 

[micromass]

Ok but does that also mean any geometrical facts (in any geometries) can be derived as theorems in projective geometry?

Yes, for those geometries for which the Erlanger program is applicable and useful.

So geometrical facts derived using either differential geometry or algebraic geometry can in theory be derived in projective geometry?

Not exactly, since differential geometry for example has very different results than the Cayley-Klein geometries. There is an overlap (for example when discussing hyperbolic geometry), but a lot of differential geometry deals with very different stuff.

 

[tgt]

Not exactly, since differential geometry for example has very different results than the Cayley-Klein geometries. There is an overlap (for example when discussing hyperbolic geometry), but a lot of differential geometry deals with very different stuff.

Is it because of the extra structure in differential geometry?