Different approches to Geometry

[ShayanJ]

I read somewhere that a Geometry is a non Empty set and a subset of its power set which has subsets with at least two elements.The elements of the first set are called points and the elements of the second set are called lines.With specifying these two sets and considering some axioms,you will get a geometry.Now I have two questions.
1-As with vector spaces(which you can define things as vectors too different from arrows in space),Can I build a geomery with e.g. the set of all 2x2 matrices?
2-What is the relationship of this approach to geometry with the manifold geometry?

thanks

 

[arkajad]

I read somewhere

Probably you would more likely get some answer if you could remember "where".

 

[micromass]

I read somewhere that a Geometry is a non Empty set and a subset of its power set which has subsets with at least two elements.The elements of the first set are called points and the elements of the second set are called lines.With specifying these two sets and considering some axioms,you will get a geometry.Now I have two questions.

That's the incidence geometry- approach to geometry.

1-As with vector spaces(which you can define things as vectors too different from arrows in space),Can I build a geomery with e.g. the set of all 2x2 matrices?

Certainly, the set of all 2x2-matrices is a vector space, and each vector space induces a geometry. The points will be the vectors and the lines will be sets of the form $u+span(v)$ with v nonzero. Thus the lines through the origin will be the one-dimensional subspaces.

As a geometry, the set of all 2x2-matrices will be isomorphic to the geometry $\mathbb{R}^4$.

 

[chiro]

Hey Shyan.

It may help you to think about geometry through distance and angle attributes.

We have a variety of terms including metrics, norms, and inner products which help define these things precisely and give the conditions that these must have in order to be actual term.

Vector spaces and linear algebra (as well as multilinear algebra) will give you the foundations for thinking about these kinds of things.

If you add say topology then you get some precise definitions for things like continuity. By adding concepts like "smooth" (it's not the best way I can describe this so maybe someone can jump in with a better definition), then you are able to look at geometries that you can apply 'calculus' to which gives you another tool to analyze these in the context of geometry.

By knowing distance and angle, of which for the smooth structures has a differential form which is written in terms of infinitesimals (like you see with your standard differential equations), then you can get an expression for distance between one point and another point "close" to that point in a given 'direction' (This depends on the parameterization of the actual geometry) and along with other calculus techniques you are able to then calculate 'distance' (or an approximation if you can't get an analytic solution) and also 'angle' if you have a valid inner product.

 

[blue_raver22]

for sharing

I find this approach to geometry to be quite interesting. It seems to be a more abstract and mathematical way of defining geometry, rather than the traditional approach of studying shapes and their properties. To answer your questions:

1. Yes, it is possible to define a geometry using the set of all 2x2 matrices. In fact, this has been done in the field of non-Euclidean geometry, where matrices are used to represent transformations in a non-flat space. This approach allows for a wider range of geometries to be studied, beyond just the traditional Euclidean geometry.

2. The relationship between this approach to geometry and manifold geometry is that they both deal with abstract spaces and their properties. Manifold geometry is a branch of mathematics that studies spaces that are locally similar to Euclidean space, but may have different global properties. This can be seen as a more advanced and specialized version of the approach you described, where the sets and axioms are more complex and specific. Both approaches are important in understanding and studying different types of geometric spaces.