# What is needed to define a geometry?

I'm not quite sure whether what I am asking is a valid question, however I have come across two different 'geometries': Euclidean and Minkowski (I have also heard of differential geometry, Riemannian geometry, elliptical geometry but I don't know much about these). In Euclidean geometry, we don't involve time whereas in Minkowski we do. In Minkowski we also have a different distance measurement...

I am not sure whether I am perhaps mixing up a 'geometry' with 'a list of properties of spacetime' in the last part...

And I apologise that my question seems to be based in physics; I am interested in geometries in general , but am just a first year general science undergraduate so my examples are simple physics-based ones.

• This is a great question! – Nefertiti Mar 29 '17 at 14:07

I do (for some reason) pretend to be a geometer so perhaps I can try and say something. There are, as you've observed, many mathematical disciplines refer to as some kind of geometry' which are all related to each other.

Since your examples fall under the umbrella of Differential/Riemannian geometry I'll stay in that subject (algebraic geometry has a somewhat different character and would need another answer to tackle it).

Geometry in this context breaks up naturally into a local part and a global part, and any 'space' (manifold) is considered as glued together out of local pieces by maps which satisfy certain properties.

The 'local part' is concerned with studying the properties of different notions of distance in the space $\mathbb{R}^n$ (in fact with different metrics which are symmetric positive definite two forms on tangent vectors - a kind of infinitesimal' notion of distance, but integrating one can recover a notion of distance between two points).

One famous example is given by endowing $\mathbb{R}^n$ with the usual (Euclidean) distance. Another (if we drop the positive definite requirement for the metric), is Minkowski space. Probably the most important local notion is that of curvature. The famous constant curvature metrics on $\mathbb{R}^2$ give the (local) studies of spherical, euclidean and hyperbolic geometries respectively.

The global component comes in when we consider more complicated spaces as made up of local patches and try and globally define the geometric objects we have in the local case. For example I can put a notion of distance on a sphere coming from it's usual embedding in $\mathbb{R}^3$ and view it as coming from patches of the plane with a metric (as it happens with positive curvature). Different types of geometry (broadly) come from the requirements that we put on the gluing maps between pieces - because these maps determine what kinds of objects are globally defined. For example in the cases I've described to get a global notion of distance you must preserve the metric (this is called being an isometry). There are many other cases, both stronger and weaker e.g. my gluing map might only need to be continuous (topological manifolds), or be complex analytic (complex geometry) or preserve a symplectic form (symplectic geometry).

So, to summarise an answer, you need a local notion of distance, called a metric, and a way of gluing local patches together which preserve it.

I don't pretend to be a geometer; from what you wrote I guess what you need is a friendly, pop understanding. There was originally one geometry, the so-called Euclidean geometry a.k.a. the plane geometry. One of the Euclid's five axioms is the so-called Parallel Postulate. Generations of mathematicians tried to prove that this axiom is actually not an axiom in the sense that it may be derived from the other four axioms. It turns out that this axiom is proved independent of the other four axioms. But, unlike Physics, which explores the universe we occupied, mathematics explores every possible universe! You may think of it this way, that when we "modify" the set of axioms by replacing exactly one of them or other reasonable actions, then we have got a new geometry that may be different than Euclidean one. For example, the sum of the interior angles of a triangle on a sphere can be proved to be $> \pi$.

In modern mathematics the concept of axiom is nothing more than a set of statements having certain nice properties that serves as a logical beginning for inference. The set of axioms for a formal system resembles to a certain degree the set of rules for a game.

When you take a look at here, I think the most common "Definitions" of a geometry are made by a system of axioms on how incidences and relations between primitives (lines, points,...) work.

For example you get the Projective Geometry essentially by altering the axiom on parallelity on lines and say parallel lines meet at the point at infinity.