What is (a) geometry?

Solution 1:

According to Klein, geometry can be viewed as the action of a group on a space, be it smooth or finite. See this. That is, a geometry on a set $X$ is a triple $(X,G,A)$, where $G$ is a group with action $A$ on $X$.

Solution 2:

Usually, geometry consists of an underlying topological space (a manifold, for example) and some structure on this space. The structure is an analogy of some tool – such as a ruler or compass – that enables you to see more than what the topology sees. It might be something that enables you, for example, to “measure angles and distances” (Riemannian geometry), or “just to measure angles” (Conformal geometry), or “to see what are lines and what are not lines” (Projective geometry), or some other, more abstract analog of a “ruler and compass”.

From what I have learned, the Cartan geometry – which defines geometry as a principal bundle over a manifold with some Cartan connection – generalizes both Kleinian and Riemannian geometry in some sense; one book where this is explained is Sharpe: Cartan's generalization of Klein's Erlangen program.

The reason why there is not a single universal definition, unlike in topology, is the immense history of geometry (2500 years, compared to 100 years of topology).