A useful example of good writing style is how Thurston introduces his geometrization conjecture in

W. P. Thurston. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982.

Here is the excerpt:

To state the conjecture formally, Thurston would have had to first define his 8 geometries. Instead, he wisely spends the introduction to explain where the conjecture comes from, in particular mentioning that it would imply the Poincare conjecture. He introduces the now famous 8 geometries in the subsequent pages, along with illuminating examples and figures. All in all, a deep technical conjecture that would take pages to state formally becomes an enjoyable read even for the non-expert.

This example teaches us how one can provide a rough overview of a result before getting to the technical details.

A useful example of good writing style is how Thurston introduces his geometrization conjecture in

W. P. Thurston. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982.

Here is the excerpt:

Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the

1.1. CONJECTURE. The interior of every compact 3-manifold has a canonical decomposition into pieces which have geometric structures.

In §2, I will describe some theorems which support the conjecture, but first some explanation of its meaning is in order.

To state the conjecture formally, Thurston would have had to first define his 8 geometries. Instead, he wisely spends the introduction to explain where the conjecture comes from, in particular mentioning that it would imply the Poincaré conjecture. He introduces the now famous 8 geometries in the subsequent pages, along with illuminating examples and figures. All in all, a deep technical conjecture that would take pages to state formally becomes an enjoyable read even for the non-expert.

This example teaches us how one can provide a rough overview of a result before getting to the technical details.