Why the interest in locally Euclidean spaces?

One of the main reasons for interest is that manifolds are homogeneous. So from a geometric point of view, we study manifolds for much the same reason we study groups.

Put in less abstract ways, there is the "classical problem" of whether or not the earth is flat. The "flat earth problem" deals with the possibility that on small scales something may look linear, but macroscopically it need not be. Manifolds are the abstract manifestation of the flat earth problem.

People do study large families of objects that are locally modelled on spaces that aren't Euclidean spaces. There's infinite-dimensional manifolds, fractal manifolds, orbifolds (locally modelled on euclidean space mod a finite group action), etc... Fibre bundles are maps that are locally projection maps. This idea resurfaces in mathematics in many different forms and many different ways.

In my opinion (because it really is an opinion), the main reason to care about euclidean spaces is that they come equipped with a lot of particularly nice structures. In particular, they are finite-dimensional Hilbert spaces. This means that they are:

• Finite-dimensional vector spaces (so we can talk about addition and scaling)
• Complete metric spaces (so we can talk about distances and limits, and limits behave as we would like them to)
• Inner product spaces (so we can talk about the notion of "angle," and thereby do all sorts of geometric things.)

And really, there's just so much geometric intuition that comes along with these ideas -- not to mention an entire calculus apparatus. After all, there are plenty of topological spaces where the above notions are not defined or otherwise fail to be true.

The reason (at least to me) to study locally euclidean spaces is that we want to study spaces that are more general than euclidean spaces, yet still retain many of their nice features. In particular, we want a place where calculus makes sense.

Areas of math which study more abstract spaces include topology and algebraic geometry. Admittedly, I'm not very well-versed in either just yet, but I'm sure practical uses (and physical models) have been found in both.

I'm interested in differential topology, so that baises my answer. $\mathbb{R}$, and Euclidean spaces generally, are interesting because you can do calculus on them. Manifolds are interesting because, not only as OrbiculaR says, you can do standard analysis in a chart, but you can sensibly extend the concepts of analysis to the entire manifold.

You can also do analysis in the p-adics, so you might wonder why aren't studying p-adic manifolds. (I don't actually know.)