# Linearity and Euclid

To begin to discuss Euclidean space we need a path-connected topological space and a notion of length for paths. If we continuously deform a path, its length varies continuously, so lengths form a continuum.

The lengths of paths can be compared; furthermore, in the Euclidean world (but not in Minkowski space, hence relativity's space-time) lengths are positive, save only that a constant path – i.e. a single point, the path starting and ending without having gone anywhere – will have zero length. This last being a degenerate case, I'm choosing to disallow the constant path as a path – or, at least, allow length to not have to apply itself to such paths.

At least on a trajectory (or open path – not ending where it started), we can mark any two (distinct) points and examine the portion of the path between these; this path is shorter than the original (provided at least one of the chosen points isn't an end-point). Further for any path shorter than the given trajectory, there is at least some portion of our trajectory which has the same length as the given shorter path. In this sense, our continuum of lengths is one-dimensional.

When one path ends where another begins, we can combine the two paths; the length of the composite must be equal to the sum of the lengths of the two parts. When a path loops round and ends where it started, we can combine it with itself arbitrarily often and infer that lengths may be multiplied by integers. This suffices to require that our one-dimensional continuum of lengths is a (one-sided) linear space over the positive scalars; in short, given some length to use as a unit, every length is simply a positive scalar multiple of this unit.

I chose to build up the discourse for a Euclidean space via stages which can, as far as possible, be re-used for a locally Euclidean space, such as arises in discussion of smooth manifolds. Consequently, I shan't jump straight from the linearity of lengths to the end-game, in which we express (globally) Euclidean spaces as linear spaces over the positive scalars. The road there is worth travelling.

## Shortest paths and geodesics

To do geometry, we need paths that are straight; they make the discussion so much more straightforward: so let me try to characterize the notion straight.

Chose two distinct points in our path-connected topological space and consider the lengths of all possible trajectory from one to the other. One can clearly pause part-way down any such trajectory, go round some loop back to the pause-point, then continue along down the trajectory: thus, for any trajectory between the points, there's a longer trajectory between them; so there's no such thing as a longest path between points. On the other hand, one may find a path than which no other is shorter (though, conceivably, there might be some other with exactly the same length): such a path is described as a shortest path between the points.

Chose any two distinct points on a shortest path between two given end-points and look at the portion of the given shortest path that connects the chosen points. If there were a shorter path between the chosen points, we could use it in place of the given portion of the given shortest path, thereby obtaining a shorter path between the original points, so the given portion must be a shortest path between the chosen points. Thus each portion of a shortest path is itself a shortest path.

The surface of a cylinder is a path-connected topological space. On it, there are loops that are locally shortest – that is, chose two points on the loop and you'll find a shortest path between them which is a portion of the loop. Chose two points on the loop and divide the loop into the two portions whose end-points they are; if these two portions have equal lengths, replace one of the points with an interior point of either portion and start again; the resulting portions won't have equal length. Chose the longer portion: it clearly isn't a shortest path between its end-points; yet all sufficiently short portions of it are shortest paths between their end-points.

I'll describe a path as locally short precisely if it is covered by some list of portions each of which is a shortest path. Every shortest path is locally short.

When Obelix cuts a cake, he cuts a few dainty thin slices for other folk, keeping the rest of the cake for himself. The interior of Obelix's slice is a path-connected topological space. When Obelix cut the cake, he cut two flat faces joining the centre to the outside; from any point on one face to any point on the other, the paths within the cake have to go round the hole left when Obelix gave slices to other folk; the shortest path between two such points goes to the center and back out, so it's made of straight pieces, but it isn't straight. If we move one of the end-points of the bent path but keep it in the plane formed by the two straight portions of the path, we can construct the shortest path between its new position and the other, unmoved, end-point; this path coincides with the original along one of the straight portions.

I'll describe a path, P, as intrinsically angled if there's some shortest path, R, sharing an end-point with P for which there is a shortest path between the other ends of P and R which: doesn't have (R or) P as a portion and isn't a portion of P but; shares a common portion with P.

I'll describe a path as a geodesic precisely if it is locally short and not intrinsically angled. When a geodesic is also a shortest path, I'll describe it as a shortest geodesic.

## Locally Euclidean blobs

Now chose a point, K, in our path-connected topological space and look at all the geodesics passing through it. To characterize locally Euclidean, I'll look at some neighbourhood of K in terms of those geodesics.

One of the crucial properties of a locally Euclidean space is that for each point, K, there is some length, d, for which there are no geodesic loops through K of length less than 2.d. We can then, in effect, look at a spherical region around K of diameter d and be sure that the sphere doesn't intersect itself.

I'll describe a path-connected topological space, U (typically an open sub-set of some larger p-c.t.s), with a positive path-metric as

iff, for every X in U, there is a unique geodesic within U connecting K to X. I'll describe K in U as a center of U precisely if U is star-shaped about K.

convex

iff U is star-shaped about every K in U.

(A Euclidean space is convex; a locally Euclidean space can be stitched together out of overlapping star-shaped patches, which can be convex if one wishes.)

bounded

iff there is some (finite) length which is greater than the length of any shortest geodesic in U. Such a length is described as an outer diameter of U; one will typically be interested in the least such.

voluminous

iff there is some positive length, r, and some K in U for which:

• for any X in U there is a shortest geodesic of length at least r which has K as an end-point and passes through X; and
• for any shortest geodesics KX and KY whose lengths are less than r/3, there is a shortest geodesic of length at least r having X as an end-point and passing through Y.

(The second condition is there to enable me to take two points near K and build a polygon around K to show that, for instance, it's not on the axis of Obelix's cake-slice.)

In such a case, I'll describe K as an inner point of U and r as an inner radius for U.

a proto-Euclidean blob

iff U is voluminous and star-shaped about some inner point of U. I'll describe a center which is also an inner point as an inner center.

So, in a proto-Euclidean blob, chose an inner center and an inner radius about that center. We can now look at the collection of points exactly that far from our center: these we can try to characterize as a unit sphere. Every point in our blob, aside from the center itself, is on exactly one geodesic out of the center and this geodesic does cut the unit sphere either on the way to the point or, if it is very near the center, after passing through the point but before leaving the blob. We are thus able to describe the each point in the blob in terms of a length (distance from some chosen inner center) and a direction (point on the surface of the unit sphere).

I'll describe a proto-Euclidean blob, U, as locally Euclidean iff ...

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