I'll get back to you when I've had a look over my old undergrad Maths for General Relativity notes.

This interest has been piqued by use of the term by Alain Badbwoy, I assume?

Edit: dur, Deleuze, obv.

OK, well I've had a quick look at my notes and also at Mathworld and a few Wikipedia entries, so I'm prepared to have a go.

A manifold is an N-dimensional space where every point can be *locally* mapped to R^N, where R means the real numbers and taking R "to the power of" N just means you're using N real numbers to describe a position - so that on an infinite, flat 2-dimensional plane, any point can be described completely with two coordinates, eg. x and y on the familiar Cartesian axes.

This flat plane is also called a 2-dimensional Euclidean space. Extend it to 3 dimensions, and it's a pretty good description of real, physical space in the absence of strong gravitational fields - i.e. the space we live in. You can add a time dimension to this 3-d Euclidean space to obtain a four-dimensional space-time, a.k.a. Minkowski space-time, which forms the basis for special relativity.

Anyway, in the middle of the 19th century Riemann, building on earlier work by Gauss, started to experiment with spaces in which the axioms of Euclidean geometry (such as "parallel lines converge at infinity" and "the shortest distance between two points is a straight line") were not assumed. This was all done in the spirit of purely mathematical enquiry, as it was not imagined at this point that there could be real physical applications of this work. So rather than just working in 'spaces', Riemann and others worked with 'manifolds', where Euclid's rules for 'flat' space don't (necessarily) hold, and the mapping of points from the manifold to R^N is perhaps non-trivial. An example that's simple to visualise is S^2, the surface of a sphere, where the x and y coordinates of 2-d Euclidean space are replaced by latitude and longitude, and the shortest line between two points is no longer straight but a geodesic, a section of a 'great circle'. Also, the surface of a sphere has intrinsic curvature, which is not shared by a flat plane. (Intrinsic curvature means that a flat sheet - a piece of paper, say - cannot be deformed onto the surface without 'crumpling': thus the cylindrical main part of a wine bottle does *not* have intrinsic curvature, but the 'shoulder' where this part meets the neck *does* have intrinsic curvature, as does a sphere.)

Now a differentiable manifold is one which is locally similar enough to Euclidean to allow calculus to be performed; in other words, it has no 'sharp' boundaries or discontinuities, ensuring that (in a given mapping to R^N) the mapping is single-valued at all points, and differentiable, i.e. continuous, or having a finite derivative with respect to some coordinate system. Another ingredient we need here is the metric, which is a function *g(x1,x2)* that gives the shortest distance between two points *x1* and *x2.* Then a Riemannian manifold is a real differentiable manifold with a non-negative metric, which allows intuitive geometrical quantities like length, area, volume and angle to be meaningful.

The maths here is probably a bit mangled, because I learnt the basics of this stuff from a physicist's POV. Someone like Slothrop may be able to give a more rigorous and accurate description. Anyway, general relativity is phrased mathematically in terms of Lorentzian manifolds, which are classified as 'pseudo-Riemannian' because the metric can be negative. This may sound nonsensical, as it describes a distance, but remember we are talking here about 'distances' in space-time, not space.

Hope this has been of some help.