PMB
That is incorrect. While its true that curvature is a local property of a manifold, it has a definite curvature at every single point on a manifold and its either zero or nonzero, a property which is independent of the coordinate system. And its for that reason I said that its an absolute property of the manifold itself. And by this I mean how these terms are defined in the mathematical literature and the physics literature, which includes the GR literature.

The only purpose of this paragraph seems to be to contradict me, but instead it only seems to contradict itself (within the first 6 words).
I am sorry you are having personal medical problems, but being grumpy doesn't help communication here.
I have no intention of getting into a futile slanging match with you and I'm sure your Physics is rock solid. However tripping through the minefield of mathematics definitions requires extreme care as almost nothing is as it seems.
A mathematical manifold is nothing more than a set of elements or points which share a common property or collection of properties.
It doesn't have to be all the items in the universe with those properties.
It doesn't have to have any structure, but it may do. (That is there need be no relations between the individual elements and no operations such as addition defined between the elements)
But each and every element much possess or be attributable to the common property of interest.
So if the common property of interest is redness then the set
{Ferrari F1 car, a tomato, Mars} is a manifold.
Not very interesting, mathematically but a set of red tomatoes might be of mathematical interest to a horticulturalist comparing redness statistically.
kiwiheretic
I guess a differentiable manifold is rougly equivalent to a surface being continuously differentiable in calculus.

PMB
As I recall, that's almost exactly what it means.

A good perceptive question and answer, which brings me to topological manifolds, which allow us to do the very desirable operations of calculus on a manifold.
The common property of a topological manifold is that every element of a topological manifolds is homeomorphic to the interior points of a sphere (otherwise called an open ball) in a Euclidian space.
This means that each point has neighbours or that the manifold may be divided into disjoint open subsets. The word open is very important.
Formally a manifold M is an n dimensional topological manifold if there is
locally a finite open cover Z that maps each point in M onto an open subset of R(^n).
The pair (M, Z) is a C(^r) manifold if Z is a C(^r) differential structure.
So we have the basis of our calculus.
I think that a one dimensional manifold is an excellent and easy to understand example, compared to the above (simplified) formal definition.
In fact it is so simple that it has been used for hundreds of years by surveyors under the guise of 'Through Chainage'.
We also used it in Kiwiheretic's recent question about Lagrangians and angles.
It is easy to see that there is no global function that will give you the circular curve connecting two roads which meet at angle theta in one dimension. You can determine the curvature from points on the connecting curve, but not from points on the straights.
Turning to two dimensional manifolds,
Naval architects and sheet metal workers have used the concept of ruled surfaces for centuries and more recently civil engineers have used this to construct the iconic cooling towers in power stations.
The curved surface of your cylinder (without the ends) is a ruled surface because it is isomorphic to the plane. The surface of a sphere is not so isomorphic as any cartographer will tell you.
This isomorphism means that you can rule straight lines on a piece of paper or card and then roll it into a cylinder with the lines parallel to the axis remaining straight.