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Well as a mathematician I'm of the view there is no such thing as a meaningless abstraction. Consider the 3D manifold embedded in the 4D Euclidean space A = cos x cos y, B = sin x cos y, C = sin y, D = t. Which gives a metric
cos^2 y, 0, 0
0, 1, 0
0, 0, 1
This shape is essentially a 3D sphere extruded through the time dimension. It is a clear shape with a clear curvature which may be calculated Ether from the metric or directly from our original definition. Other manifolds may have the same metric and internally you can't tell the difference but externally it may have a different shape. There are different measures of curvature. Essentially when we say curvature we are talking about a generalisation of Gaussian curvature so rather than a single number we have a curvature tensor which reduces to a single number in the case of 3D surfaces.
The Gaussian curvature of a cylinder is 0 just the same as a flat sheet. Essentially an ant on a cylinder can not tell the cylinder from the sheet.
The properties of this metric are internal and to an ant only capable of seeing the 'surface' the space around the manifold appears to be non existent but that does not mean it does not exist. In fact the ekpyrotic theory of how the universe started relies on the space around the manifold not being empty.
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