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(Positive) Curvature.
When we draw a triangle on a plane using our sticky tape, the sum of the angles is 180°. This is a Euclidean surface. We say that it contains no curvature. It is truly a flat surface. The sum of the angles of our triangle is the Euclidean sum. When we drew the triangle on our cone, our "posicone," and when the vertex S was outside it, the sum was still 180°. Conversely, when the vertex was inside, the sum was 180° plus the angle q (the cut we made to construct our posicone; see figure (8)).
This vertex is a peculiar point on the surface, a conical one, and we say that it contains some (positive) concentrated curvature. It is a point of concentrated (positive) curvature.
Now we can make two cuts, corresponding to angles q1 and q2. See figure (13). We then obtain a strange surface with two conical points, S1 and S2. See figure (14).
(13)
(14)
Now you can draw as many geodesic triangles as you want, corresponding to different cases.
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If they contain no conical vertex, the sum of the angles is 180°.
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If they contain the vertex S1, the sum is 180° plus q1.
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If they contain both vertices, q1 and q2, the sum is 180° + q1 + q2.
(15)
Imagine now that you can make a large number of tiny posicones and glue them together, as shown in figure (16). Each tiny posicone corresponds to an elementary angle Dq. You can arrange these mini-cones in a regular way. I mean that the distance between a vertex and the vertices of neighboring mini-cones would be almost constant everywhere.
(16)
If your mini-cones become smaller and smaller, as do their associated elementary angles Dq, you will construct a portion of a regular surface with constant curvature density.
A sphere is a surface with constant local curvature density. More simply, we say that the sphere is a surface of constant curvature.
If you arrange your mini-cones differently, you can construct a surface with variable local curvature density. For example, an egg. A hen’s egg is a surface with variable local curvature density. But a ping-pong ball is a surface with constant curvature density. This is how the hen recognizes its egg and distinguishes it from a ping-pong ball. It draws geodesics with sticky tape, and so on...
In fact, the hen does not physically draw geodesics on the object. It does so mentally.
(17)
In general relativity, mass density r is identified with local curvature.
Of course, the arena of general relativity is not a 2D surface. You can imagine a 3D hypersurface. You can imagine a 3D hypersphere. But who can imagine a 4D hypersurface?
Incidentally, the 4D curvature of the 4D hypersurface called the "universe" has special features that we are not going to explore here. This shows that didactic models have limitations. But they are good for stimulating the imagination and opening the mind to somewhat different worlds.
