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Conversely, real intrinsic singularities exist on surfaces.
They are true geometric singularities: (55)
(56)
(57)
And so on...
Incidentally, a fold is a peculiar region of a surface where
linear curvature is concentrated. In figure (57), on the left, we have negative
linear curvature. On the right, positive linear curvature.
We used two portions of a sphere in each sub-figure. The
global object has the same topology as the sphere, which means that its total
angular curvature is 4π
.
Suppose the object on the left was built with two portions
of spheres containing, each, an angular curvature of 3π.
3π +
3π = 6π
This is too much. Then the (negative) linear curvature
must balance this, in order to give the final required value of 4π
:
As a conclusion, our fold contains a negative curvature
:
- 2π
.
This curvature is uniformly distributed along the circular
curve, along the fold.
Return to figures (57). We have drawn triangles,
constructed with geodesic lines. But you can cross a fold without any problem
using a (narrow) strip of sticky tape. You know how to compute, to predict the sum of
the three angles of the triangle. You just need to compare the area of the
triangle to the area of the sphere. The excess of curvature is:
(58)
But you have to take into account the (negative or positive)
curvature contained in the portion of the fold, in the arc mn. It is:
(59)
Suppose a sort of lens, on the right of figure (57), is
built from two portions of a sphere, each containing an angular curvature of π
. Therefore, if we ignore the fold, this set of two portions
of a sphere contains an angular curvature of 2π
. But this lens has a spheroidal topology. The angular curvature
contribution must be 2π.
Then:
2π +
2π = 4π
( total curvature of the sphere )
You can also predict the sum of this strange triangle,
formed from three geodesic lines. The arc mn contains the linear angular curvature
:
(60)
When measuring the amount of angular curvature contained
in the fold, inside the triangle, one can gauge the departure from the Euclidean
sum, which is π.
You see that you can handle these curvature problems,
on surfaces, relatively easily.
A surface may possess conical points, or fold lines. They
are intrinsic singularities, not these artificial singularities, due to some
choice of coordinates. Notice that we can smooth the fold. Then we get a peanut
shape:
(61)
This is similar to smoothing the point-like summit of
a cone (concentrated angular curvature), transforming the object into a blunt
cone (angular curvature displayed over a portion of a sphere).
Suppose the two portions of sphere, above, in figure (61)
represent 2/3 of a sphere, each, i.e., a curvature of
(62)
The grey portion of the "peanut" contains negative curvature.
Precisely:
(63)
