To understand the concept of a singularity in curved space, we need to first understand intrinsic curvature.
Whereas "extrinsic" curvature deals with how this object looks to us,
"Intrinsic" curvature deals with how this object would look to two dimensional beings living on its surface.
Suppose they are forbidden from going to the tip of the cone.
As far as the two dimensional beings living on this surface can tell,
this space is flat so long they don’t go all the way around the tip.
In flat space, if we move a vector around and then return it to its starting position,
the vector will always still be pointing in its original direction.
If the two dimensional beings want to determine that their space is not flat,
they can do this by going all the way around the tip of the cone and returning to where they started.
This is because their vector is now no longer pointing in its original direction.
In general, the rule for moving a vector along a curved surface is the following.
Each time a vector is moved by a tiny amount, the vector is rotated along the line perpendicular to the flat surface tangent to its new location.
If the two dimensional beings go all the way around the tip of the cone in the "counterclockwise" direction, their vector is rotated "counterclockwise."
If they go all the way around the tip in the "clockwise" direction, their vector is rotated "clockwise."
Because the vector rotates in the same direction in which they travel, we say that the area has a positive intrinsic curvature.
The angle by which their vector rotates is exactly equal to the angle of the missing section of this circle.
Let’s consider a new example.
In this new example, we will add an extra section to the circle.
The angle of the extra section that we add to the circle is exactly equal to the angle by which the vector rotates when it travels all the way around the surface.
This time, the vector rotates opposite to the direction of travel.
When the vector rotates opposite to the direction of travel, we say that the area has a negative intrinsic curvature.
Here is another way to think about positive and negative intrinsic curvature.
When the two circles curve in opposite directions, this surface has a negative intrinsic curvature.
The amount of curvature at the point where the two circles meet is given by the following equation.
When the two circles curve in same direction, this surface has a positive intrinsic curvature.
The amount of curvature at the point where the two circles meet is given by the following equation.
This surface has regions with positive curvature, because it has points where the two circles curve in the same direction.
And this surface also has regions with negative curvature, because it has points where the two circles curve in opposite directions.
This region has negative curvature.
This means that the vector rotates opposite to the direction of travel.
If a vector travels "counter-clockwise" around this area, the vector is rotated "clockwise."
This other region has positive curvature.
If a vector travels "counter-clockwise" around this area, the vector is rotated "counter-clockwise."
The vector rotates the same way as the direction of travel, as expected for a region of positive curvature.
If a vector travels around an area composed of the sum of two smaller areas,
then the amount by which the vector will rotate is the sum of the rotations associated with each individual area.
In this example, the vector rotates 90 degrees plus 90 degrees, which is equal to 180 degrees.
If the vector rotates around the region composed of this area with positive curvature and this area with negative curvature, the effects of the two areas will cancel each other out.
The vector rotates by 90 degrees plus negative 90 degrees, meaning that the vector returns pointing in its original direction.
As it turns out, if we add up all the positive and negative curvatures everywhere on a donut, we get a total curvature of zero.
The curvature of any surface at any point can be defined as follows.
The intrinsic curvature at a point is proportional to the angle by which a vector rotates when it travels around an area, divided by the size of the area, as the size of the area approaches zero.
This gives us the same answer as the definition involving the two circles.
However, the definition involving the two circles only works for a two dimensional surface embedded in a three dimensional space.
For higher dimensions, we need the more general definition of intrinsic curvature, based on how much a vector rotates when it travels around an area and then returns to its original position.
Consider the intrinsic curvature of the following surface.
We can visualize this by using our definition of curvature involving the two circles.
A line can be thought of as a circle with a radius of infinity.
Therefore, our equation tells us that the intrinsic curvature is zero at every point on this surface.
This means that two dimensional beings living on this surface would never be able to tell that that their space is curved.
Similarly, this visualization also tells us that a cone has zero intrinsic curvature at every point except at the tip.
This is why if we don’t go around the tip, our vector is never rotated.
For our vector to be rotated when we return to the starting position, we need to go all the way around the tip of the cone, because the tip is the only point on the surface with non-zero intrinsic curvature.
The intrinsic curvature at a particular point depends on the angle by which the vector is rotated, divided by the size of the area around which it travels, as the size of the area approaches zero.
But unlike smooth surfaces, in the case of the tip of a cone, even as the size of the area around the tip approaches zero, the angle by which the vector is rotated stays the same.
Therefore, the curvature at the tip of the cone is infinite.
If we double the angle of the missing section of the circle, the angle by which the vector rotates is also doubled.
The curvature at the tip of any cone is infinity, but not all infinities are created equal.
Suppose we increase the angle of the missing section of the circle to 180 degrees.
The amount by which the vector is rotated has now increased to 180 degrees.
For any smooth surface, as the size of the area approaches zero, the angle by which the vector rotates also approaches zero, giving us a finite value for the curvature.
But this is not the case for the tip of a cone, and we get a curvature of infinity.
As long as the two dimensional beings living on this surface don’t actually go through the tip, they can analyze their space without having to resolve the mathematics of infinity.
If they go around the tip, then replacing the area they never enter with a smooth surface will not affect the amount by which their vector is rotated.
They can just think of the path they followed as being on a sphere tangent to the cone at this location.
The closer the path is to the tip of the cone, the smaller the radius of the sphere.
If they actually go to the tip, they would need a sphere with a radius of zero, which has a curvature of infinity.
A point with infinite curvature, such as the tip of a cone, is an example of what we refer to as a "singularity."
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