Incoming links: Real projective plane
So simple!! You can either:
- cut two holes and glue a handle. This is easy to visualize as it can be embedded in : you just get a Torus, then a double torus, and so on
- cut a single hole and glue a Möbius strip in it. Keep in mind that this is possible because the Möbius strip has a single boundary just like the hole you just cut. This leads to another infinite family that starts with:
A handle cancels out a Möbius strip, so adding one of each does not lead to a new object.
You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!
Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.
Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.
A unique projective space can be defined for any vector space.
The definition is to take the vector space, remove the zero element, and identify all elements that lie on the same line, i.e.
The most important initial example to study is the real projective plane.
It good to think about how Euclid's postulates look like in the real projective plane:
- two parallel lines on the plane meet at a point on the sphere!Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The parallel postulate does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.One thing to note however is that ther real projective plane does not have angles defined on it by definition. Those can be defined, forming elliptic geometry through the projective model of elliptic geometry, but we can interpret the "parallel lines" as "two lines that meet at a point at infinity"
- points in the real projective plane are lines in
- For every two projective points there is a single projective line that passes through them.Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.
Unlike the real projective line which is homotopic to the circle, the real projective plane is not homotopic to the sphere.
The topological difference bewteen the sphere and the real projective space is that for the sphere all those points in the x-y circle are identified to a single point.
One more generalized argument of this is the classification of closed surfaces, in which the real projective plane is a sphere with a hole cut and one Möbius strip glued in.
To see that the real projective plane is not simply connected space, considering the lines through origin model of the real projective plane, take a loop that starts at and moves along the great circle ends at .
Note that both of those points are the same, so we have a loop.
Now try to shrink it to a point.
There's just no way!