Embedding Real Projective Plane RP2 into R4

  • Thread starter sin123
  • Start date
In summary: This is exactly what we need to show in order to prove that the image of $F$ is a manifold.In summary, the map $F$ given by $F(x,y,z)=(x^2-y^2,xy,xz,yz)$ is an embedding of the real projective plane into $\mathbb{R}^4$, and its image is a smooth 3-dimensional submanifold of $\mathbb{R}^4$.
  • #1
sin123
14
0
P.S. I carelessly posted this question in the Topology&Geometry forum first:
https://www.physicsforums.com/showthread.php?t=341773

Homework Statement



Let F be a map from S2 in R3 into R4, given by

[tex] F(x,y,z) = (x^2 - y^2, xy, xz, yz) \. [ = (a,b,c,d)] [/tex]

Eventually I am supposed to show that this is an embedding of the real projective plane, but first I am asked to verify that the image of this map is a manifold at all. And that proved trickier than it looked like.

I know two ways of verifying that something is indeed a manifold.

1) Find local diffeomorphisms, taking a neighborhood of the manifold into R4 such that points on the manifold land in a copy of R2 inside R4

2) Show that the manifold is the level set of some function, where the derivative of that function has full rank at every point inside the level set.

Homework Equations



I am trying to go with approach (1). I can write down rules for the required functions, but I am not sure how to find neighborhoods that work.

The Attempt at a Solution



I have to deal with 6 cases, depending on where in the image my point lies. For example, if I start with a point p=F(x,y,z) where z2 is not 0 or 1/2, I get

[tex]a = \frac{2c^2 - 2 d^2}{1 \pm \sqrt{1 - 4 (c^2 + d^2)}},
b = \frac{2cd}{1 \pm \sqrt{1 - 4 (c^2 + d^2)}}[/tex]

where I use a plus in the denominator if z2 < .5 and a minus if z2 > .5. Once I have a and b in terms of c and d, I could pick a map as follows:

[tex]g(a,b,c,d) = (a - \frac{2c^2 - 2 d^2}{1 \pm \sqrt{1 - 4 (c^2 + d^2)}}, b - \frac{2cd}{1 \pm \sqrt{1 - 4 (c^2 + d^2)}}, c,d)[/tex]

This map returns (0,0,c,d) iff (a,b,c,d) was on the manifold in a small enough neighborhood of p. My problem is that I need to pin down an open set in R4 that does not accidentally contain a point on the manifold that is "on a different part" of the manifold than p.

I also tried working in polar coordinates but I had a very hard time tracking cases and solving for variables in a couple of instances, and it didn't seem to make the search for appropriate neighborhoods any easier.Any suggestions? How do I find and verify my neighborhoods?
 
Physics news on Phys.org
  • #2
A:The image of this map is a 3-dimensional submanifold of $\mathbb{R}^4$. To see this, note that the equation $F(x,y,z)=(a,b,c,d)$ defines an implicit surface in $\mathbb{R}^3$, which is given by $$x^2-y^2=a, \quad xy=b, \quad xz=c, \quad yz=d.$$ This surface is not just any surface; it is a so-called quadric surface. Quadric surfaces are well known to be smooth (in fact, they are even algebraic varieties), and so this surface is a smooth 3-dimensional submanifold of $\mathbb{R}^3$.For any $(a,b,c,d)\in\mathbb{R}^4$, this equation defines a point $(x,y,z)\in\mathbb{R}^3$, and so we can think of the map $F$ as a smooth map from $\mathbb{R}^3$ to $\mathbb{R}^4$. Since the surface is smooth, this means that its image under $F$ is also a smooth 3-dimensional submanifold of $\mathbb{R}^4$.
 

1. What is RP2 into R4 embedding?

RP2 into R4 embedding is a mathematical concept that involves mapping a 2-dimensional projective space (RP2) into a 4-dimensional Euclidean space (R4). It is a method used in topology and geometry to study the properties of higher dimensional spaces.

2. How is RP2 into R4 embedding used in scientific research?

RP2 into R4 embedding is used in various fields of study such as physics, computer science, and biology. It is used to model and analyze complex systems, as well as to study the behavior and interactions of objects in higher dimensions.

3. What are the applications of RP2 into R4 embedding?

The applications of RP2 into R4 embedding are diverse and include computer graphics, robotics, and data visualization. It is also used in the study of knots and surfaces, as well as in the development of algorithms for optimization and machine learning.

4. What are the challenges of RP2 into R4 embedding?

One of the main challenges of RP2 into R4 embedding is the visualization and interpretation of higher dimensional objects. It can also be computationally intensive and may require advanced mathematical techniques to fully understand and analyze.

5. How does RP2 into R4 embedding relate to other mathematical concepts?

RP2 into R4 embedding is related to other concepts in topology and geometry, such as homotopy and homology. It is also closely linked to the study of manifolds and their embeddings in higher dimensional spaces.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
265
  • Calculus and Beyond Homework Help
Replies
14
Views
571
  • Calculus and Beyond Homework Help
Replies
4
Views
635
  • Calculus and Beyond Homework Help
Replies
3
Views
116
  • Calculus and Beyond Homework Help
Replies
21
Views
754
  • Calculus and Beyond Homework Help
Replies
7
Views
132
  • Calculus and Beyond Homework Help
Replies
3
Views
784
  • Calculus and Beyond Homework Help
Replies
3
Views
221
  • Calculus and Beyond Homework Help
Replies
6
Views
517
  • Calculus and Beyond Homework Help
Replies
3
Views
488
Back
Top