ATRACTOR

\(\DeclareMathOperator{\sen}{sen}\)

Consider in the vector space \(M\) of dimension \(9\) defined by the matrices \(3\times3\), the usual inner product when identified with \(\mathbb{R}^{9}\).

Let \(M_{sim}\) be the vector subspace of dimension \(6\) of \(M\) defined by the symmetric matrices and \(M_{sim1}\) an affine subspace of dimension \(5\) of this latter space defined by the symmetric spaces of trace \(1\).

The projective space \(\mathbb{P}^{2}(\mathbb{R})\) is diffeomorphic to the subset \(P^{2}(\mathbb{R})\) of \(M_{sim1}\) through the diffeomorphism \(\varphi\) that to each vector subspace of dimension \(1\) of \(\mathbb{R}^{3}\) corresponds a matrix of the orthogonal projection of \(\mathbb{R}^{3}\) onto that subspace. If \((x,y,z)\in\mathbb{R}^{3}\) has norm \(1\), the matrix that corresponds to the vector subspace generated by \((x,y,z)\) is the matrix \[\left(\begin{array}{ccc} x^{2} & xy & xz\\ yx & y^{2} & yz\\ zx & zy & z^{2} \end{array}\right).\]

In fact, the image \(P^{2}(\mathbb{R})\) of the projective space is contained in the intersection with \(M_{sim1}\) of the spherical hypersurface \(S\) of center \(O\) and radius \(1\) of \(M_{sim}\).

One of the elements of \(S\cap M_{sim}\) that does not belong to \(P^{2}(\mathbb{R})\) is the matrix \[A=\left(\begin{array}{ccc} \frac{1}{3} & -\frac{1}{3} & -\frac{1}{3}\\ -\frac{1}{3} & \frac{1}{3} & -\frac{1}{3}\\ -\frac{1}{3} & -\frac{1}{3} & \frac{1}{3} \end{array}\right).\]

The stereographic projection \(\psi\), from \(A\), is a diffeomorphism of \(S\setminus\left\{ A\right\}\) over the vector subspace \(F\) of dimension \(5\) of \(M_{sim}\) defined by the orthogonal symmetric matrices of \(A\) and is explicitely defined by \[\psi(X)=A+\frac{1}{1-\left\langle X,A\right\rangle }(X-A).\]

It applies \(P^{2}(\mathbb{R})\) in the affine subspace \(F\cap M_{sim1}\) of dimension \(4\) from \(M_{sim}\), defined by the matrices \[Y=\left(\begin{array}{ccc} y_{1,1} & y_{1,2} & y_{1,3}\\ y_{2,1} & y_{2,2} & y_{2,3}\\ y_{3,1} & y_{3,2} & y_{3,3} \end{array}\right)\] that satisfy the conditions \(y_{1,1}+y_{2,2}+y_{3,3}=1\) and \(y_{1,2}+y_{1,3}+y_{2,3}=\frac{1}{2}\).

Having gotten to this point, we start a new stage: to try to immerse the projective space in \(\mathbb{R}^{3}\) (naturally with self-intersections, but with as few as possible) we compose the diffeomorphism \(\psi\circ\varphi\) with a reasonable affine morphism from \(F\cap M_{sim1}\) into \(\mathbb{R}^{3}\). For a second try we consider and affine morphism that to the matrix \(Y\) corresponds \((y_{1,1},y_{1,2}+y_{2,3},y_{3,3})\). Yet, after the composition with a translation of \(\mathbb{R}^{3}\) (which is equivalent to ignore the term \(A\) of \(\psi(X)\)), we obtain a morphism from \(\mathbb{P}^{2}(\mathbb{R})\) that to each vector subspace generated by a vector \((x,y,z)\) of norm \(1\), with corresponding matrix \[X=\left(\begin{array}{ccc} x^{2} & xy & xz\\ yx & y^{2} & yz\\ zx & zy & z^{2} \end{array}\right),\] associates the image, through the affine morphism, of the reffered matrix \[ \frac{1}{1-\left\langle X,A\right\rangle }(X-A) = \\ =\frac{3}{3-(x^{2}-2xy-2xz+y^{2}-2yz+z^{2})}\left(\begin{array}{ccc} x^{2}-\frac{1}{3} & xy+\frac{1}{3} & xz+\frac{1}{3}\\ yx+\frac{1}{3} & y^{2}-\frac{1}{3} & yz+\frac{1}{3}\\ zx+\frac{1}{3} & zy+\frac{1}{3} & z^{2}-\frac{1}{3} \end{array}\right)=\\ =\frac{3/2}{1+xy+xz+yz}\left(\begin{array}{ccc} x^{2}-\frac{1}{3} & xy+\frac{1}{3} & xz+\frac{1}{3}\\ yx+\frac{1}{3} & y^{2}-\frac{1}{3} & yz+\frac{1}{3}\\ zx+\frac{1}{3} & zy+\frac{1}{3} & z^{2}-\frac{1}{3} \end{array}\right) \]

Finally, consider the projective space as the image of the northen hemisphere of the unitary sphere with center the origin of \(\mathbb{R}^{3}\), parametrized by the latitude \(u\in[0,\frac{\pi}{2}]\) and the longitude \(v\in[0,2\pi]\), by \[\begin{array}{ccl} x & = & \cos(u)\cos(v)\\ y & = & \cos(u)\sen(v)\\ z & = & \sen(u), \end{array}\] which leads us to the parametrization of the image of the projective space, that to \((u,v)\) associates the element of \(\mathbb{R}^{3}\) with the three coordinates \[\begin{array}{c} \frac{3(\cos^{2}(u)\cos^{2}(v)-\frac{1}{3})}{2(\cos(v)\sen(v)\cos^{2}(u)+\cos(v)\sen(u)\cos(u)+\sen(u)\sen(v)\cos(u))}\\ \\ \frac{3(\cos(v)\sen(v)\cos^{2}(u)+\sen(u)\sen(v)\cos(u)-\frac{2}{3})}{2(\cos(v)\sen(v)\cos^{2}(u)+\cos(v)\sen(u)\cos(u)+\sen(u)\sen(v)\cos(u))}\\ \\ \frac{3(\sen^{2}(u)-\frac{1}{3})}{2(\cos(v)\sen(v)\cos^{2}(u)+\cos(v)\sen(u)\cos(u)+\sen(u)\sen(v)\cos(u))} \end{array}\]