This note introduces a \(1\)-parameter of cubic curves naturally associated to the sphere \(S^4\) considered in the unique \(5\)-dimensional irreducible representation space of \(SO(3)\). Eight examples are discussed with the last two being elliptic curves. Also, two conics are defined naturally in our setting by a special basis of the Lie algebra \(sl(3, \mathbb{R})\).
The paper [1] of Nigel Hitchin starts with a very interesting approach of the unit sphere \(S^4\). More precisely, this sphere is not considered directly in the Euclidean space \(\mathbb{E}^5:=(\mathbb{R}^5, g_{can})\) but is the unit sphere in a five-dimensional linear space of real \(3\times 3\) matrices. Concretely, the space is the intersection \(Sym_0(3):=Sym(3)\cap sl(3, \mathbb{R})\) where \(Sym(3)\) is the six-dimensional space of symmetric matrices while \(sl(3, \mathbb{R})\) is the usual Lie algebra of the special Lie group \(SL(3, \mathbb{R})\) of dimension \(8\). The space \(Sym_0(3)\) arises naturally in the Cartan decomposition: \[sl(3, \mathbb{R})=Sym_0(3)\oplus so(3);\quad dim: 8=5+3.\]
Considering a diagonal matrix as being defined by \(diag(\lambda _1, \lambda _2, \lambda _3)\) we use the traceless property of elements of \(sl(3, \mathbb{R})\) in order to define the cubic curve \(\mathcal{C}: y^2=(x-\lambda _1)(x-\lambda _2)(x-\lambda _3)\). Hence, the present work focuses on the study of the \(1\)-parameter family of cubics \(\mathcal{C}=\mathcal{C}(u)\), \(u\in [0, 2\pi )\) provided by \(S^4\) as unit sphere in \(Sym(3)\cap sl(3, \mathbb{R})\) endowed with the inner product \(Trace(B_1B_2)\). So, we compute the Weierstrass coefficients \(p, q\) and the discriminant \(\Delta\) of \(\mathcal{C}(u)\). Also, a main subject is that of examples and hence an elliptic curve is discussed after two singular cubics. We finish the first part with a non-existence result of Euclidean cubic polynomials for our setting.
In the second part of this paper we consider conics naturally provided by the symmetric matrices from a specific basis of \(sl(3, \mathbb{R})\). We point out that this research continues that of [2] where cubics and conics are geodesically associated to the points of a geometric surface.
Nigel Hitchin introduces in [1] a \(1\)-parameter family of Einstein metrics on the standard sphere \(S^4\). The parameter is an integer \(k\geq 3\) and each metric is self-dual, \(SO(3)\)-invariant, of positive scalar curvature, and noncompact but admits a compactification as a metric with conical singularities. In fact, for \(k=3\), there is no singularity at all, and the metric is the standard one on the \(4\)-sphere.
The underlying sphere \(S^4\) is considered as the unit sphere in the \(5\)-dimensional linear space \(Sym_0(3):=Sym(3)\cap sl(3, \mathbb{R})\) of symmetric and traceless \(3\times 3\) real matrices \(B\), with \(SO(3)\) acting by conjugation. The invariant inner product of \(Sym_0(3)\) is defined as \(Trace(B_1B_2)\) and hence \(S^4\) corresponds to the matrices \(B\in Sym_0(3)\) with eigenvalues \(\lambda _1, \lambda _2, \lambda _3\in \mathbb{R}\) satisfying: \[\left\{ \begin{array}{ll} \lambda _1+\lambda _2+\lambda _3=0, \\ \lambda _1^2+\lambda _2^2+\lambda _3^2=1. \end{array} \right. \label{eq2.1} \tag{1}\]
The intersection of the plane \(\Pi: \lambda _1+\lambda _2+\lambda _3=0\) with the sphere \(S^2\) provided by the second equation (1) yields the ellipse: \[E: (\lambda _2-\lambda _1)^2+3(\lambda _1+\lambda_2)^2=2, \label{eq2.2} \tag{2}\] having the eccentricity: \[\mathfrak{e}=\sqrt{\frac{2}{3}}\simeq 0.81649. \label{eq2.3} \tag{3}\]
The inverse \(\frac{1}{\mathfrak{e}}=\sqrt{\frac{3}{2}}\simeq 1.2247\) is the eccentricity of the self-complementary hyperbolas, conform [3], while the intersection of the plane \(\Pi\) with the general rotational ellipsoid \(\mathcal{E}: \frac{\lambda _1^2}{a^2}+\frac{\lambda _2^2}{b^2}+\frac{\lambda _3^2}{c^2}=1\), for \(a, b, c>0\), is the ellipse: \[E(a, b, c): \left(\frac{1}{a^2}+\frac{1}{c^2}\right)\lambda _1^2+\frac{2}{c^2}\lambda _1\lambda _2+\left(\frac{1}{b^2}+\frac{1}{c^2}\right)\lambda _2^2-1=0,\] with the eccentricity: \[\mathfrak{e}^2=\mathfrak{e}^2(a, b, c)=\frac{2\sqrt{(a^2-b^2)^2c^4+4a^4b^4}}{2a^2b^2+c^2(a^2+b^2)+\sqrt{(a^2-b^2)^2c^4+4a^4b^4}}.\]
It results that the eigenvalues \(\lambda\) defines a curve on the sphere \(S^2\): \[C: \bar{r}(\varphi )=(\lambda _1, \lambda _2, \lambda _3)(\varphi ):=\left(\frac{1}{2}(\mathfrak{e}\sin \varphi -\sqrt{2}\cos \varphi ), \frac{1}{2}(\mathfrak{e}\sin \varphi +\sqrt{2}\cos \varphi ), -\mathfrak{e}\sin \varphi \right),\quad \varphi \in [0, 2\pi ). \label{eq2.4} \tag{4}\]
We point out that in [1, p. 192] the eigenvalues are expressed with rational functions of \[t=\Big|\tan\left(\frac{3\pi }{4}+\frac{\varphi }{2}\right)\Big|=\Big|\frac{\tan \frac{\varphi }{2}-1}{\tan \frac{\varphi }{2}+1}\Big|,\] but we prefer a trigonometrical approach.
Due to the first identity (1) we introduce now a parametrized cubic curve naturally associated to this setting: \[\mathcal{C}(\varphi ): y^2=P_{\varphi }(x):=(x-\lambda _1(\varphi ))(x-\lambda _2(\varphi ))(x-\lambda _3(\varphi )),\quad \varphi \in [0, 2\pi ), \label{eq2.5} \tag{5}\] and then we have immediately the main theoretical result of this note:
Proposition 1. The Weierstrass coefficients \(p\), \(q\) of the cubic curve \(\mathcal{C}(\varphi ): y^2=x^3+px+q\) are: \[p=constant=-\frac{1}{2},\quad q=q(\varphi )=-\frac{\sqrt{6}}{18}\sin (3\varphi )\in \left[q_{min}=-\frac{1}{3\sqrt{6}}, q_{max}=\frac{1}{3\sqrt{6}}\right]. \label{eq2.6} \tag{6}\]
The discriminant \(\Delta :=4p^3+27q^2\) of \(\mathcal{C}(\varphi )\) is: \[\Delta (\varphi )=-\frac{1}{2}\cos ^2 (3\varphi )\in \left[-\frac{1}{2}, 0\right]. \label{eq2.7} \tag{7}\]
The discriminant map \(\Delta\) has an unique fixed point (\(\Delta (\varphi )=\varphi\)), namely \(\varphi _0\simeq -0.257079\).
Proof. We have directly: \[\Delta =27\cdot \frac{6}{18^2}\sin ^2(3\varphi )-\frac{1}{2}=\frac{1}{2}(\sin ^2(3\varphi )-1). \tag{8}\]
The value of the fixed point is obtained by using WolframAlpha as solution \(\cos ^2(3\varphi )=-2\varphi\). It is easy to prove the uniqueness of this fixed point: let us consider the function \(f:\mathbb{R}\rightarrow{R}\), \(f(x):=2x-\cos ^2(3x)\). This function is smooth and it has the derivative \(f^{\prime }(x)=2-3\sin (6x)\). For \(x\in \left[-\frac{1}{2}, 0\right]\) it results \(6x\in [-3, 0]\subset (-\pi , 0]\) and hence \(\sin (6x)<0\) which means that the restriction of \(f\) to the interval \(\left[-\frac{1}{2}, 0\right]\) is strictly increasing. ◻
Remark 1. i) For the sake of completeness we provide also the expressions in \(t\): \[q(t)=\frac{(1-t^2)(1-14t^2+t^4)}{3\sqrt{6}(1+t^2)^3},\quad \Delta (t)=-\frac{2t^2(t^2-3)^2(3t^2-1)^2}{(1+t^2)^6},\quad t\in [0, +\infty ]. \label{eq2.8} \tag{9}\]
ii) We have the periodicity: \(\Delta (\varphi +\pi )=\Delta (\varphi )\) which means that the function \(\Delta (\cdot )\) corresponds to a function on the projective space \(\mathbb{R}P^1\). Indeed, with \(\left(\cos \varphi =\frac{u}{\sqrt{u^2+v^2}}, \sin \varphi =\frac{v}{\sqrt{u^2+v^2}}\right)\) we have the function: \[\Delta :\mathbb{R}P^1\rightarrow \mathbb{R},\quad [u, v]\rightarrow \Delta ([u, v])=-\frac{u^2(u^2-3v^2)^2}{2(u^2+v^2)^3}. \label{eq2.9} \tag{10}\]
iii) Due to the expression of the coefficient \(q\) we note the trigonometrical identity: \[\sin (3\varphi )=4\sin \varphi \sin \left(\varphi+\frac{\pi }{3}\right)\sin \left(\varphi+\frac{2\pi }{3}\right), \tag{11}\] which was used in [4] in order to prove the famous Morley’s theorem (1899).
iv) A very useful remark of the anonymous referee is that the \(1\)-parameter cubic curve \(\mathcal{C}(\varphi )\) is based only on the spectral data \((\lambda _1, \lambda _2, \lambda _3)\). Hence, it remains a (very interesting) open problem if the dependence of \(3\varphi\) captures some geometrical or representation-theoretic information(s) about the Hitchin’s matrix-model viewpoint of the initial round sphere \(S^4\).
Our main interest is in studying remarkable examples. Firstly, we remark that are six singular cases provided by the vanishing of \(\Delta\) from (7); in fact, for the singular angles \(3\varphi \in \Big\{\frac{(2k+1)\pi }{2}; k=0,…,5\Big\}\) we derive only two singular curves while a periodicity of \(3\) in the values of the parameter \(k\) yields antipodal points on \(S^2\).
Example 1. For \(k=1\) we have \(\varphi =\frac{\pi }{2}\) (or \(t=0\)), the cubic curve \(\mathcal{C}\left(\frac{\pi }{2}\right)=\mathcal{C}_1: y^2=x^3-\frac{x}{2}+\frac{1}{3\sqrt{6}}\) is singular and corresponds to the point \(\bar{r}\left(\frac{\pi }{2}\right)=\mathfrak{e}\left(\frac{1}{2}, \frac{1}{2}, -1\right)\in S^2\).
Example 2. Similarly, for \(k=4\) we have \(\varphi =\frac{3\pi }{2}\) (or \(t=+\infty\)), the cubic curve \(\mathcal{C}\left(\frac{3\pi }{2}\right)=\mathcal{C}_2: y^2=x^3-\frac{x}{2}-\frac{1}{3\sqrt{6}}\) is singular and corresponds to the point \(\bar{r}\left(\frac{3\pi }{2}\right)=-\bar{r}\left(\frac{\pi }{2}\right)\).
Example 3. For \(k=0\) we have \(\varphi =\frac{\pi }{6}\) (or \(t=\frac{1}{\sqrt{3}}\)) and the cubic curve \(\mathcal{C}\left(\frac{\pi }{6}\right)=\mathcal{C}_2\) corresponds to the point \(\bar{r}\left(\frac{\pi }{6}\right)=\frac{1}{2}\left(\frac{\mathfrak{e}-\sqrt{6}}{2}, \frac{\mathfrak{e}+\sqrt{6}}{2}, -\mathfrak{e}\right)\in S^2\).
Example 4. Similarly, for \(k=3\) we have \(\varphi =\frac{7\pi }{6}\) (or \(t=\sqrt{3}\)) and the cubic curve \(\mathcal{C}\left(\frac{7\pi }{6}\right)=\mathcal{C}_1\) corresponds to the point \(\bar{r}\left(\frac{7\pi }{6}\right)=-\bar{r}\left(\frac{\pi }{6}\right)\).
Example 5. For \(k=2\) we have \(\varphi =\frac{5\pi }{6}\) and the cubic curve \(\mathcal{C}_1\) corresponds to the point \(\bar{r}\left(\frac{5\pi }{6}\right)=\mathfrak{e}\left(1, -\frac{1}{2}, -\frac{1}{2}\right)\in S^2\).
Example 6. For \(k=5\) we have \(\varphi =\frac{11\pi }{6}\) and the cubic curve \(\mathcal{C}_2\) corresponds to the point \(\bar{r}\left(\frac{11\pi }{6}\right)=-\bar{r}\left(\frac{5\pi }{6}\right)\).
Example 7. For \(\varphi =0\) (or \(t=1\)) we have the elliptic curve \(\mathcal{C}(0): y^2=x^3-\frac{x}{2}\) with \(\Delta =-\frac{1}{2}\) and correspond to the equatorial point \(\bar{r}(0)=\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right)\in S^2\). This elliptic curve appears in the LMFDB Database as https://www.lmfdb.org/EllipticCurve/Q/256/c/1 with the equation: \[\mathcal{C}(0): Y^2=X^3-8X,\quad \Delta =-2^{11}=-2048,\quad X=2^2x,\quad Y=2^3y , \label{eq2.10} \tag{12}\] and is a CM-elliptic curve. We remark that Susumo Okubo showed in [5] that the space of \(3\times 3\) traceless complex matrices can be endowed with a multiplication, derived from the usual matrix multiplication, in such a way that it becomes a non-unital composition algebra.
In [6] we introduce the notion of Euclidean polynomial as being a monic polynomial for which the sum of squares of its roots is equal with the sum of squares of its proper coefficients; for the monic polynomial \(P_{\varphi }\) of (5) this means that \(p^2+q^2=\lambda _1^2+\lambda _2^2+\lambda _3^2=1\) which means that \(q^2=\frac{3}{4}\). We obtain a non-existence result as follows:
Proposition 2. There are no Euclidean polynomials \(P_{\varphi }\) of (5)-type.
Proof. From \(q^2=\frac{3}{4}\) it results \(q_{\pm }=\pm \frac{\sqrt{3}}{2}\) but \(q_{+}>q_{max}\). The value \(q_{-}\) is greater than \(q_{min}\) but the cubic equation provided by the expression of \(q\) from (6): \[X\left(\frac{4}{3}X^2-1\right)=\frac{2}{e}q_{-}=-\frac{3}{\sqrt{2}}, \label{eq2.11} \tag{13}\] has only one real solution \(X\simeq -1.3796<-1\) and hence it can not be a \(\sin \varphi\). ◻
In this section we firstly study the Frenet geometry of curve \(C\) considered as curve in the Euclidean space \(\mathbb{E}^3:=(\mathbb{R}^3, \langle \cdot, \cdot \rangle _{canonic})\): \[C: \bar{r}(\varphi )=\mathfrak{e}\left(-\cos \left(\varphi +\frac{\pi }{6}\right), \cos \left(\varphi -\frac{\pi }{6}\right), -\sin \varphi \right)\in S^2. \label{eq3.1} \tag{14}\]
Its tangent vector field is: \[T(u)=\bar{r}^{\prime }(\varphi )=\mathfrak{e}\left(\sin \left(\varphi +\frac{\pi }{6}\right), -\sin \left(\varphi -\frac{\pi }{6}\right), -\cos \varphi \right)\in S^2. \label{eq3.2} \tag{15}\]
Its binormal vector field is constant since \(\bar{r}\) belongs to the plane \(\Pi: \lambda _1+\lambda _2+\lambda _3=0\): \[B(u)=\bar{r}^{\prime }(\varphi )\times \bar{r}^{\prime \prime }(\varphi )=-\frac{\mathfrak{e}}{\sqrt{2}}(1, 1, 1)\in S^2. \label{eq3.3} \tag{16}\]
Then its torsion is zero while the curvature function is constant: \[k(\varphi )=\frac{\|\bar{r}^{\prime }(\varphi )\times \bar{r}^{\prime \prime }(\varphi )\|}{\|\bar{r}^{\prime }(\varphi )\|^3}=1, \label{eq3.4} \tag{17}\] which confirm that in the given plane \(\Pi\) this curve is the unit circle centered in \(O(0, 0, 0)\). Its normal vector field is: \[N(\varphi )=B(\varphi )\times T(\varphi )=\mathfrak{e}\left(\cos \left(\varphi +\frac{\pi }{6}\right), -\cos \left(\varphi -\frac{\pi }{6}\right), \sin \varphi \right)=-\bar{r}(\varphi )\in S^2. \label{eq3.5} \tag{18}\]
We have studied recently other classes of spherical curves in both \(S^2\) and \(S^3\) in [7], [8] and [9].
Example 8. We point out that an interesting problem is when the matrix \(Frenet(\varphi )=(T(\varphi ), N(\varphi ), B(\varphi ))\in SO(3)\) is symmetric. A straightforward computation yields the characterization \(Frenet(\varphi )\in Sym(3)\) if and only if \(\varphi =-\frac{\pi }{4}\) when: \[Frenet \left(-\frac{\pi }{4}\right)=\frac{1}{\sqrt{3}}\left( \begin{array}{ccc} \frac{1-\sqrt{3}}{2} & \frac{1+\sqrt{3}}{2} & -1 \\ \frac{1+\sqrt{3}}{2} & \frac{1-\sqrt{3}}{2} & -1 \\ -1 & -1 & -1 \\ \end{array} \right),\quad Trace Frenet \left(-\frac{\pi }{4}\right)=-1. \label{eq3.6} \tag{19}\]
The corresponding elliptic curve is: \[\mathcal{C}\left(-\frac{\pi }{4}\right): y^2=x^3-\frac{x}{2}+\frac{1}{6\sqrt{3}},\quad \Delta \left(-\frac{\pi }{4}\right)=-\frac{1}{4}. \label{eq3.7} \tag{20}\]
Secondly, we associate two conics to our linear space \(Sym_0(3)\). To do this, we recall after [10, p. 13] that the semisimple Lie algebra \(sl(3, \mathbb{R})\) is spanned by the matrices: \[E_1=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right),\quad E_2=\frac{1}{2}\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \\ \end{array} \right),\quad E_3=-E^t_1,\quad D=\frac{1}{6}\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \\ \end{array} \right), \tag{21}\] \[P_1=\left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right),\quad P_2=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} \right),\quad R_1=-P^t_2,\quad R_2=P_1^t, \label{eq3.8} \tag{22}\] where the subscript \(t\) means the transposition map. The motivation for this choice of basis consists in the presence of \(E_2\) and \(D\) as elements in \(Sym_0(3)\) .
With the approach of [3] a matrix \(\Gamma \in Sym(3)\) defines naturally a conic in the Euclidean plane \(\mathbb{E}^2\) of coordinates \((x, y)\) through the equation: \[\left( \begin{array}{ccc} x & y & 1 \\ \end{array} \right)\cdot \Gamma \cdot \left( \begin{array}{c} x \\ y \\ 1 \\ \end{array} \right)=0. \tag{23}\]
In conclusion, there are two associated conics:
i) \(E_2\) is the degenerate hyperbola consisting in the pair of bisectrices \(B_1: y=x\), \(B_2: y=-x\),
ii) \(D\) is the circle centered in the origin \(O(0, 0)\in \mathbb{R}\) and having the radius \(R=\sqrt{2}\).
The symmetric matrix (19) yields a hyperbola with the eccentricity: \[\tilde{\mathfrak{e}}=\sqrt{1+\frac{1}{\sqrt{3}}}\simeq 1.2559. \label{eq3.9} \tag{24}\]
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