On the structure of the homogeneous space SO(5)/SO(3) (Q2799778)

Consider the linear isomorphism \(\sigma\) between \(\mathbb{R}^5\) and the space \(\mathrm{Symm}_{3,0}\) of all symmetric, traceless \(3\times3\)-matrices given by NEWLINE\[NEWLINEX=(x_1,\ldots,x_5)\in\mathbb{R}^5\leftrightarrow\sigma(X)= \begin{pmatrix} \frac{x_1}{\sqrt{3}}-x_4&x_2&x_3\\ x_2&\frac{x_1}{\sqrt{3}}+x_4&x_5\\ x_3&x_5&-\frac{2\,x_1}{\sqrt{3}}\end{pmatrix}NEWLINE\]NEWLINE and let \(\mathrm{SO}(3)\) act on \(\mathrm{Symm}_{3,0}\) by conjugation. The pull-back of this action under \(\sigma\) gives an irreducible representation of \(\mathrm{SO}(3)\) in \(\mathbb{R}^5\), studied earlier by \textit{M. Bobieński} and \textit{P. Nurowski} [J. Reine Angew. Math. 605, 51--93 (2007; Zbl 1128.53017)]. It leads to a nonstandard embedding of \(\mathrm{SO}(3)\) in \(\mathrm{SO}(5)\) and to a nonstandard factor space \(M^7=\mathrm{SO}(5)/\mathrm{SO}(3)\), which is the main subject of this paper. Considering the orthogonal composition \(\mathrm{so}(5)=\mathrm{so}(3)\oplus V\) with respect to the scalar product \(\langle A,B\rangle=-\mathrm{trace}(A\cdot B)/10\) it turns out that the commutator of two vectors of \(V\), followed by an orthogonal projection gives a kind of vector product on \(V\) and, consequently, on \(M^7\). The sectional curvature of \(M^7\) is shown to be constant, \(K=1/4\), as well as the Ricci curvature, \(\mathrm{Ric}=3/2\), and the scalar curvature, \(S=21/2\).NEWLINENEWLINEFor the entire collection see [Zbl 1298.53003].