Homogeneous Berger space and deformations of the $\mathrm{SO(3)}$-structure by its geodesic on $5$-dimension Lie groups

A. G. Sedykh   ; A. S. Berezina

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2014), pp. 35-45

Résumé

An irreducible $\mathrm{SO(3)}$-structure can be defined by means of a symmetric tensor field $T$ of type $(0,3)$ on a manifold $M$. Definition 1. An $\mathrm{SO(3)}$ structure on a $5$-dimensional Riemannian manifold $(M, g)$ is a structure defined by means of a rank $3$ tensor field $T$ for which the associated linear map $X\to T_X\in End(TM)$, $X\in TM$, satisfies the following condition: symmetricity, i. e. $g(X,T_Y Z) = g(Z,T_Y X) = g(X,T_Z Y)$, the trace $tr(T_X) = 0$, for any vector field $X \in TM$, $$ T_X^2X=g(X,X)X. $$ In any tangent space, it is possible to choose an adapted basis $\{e_1,e_2,e_3,e_4,e_5\}$ in which metrics $g$ and tensor $T$ have the canonical form $g_{ij}=\delta_{ij}$ and $$ \begin{gathered} T=\frac12e^1\left(6(e^2)^2+6(e^4)^2-2(e^1)^2-3(e^2)^2-3(e^5)^2\right)+\\ +\frac{3\sqrt3}2e^4\left((e^5)^2-(e^3)^2\right)+3\sqrt3e^2e^3e^5. \end{gathered} $$ Her, $\{e_1,e_2,e_3,e_4,e_5\}$ is the dual coframe. Polarising the expression yields components of $T$: $$ \begin{gathered} t_{111}=-1,\quad t_{122}=1, \quad t_{144}=1, \quad t_{133}=-\frac12,\quad t_{155}=-\frac12,\\ t_{433}=-\frac{\sqrt3}2,\quad t_{455}=\frac{\sqrt3}2,\quad t_{235}=\frac{\sqrt3}2. \end{gathered} $$ Thus, an irreducible $\mathrm{SO(3)}$-structure on a manifold is a Riemannian structure $g$ and a tensor field $T$ possessing properties 1–3. Theorem 1. The stabilizer of $T_{ijk}$ is an irreducible $\mathrm{SO(3)}$ embedded into $\mathrm{O(5)}$. Since the stabilizer $T_{ijk}$ is an irreducible $\mathrm{SO(3)}$, its orbit under the action of $\mathrm{O(5)}$ is a 7-dimension homogeneous space $\mathrm{O(5)/SO(3)}$. A homogeneous Berger space $\mathrm{SO(5)/SO(3)}$ is topologically equivalent to an $\mathrm{S^3}$ fiber bundle over $\mathrm{S^4}$. With respect to the biinvariant scalar product $\langle A,B\rangle=-\frac1{10}tr(AB)$ on $\mathrm{SO(5)}$, a decomposition of the Lie algebra $\mathrm{so(5)}$ into a direct sum $\mathrm{so(5)} = \mathrm{so(3)} + V$ of the Lie algebra and $\mathrm{ad(SO(3))}$ of an invariant space $V$ has been obtained. Examples of deformations of the structural tensor $T$ by geodesics $g_t$ of the homogeneous space $\mathrm{SO(5)/SO(3)}$ are considered, the covariant divergence of the obtained structure tensor is calculated, and the property of nearly integrability is investigated.

Détail

Keywords: special $\mathrm{SO(3)}$ structure, homogeneous Berger space
Mots-clés : Lie group.

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