On the dimension of the Lie group of automorphisms of a paracontact metric manifold

V. I. Pan'zhenskii   ; O. P. Surina

Čebyševskij sbornik, Tome 25 (2024) no. 5, pp. 244-253

Résumé

It has been proved that the dimension of the Lie group of automorphisms of a $(2n+1)$-dimensional smooth manifold endowed with the paracontact metric structure $(\eta,\xi,\varphi, g)$ does not exceed $(n+1)^2$, where $\eta$ is a differential $1$-form defining the contact $2n$-dimensional distribution $H=\ker \eta$, $\xi$ is a Reeb vector field, $\varphi$ is a structural endomorphism, $g $ is a pseudo-Riemannian metric whose restriction to the contact distribution $H$ has signature $(n,n)$. The analysis of the conditions for the invariance of the paracontact metric structure with respect to infinitesimal automorphisms, as well as using the Darboux atlas, in each chart of which the contact form $\eta$ has a canonical form, allows us to state that the isotropy group induced by the stationary subgroup of the point $p (0,..., 0)$, rotates only vectors lying in the contact plane $H_p$, and leaves invariant the pseudo-Euclidean metric and the symplectic structure defined by the differential $2$-form $\Omega=d\eta$. So the maximum dimension of the Lie algebra of the isotropy group is $n^2$. Since the dimension of the translation subgroup does not exceed the dimension of the manifold, the dimension of the automorphism group does not exceed $n^2+2n+1$. The paper also proves that the maximum dimension of the Lie algebra of infinitesimal automorphisms is equal to $(n+1)^2$. An example of a paracontact metric manifold admitting a Lie algebra of infinitesimal automorphisms of maximum dimension is the generalized Heisenberg group endowed with a canonical para-Sasakian structure. The basis vector fields of this algebra are found.

Détail

Mots-clés : paracontact metric structure
Keywords: automorphisms, infinitesimal automorphisms, Heisenberg group.