Classification of 3-dimensional left-invariant almost paracontact metric structures
Recall that an almost paracontact structure on a differential manifold \(M\) of dimension \(2n+1\) is a triple \((\varphi,\xi,\eta)\), where \(\varphi\) is a \((1,1)\)-tensor, \(\xi\) is a vector field and \(\eta\) is a \(1\)-form satisfying the following compatibility conditions \[ \varphi(\xi) = 0, \quad \eta\circ\varphi = 0, \quad \eta(\xi)= 1, \quad \varphi^2 = id - \eta\otimes\xi, \] and the restriction of \(\varphi\) on the horizontal distribution \(\ker \eta\) is an almost paracomplex structure, that is, the eigensubbundles \(T^+\), \(T^-\) corresponding to the eigenvalues \(1,-1\) of \(\varphi\), respectively, have equal dimension \(n\). An almost paracontact metric in an almost paracontact manifold is a pseudo-Riemannian metric \(g\) compatible with the almost paracontact structur \(a\), that is, \[ g(\varphi X,\varphi Y)=-g(X,Y)+\eta(X)\eta(Y), \quad X,Y \in \Gamma(TM). \] In this case \((\varphi,\xi,\eta,g)\) is said to be an almost paracontact metric structure. The authors study left-invariant almost paracontact metric structures on Lie groups, or equivalently on its Lie algebras. In particular they are interested in almost paracontact metric structure satisfying an extra condition, that is, \(\xi\in\ker\eta\). An almost paracontact structure \((\varphi,\xi,\eta)\) with this property is said to be natural. They obtain a complete classification of 3-dimensional simply connected Lie groups admitting a left-invariant natural almost paracontact metric structure, up to isomorphism (see Theorem 4.3 for the unimodular case and Theorem 4.5 for the non-unimodular case). They also study special cases of natural almost paracontact metric structure, namely normal structures and almost paracosymplectic structures. And they classify 3-dimensional simply connected Lie groups admitting these kind of structures in Theorem 5.1 and Theorem 5.4 respectively.
scientific article; zbMATH DE number 6854795
scientific article