Some Geometric Structures on the Tangent Bundle of a Riemannian Manifold

Abstract

Abstract. It is defined a new almost complex structure with Norden metric (hyperbolic metric) on the tangent bundle TM of an n-dimensional Riemannian manifold M. Next, the conditions under which the considered almost complex structure with Norden metric belongs to one of the eight classes of almost complex manifolds with Norden metric obtained by G. T. Ganchev and D. V. Borisov in the classification from [2] there are studied. IntroductionIt is well known that the tangent bundle T : TM -> M of a Riemannian manifold (M,g) can be equipped with the structure of almost Kaehlerian manifold by using the Sasaki metric and an almost complex structure defined by splitting of the tangent bundle to TM into the vertical and horizontal distributions VTM, HTM (the last one being determined by the Levi Civita connection on M) (see [1], [15]). However this structure is Kaehler only in the case where the base manifold is locally Euclidean (see [1]). The Sasaki metric is rather rigid and it should be interesting to get another Riemannian or pseudo-Riemannian metrics on TM, having some better properties. One possibility is to consider some (pseudo-)Riemannian metrics involving the natural lifts of the Riemannian metric g on M (for the definition and the expression of the natural 1-st order lifts of the Riemannian metric g to TMA particular case of such a Riemannian metric which can be considered is a slight generalization of the Sasaki metric and it has been studied by V. Oproiu in [9]. The Riemannian metric G on TM considered in [9] has been defined by using the Levi Civita connection of the Riemannian 216N. Papaghiuc metric g on M and two smooth real valued functions u(t), v(t) depending on the energy density only and such that u(t) > 0 and u(t) + 2tv(t) > 0 for all t € [0, oo). The Sasaki metric can be obtained from G in the case where u(t) = 1, v(t) = 0 for all t € [0, oo). Next, V. Oproiu has considered an almost complex structure J on TM, related to the metric G and has studied the conditions under which (TM, G, J) is a Kaehler-Einstein manifold. Note that the case when the Riemannian metric G on TM is defined by using a certain Lagrangian L on the base manifold (M,g) depending on the energy density only, i.e. the case when v(t) = u'(t), has been studied by V. Oproiu and the present author in [11]. Inspired from [9] and [8], in [12] and [13] we have considered another special natural 1-st order lift G of g which defines a pseudo-Riemannian metric on TM (so that, generally, G is no longer obtained as the complete lift by using a Lagrangian on M, see [7]). This new metric G has been defined by using also the Levi Civita connection of the Riemannian metric g and two real valued function u(t), v(t) such that u(t) > 0 and u(t) + 2tv(t) > 0 for all t 6 [0, oo). Next, we have studied the necessary and sufficient conditions for the pseudo-Riemannian manifold (TM, G) to be Ricci flat and respectively, locally symmetric.In the present note, we consider the same pseudo-Riemannian metric G defined in [12], [13] and we define an almost ...

Some Classes of Almost Anti-Hermitian Structures on the Tangent Bundle

“…The space F of tensor fields F of type (0,3) with the properties (12) has been decomposed (see [1]) into a direct sum of three irreducible components F 1 , F 2 , F 3 which are invariant under the natural representation of the semi-unitary (pseudounitary) group on F, induced from its standard representation on the model space of T M. From this decomposition one obtains eight classes of almost anti-Hermitian manifolds. We shall describe these classes and their characterizations.…”

“…Hence, we have: Other examples of anti-Hermitian manifolds which belong to one of the eight classes given in the classification in [1] have been obtained in [11] and [12].…”

Liouville and geodesic Ricci solitons

“…On the tangent bundle of a Riemannian manifold there are lots of very interesting metrics [4,10], but we restrict to the pseudo-Riemannian metric of [6] and [7]; see also [8] and [9]. A strong motivation of this choice is the fact that the paper [6] contains a computation of the Ricci tensor of this metric, useful for our study.…”