A Ricci flat pseudo-Riemannian metric on the tangent bundle of a Riemannian manifold

Abstract

Abstract. We consider a certain pseudo-Riemannian metric G on the tangent bundle T M of a Riemannian manifold (M, g) and obtain necessary and sufficient conditions for the pseudo-Riemannian manifold (T M, G) to be Ricci flat (see Theorem 2). A Riemannian metric G on T M has been defined by using the Levi-Civita connection of g 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, ∞). He has also considered an almost complex structure J on T M , related to the metric G and has studied the conditions under which (T M, G, J) is a Kähler Einstein manifold. Note that in [10], the author excludes some important cases which appeared, in a certain sense, as singular cases. These singular cases have been studied by V. Oproiu and the present author in [11], [9], [12]. Note also that one of the important cases studied in [11] is when the Riemannian metric G on T M 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)). On the other hand, in [8], V. Oproiu has studied a pseudo-Riemannian structure on the tangent bundle of a Lagrange manifold M , considering the pseudo-Riemannian metric G on T M as being the complete lift of a quadratic form defined by the Lagrangian L considered.

Some Geometric Structures on the Tangent Bundle of a Riemannian Manifold

“…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).…”

“…has the signature (n, n) and both distributions VTM, HTM are isotropic. In the An almost complex structure J may be defined on TM by and R^-are the local coordinate components of the curvature tensor field of V on M, we obtain (see [12], [13]) PROPOSITION …”

“…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 complex structure J on TM, related to the pseudo-Riemannian metric G. Recall that a Norden metric (or an hyperbolic metric) on an almost complex manifold (M, J) is a pseudo-Riemannian metric G on M such that G(JX, JY) = -G(X, Y), for arbitrary vector fields X, Y on M. In [2] the almost complex manifolds with Norden metric have been classified in eight classes. In this paper, we show that the considered pseudo-Riemannian metric G and the almost complex structure J on TM define an almost complex manifold with Norden metric (TM,G,J) (Proposition 1).…”

“…In the present note, we consider the same pseudo-Riemannian metric G defined in [12], [13] and we define an almost complex structure J on TM, related to the pseudo-Riemannian metric G. Recall that a Norden metric (or an hyperbolic metric) on an almost complex manifold (M, J) is a pseudo-Riemannian metric G on M such that G(JX, JY) = -G(X, Y), for arbitrary vector fields X, Y on M. In [2] the almost complex manifolds with Norden metric have been classified in eight classes. In this paper, we show that the considered pseudo-Riemannian metric G and the almost complex structure J on TM define an almost complex manifold with Norden metric (TM,G,J) (Proposition 1).…”

“…The following pseudo-Riemannian metric of natural 1-st order lift type may be considered on TM (see [12], [13])…”