A Kähler structure on the nonzero tangent bundle of a space form
Let \((M,g)\) be a Riemannian manifold. Consider a (natural) Lagrangian \(L\) on the tangent bundle \(TM\) of \(M\), defined by a differentiable real valued function of one real variable, depending on the energy density \(t=\frac 12 \| v\| ^2\), \(v\in TM\). The authors use an analogy with the well-known construction of the hyperkähler structure on the cotangent bundle of a complex space form of positive holomorphic sectional curvature [see \textit{E. Calabi}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 269-294 (1979; Zbl 0431.53066); \textit{A. Besse}, `Einstein manifolds' (Springer Verlag, Berlin) (1987; Zbl 0613.53001)] to define a Kähler structure on the tangent bundle of \((M,g)\). The usual nonlinear connection determined by the Euler-Lagrange equations associated to the Lagrangian \(L\) coincides with the nonlinear connection defined by the Levi-Civita connection of \(g\). Then a Riemannian metric \(G\) on the tangent bundle \(TM\) is considered such that the vertical and horizontal distributions \(VTM, HTM\) (defined by the Levi-Civita connection of \(g\)) are orthogonal to each other but they are no longer isometric (i.e., the considered metric \(G\) is no longer a metric of Sasaki type). It can also be defined an almost complex structure \(J\) on \(TM\) such that \((TM,G,J)\) is an almost Kähler manifold (just like in the case of the Sasaki metric). If the base manifold \((M,g)\) has positive constant sectional curvature, then there exists a Lagrangian \(L\) (obtained as solution of a second order ordinary differential equation) on the subset \(T_0M\) of nonzero tangent vectors to \(M\) such that the structure \((T_0M,G,J)\) is Kählerian. Unfortunately, the obtained Lagrangian is not smooth on the zero section of \(TM\). It is shown that the obtained Kählerian structure on \((T_0M,G,J)\) cannot be Einstein and cannot have constant holomorphic sectional curvature.
scientific article; zbMATH DE number 1339068
scientific article