A locally symmetric Kähler Einstein structure on the tangent bundle of a space form
The tangent bundle of a Riemannian manifold \((M,g)\) can be organized as an almost Kählerian manifold by using the Sasaki metric and an almost complex structure defined by the splitting of the tangent bundle \(TM\) into the vertical and horizontal distributions \(VTM\) and \(HTM\), the last one being determined by the Levi-Civita connection on \(M\). This structure is Kähler only in the case where the base manifold is locally Euclidean. In this paper, the author is interested to find a Kähler Einstein structure on the tangent bundle of a space form. The Sasaki metric is changed and also the almost complex structure on \(TM\). Considering the energy density \(t={1\over 2} g_{ij}(x) y^iy^j\) defined on \(TM\) by the Riemannian metric \(g\) of \(M\) and a real smooth function \(v:[0,+\infty)\to \mathbb{R}\), the following symmetric \(M\)-tensor field of type \((0,2)\) is defined on \(TM: G_{ij}= g_{ij}+ v(t) g_{0i}g_{0j}\), where \(g_{0i}= g_{hi} y^h\). The inverse of the matrix \((G_{ij})\) is \((H^{k\ell})\), where \(H^{k\ell}= g^{k\ell}+ wy^k y^\ell\) with \(w=- {v\over 1+ 2tv}\). By ``lowering'' the indices, the following Riemannian metric on \(TM\) is considered: \(G= G_{ij} dx^i dx^j+ H_{ij}\dot\nabla y^i\dot\nabla y^j\), where \(\dot\nabla y^i= dy^i+ \Gamma^i_{j0} dx^j\). An almost complex structure \(J\) is defined by \(J{\delta\over\delta x^i}= G^k_i{\partial\over\partial y^k}\); \(J{\partial\over\partial y^i}= -H^k_i{\delta\over\delta x^k}\). It is proved that \((TM,G,J)\) is an almost Kählerian manifold. In the next section, the integrability of the almost complex structure defined by \(J\) on \(TM\) is studied. The following result is obtained: The almost complex structure \(J\) on \(TM\) is integrable if and only if \((M,g)\) has constant sectional curvature \(c\) and \(v=-c\). If \(c<0\) a Kähler structure is defined on all of \(TM\). In the case where the constant \(c\) is positive one has a Kähler structure in the tube around the zero section in \(TM\) defined by \(t<{1\over 2c}\). In the last section, the author studies the property of \((TM,G)\) to be Einstein. After the computations, the components of the Ricci tensor field of the Levi Civita connection \(\nabla\) of \(G\) are found. The author also finds that the Ricci tensor is proportional to \(G\), hence \((TM,G)\) is an Einstein manifold.
scientific article; zbMATH DE number 1460284