Bochner-Kähler metrics (Q2719033)

The curvature function of a Kähler manifold \(M\) of complex dimension \(n\) can be written as a sum of the scalar curvature, the traceless Ricci tensor, and the Bochner tensor. If the latter vanishes the manifold \(M\) is called a Bochner-Kähler manifold, endowed with a Bochner-Kähler metric, which is supposed here to be of class \(C^5\), thus real-analytic. By means of the unitary coframe bundle \(P\), the principal right \(U(n)\)-bundle over \(M\), and the exterior differential calculus the structure equations for a Bochner-Kähler manifold are deduced. They involve three functions \(H,T,V\) and the map \((H,T,V): P\to iu(n)\oplus \mathbb{C}^n\oplus \mathbb{R}\), known as the structure function, and allow simple proofs of some results by \textit{M. Matsumoto} [Tensor, New Ser. 20, 25-28 (1969; Zbl 0174.25001); 27, 291-294 (1973; Zbl 0278.53046), with \textit{S. Tanno}], and by \textit{S. Tachibana} and \textit{R. C. Liu} [Kodai Math. Semin. Rep. 22, 313-321 (1970; Zbl 0199.25303)]. But a complete local description of Bochner-Kähler metrics is not known until now and is considered far from trivial.NEWLINENEWLINENEWLINEIn the present paper by means of \textit{É. Cartan}'s existence and uniqueness theorem (discussed in the Appendix A of the paper) it is shown that for any \((H_0,T_0,V_0)\) there exists a Bochner-Kähler structure on a neighborhood \(U\) of \(0\in \mathbb{C}^n\) whose unitary coframe bundle \(\pi: P\to U\) contains a \(u_0\in P_0= \pi^{-1}(0)\) for which \(H(u_0)= H_0\), \(T(u_0)=T_0\), and \(V(u_0)=V_0\). Any two real-analytic Bochner-Kähler structures with this property are isomorphic on a neighborhood of \(0\in \mathbb{C}^n\). Finally any Bochner-Kähler structure that is \(C^5\) is real-analytic.NEWLINENEWLINENEWLINEAccording to a corollary the set of isomorphism classes of germs of Bochner-Kähler structures in dimension \(n\) is in one-to-one correspondence with the elements of a `chamber' in \(W\subset iu(n)\oplus \mathbb{C}^n\oplus \mathbb{R}\). This shows that the space of isometry classes of germs of Bochner-Kähler metrics in complex dimension \(n\) can be naturally regarded as a closed semi-algebraic subset \(F_n\subset \mathbb{R}^{2n+1}\) and there is a corresponding mapping \(f:M\to F_n\).NEWLINENEWLINENEWLINEElements \(v_1,v_2\in F_n\) are said to be analytically connected, if there is a connected Bochner-Kähler manifold for which \(f(M)\) contains both \(v_1\) and \(v_2\). This is an equivalence relation with equivalence classes \([v]\subset F_n\). For a simply-connected \(M\) it is shown that the Lie algebra \({\mathfrak g}\) of Killing fields of a Bochner-Kähler structure on \(M\) has dimension at least \(n\). The precise dimensions are computed for each analytically connected equivalence class \([v]\subset F_n\). Also the dimension of the orbit of the local isometry pseudogroup and the cohomogeneity \(m\) are computed. The ultimate conclusion is that a Bochner-Kähler metric always possesses a rather high degree of infinitesimal symmetry. The author considers as the perhaps greatest surprise that the Lie algebra \({\mathfrak g}\) contains a canonical central subalgebra whose dimension is equal to the cohomogeneity \(m\).NEWLINENEWLINENEWLINEA polynomial embedding \(i_v[v] \to\mathbb{R}^m\) is given as a convex polytope so that the interior of \(i_v([v])\) carries a canonical Riemannian metric which is related to the metrics considered by \textit{V. Guillemin} [J. Differ. Geom. 40, 285-309 (1994; Zbl 0813.53042)]. The case of compact \([v]\) is studied and as a corollary the result of \textit{Y. Kamishima} [Acta Math. 172, 299-308 (1994; Zbl 0828.53059)] is obtained that the only compact Bochner-Kähler manifolds are the compact quotients of the known symmetric ones. At the end of the paper some discussions are made about nontrivial complete Bochner-Kähler metrics on orbifolds.