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Bochner-Kähler metrics. (English) Zbl 1006.53019
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 M. Matsumoto [Tensor, New Ser. 20, 25-28 (1969; Zbl 0174.25001); 27, 291-294 (1973; Zbl 0278.53046), with S. Tanno], and by S. Tachibana and 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.
In the present paper by means of 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.
According 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 $$f:M\to F_n$$.
Elements $$v_1,v_2\in F_n$$ 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 $$m$$.
A 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 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 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.

##### MSC:
 53B35 Local differential geometry of Hermitian and Kählerian structures 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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