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On a differential equation characterizing a Riemannian structure of a manifold. (English) Zbl 0534.53037

In this paper certain complete Riemannian manifolds (M,g) are characterized by the existence of a nontrivial solution f of the equation \(Hess(f)+k\cdot f\cdot g=0\) where k is a constant and Hess denotes the Hessian form. For \(k>0\) this is due to M. Obata [J. Math. Soc. Japan 14, 333-340 (1962; Zbl 0115.393)] and has been established independently for arbitrary k by Y. Tashiro [Trans. Am. Math. Soc. 117, 251-275 (1965; Zbl 0136.177)]. This covers theorems A, B, C, D of the present paper. It should be noted that Corollaries E and F are true locally, i.e. without assuming completeness (Cor. F follows trivially from the so-called Ricci identity). A more general local version of corollary E has been obtained by A. Fialkov [ibid. 45, 443-473 (1939; Zbl 0021.06501) in particular p. 471]. All these results make use of the warped product metric on the product \(\bar M\times {\mathbb{R}}\) where \(\bar M\) is a model for every regular f-level in M. In theorem G the author reformulates these results for the special case where (M,g) is an Einstein manifold. In this case the situation has been analyzed locally by H. W. Brinkmann [Math. Ann. 94, 119-145 (1925; JFM 51.0568.03)] who did not, however, formulate global theorems.
Reviewer: W.Kühnel

MSC:

53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53B20 Local Riemannian geometry
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