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


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
Full Text: DOI