I need a paper which wrote by Tanno where he prove that a equations $f_{ijk} + k(2f_{k}g_{ij} + f_{i}g_{jk} + f_{j}g_{ik})$ can be solved if and only if on sphere. But I can not find it on Internet even I spend a week. The paper is S. Tanno, Differential equations of order 3 on Riemannian manifolds, (technical report).
Is there anyone can download this paper? or is there any other proof and references of this equation?
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1$\begingroup$ You wrote an expression, but said it was an equation; do you mean for the expression to equal $0$? $\endgroup$– user44191Apr 22, 2021 at 4:52
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$\begingroup$ yes. I forgot it. $\endgroup$– 管山林Apr 23, 2021 at 2:24
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I am not able to locate Tanno's technical report either. However I have found a different paper by Tanno ([3], freely accessible on Project Euclid, the DOI is below) that provides additional information.
On pages 1 and 2 of [3] we learn that Obata [2] was the first to study this system of equations and to announce the theorem, but that the outline of the proof was incomplete. (And so Tanno provided a complete proof in his technical report.)
Tanno also adds that D. Ferus subsequently provided in [1] "an elegant proof" of the result (so, supposedly, a different proof). Unfortunately this is in a preprint that I was not able to locate on the web, but maybe Ferus has a subsequent publication carrying the proof.
[1] FERUS D. "A characterization of Riemannian symmetric spaces of rank one." (pre-print).
[2] OBATA M. "Riemannian manifolds admitting a solution of a certain system of differential equations," Proc. U.S.-Japan Sem. in Differential Geom., Kyoto, Japan, 1965, 101-114.
[3] Shukichi TANNO. "Some differential equations on Riemannian manifolds." Journal of the Mathematical Society of Japan, 30(3) 509-531 July, 1978. https://doi.org/10.2969/jmsj/03030509
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$\begingroup$ Thank you. My teacher said these equations of proof may be still a problem so that he ask me to read it. If we can not find the proof then we can not sure it is solved by them or not. $\endgroup$– 管山林Apr 23, 2021 at 2:27