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Correction to: Annals of Global Analysis and Geometry (2022) 62:507–532 https://doi.org/10.1007/s10455-022-09863-z
It has been observed that the proof of Lemma 5.5 in the paper [3] is not complete. It was pointed out in [1] by Prof. S. Borghini and L. Mazzieri, and in [2] by Prof. G. Catino, D. Dameno and P. Mastrolia. In the proof, a particular frame of vector fields is constructed in order to establish formulas for \(|\nabla \omega |^2\) and \(|\delta \omega |^2\). The derived formula for \(\delta \omega \) might not be correct as they pointed out because the Lie bracket of these frames does not necessarily vanish.
Consequently, Theorem 1.1 in [3] is still valid. But it remains unclear whether or not Theorem 1.2 is true.
References
Borghini, S., Mazzieri, L.: Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric \(2\)-tensor fields, Preprint
Catino, G., Dameno, D., Mastrolia, P.: Private Communication
Yun, G., Hwang, S.: Besse conjecture with positive isotropic curvature. Ann. Global Anal. Geom. 62, 507–532 (2022)
Acknowledgements
The authors wish to thank Professors S. Borghini, G. Catino and L. Mazzieri for pointing out comments mentioned above.
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Hwang, S., Yun, G. Correction to: Besse conjecture with positive isotropic curvature. Ann Glob Anal Geom 63, 15 (2023). https://doi.org/10.1007/s10455-023-09892-2
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DOI: https://doi.org/10.1007/s10455-023-09892-2