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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2020; 57(2): 539-552

Online first article July 17, 2019      Printed March 1, 2020

https://doi.org/10.4134/JKMS.j190201

Copyright © The Korean Mathematical Society.

Gradient almost Ricci solitons with vanishing conditions on Weyl tensor and Bach tensor

Jinseok Co, Seungsu Hwang

Chung-Ang University; Chung-Ang University

Abstract

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In this paper we consider gradient almost Ricci solitons with weak conditions on Weyl and Bach tensors. We show that a gradient almost Ricci soliton has harmonic Weyl curvature if {it has fourth order divergence-free Weyl tensor, or it has divergence-free Bach tensor. Furthermore, if its Weyl tensor is radially flat, we prove such a gradient almost Ricci soliton is locally a warped product with Einstein fibers}. Finally, we prove a rigidity result on compact gradient almost Ricci solitons satisfying an integral condition.

Keywords: Gradient almost Ricci soliton, harmonic Weyl curvature, conformally flat

MSC numbers: Primary 53C25; Secondary 58E11

Supported by: The second author was supported by the National Research Foundation of Korea (NRF-2018R1D1A1B05042186).

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