Equality in the Bishop Gromov theorem

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How to work out the equality condition in the Bishop-Gromov theorem? i.e. when does the ratio of volumes not strictly decrease?

Reference :

http://staff.ustc.edu.cn/~wangzuoq/Courses/16S-RiemGeom/Notes/Lec24.pdf

Look at the remark at the end of the proof.