What is the legal ruling if the Imam prostrates immediately after reciting the Fatiha and a Sura, without bowing, and the congregants bow and then follow the Imam in prostration, and then the Imam performs an additional rak'ah in place of the one in which he forgot to bow?

If the Imam is certain of his correctness, he is not obligated to heed the reminders of the congregants. If it later becomes clear to him that he missed a bowing (ruku') and it is too late to rectify it, he should perform an additional unit (rak'ah) in place of the one whose bowing was missed, and then perform the prostrations of forgetfulness (sujud al-sahw). His prayer is valid even if he omits the prostrations of forgetfulness. As for the congregants, they were correct in alerting the Imam, performing the missed bowing, and waiting for the Imam until they conclude the prayer with his salutations. Therefore, the prayer is valid for both the Imam and the congregants, and no one needs to repeat it.