There are various metrics for researching error-correcting codes. Especially, high-density data storage system gives the existence of inconsistency for the reading and writing process. The symbol-pair metric is motivated for outputs that have overlapping pairs of symbols in a certain channel. The Rosenbloom-Tsfasman (RT) metric is introduced since there exists a problem that is related to transmission over several parallel communication channels with some channels not available for the transmission. In this paper, we determine the minimum symbol-pair weight and RT weight of repeated-root cyclic codes over $\mathfrak R=\Bbb F_{p^m}[u]/\langle u^4\rangle$ of length $n=p^k$. For the determination, we explicitly present third torsional degree for all different types of cyclic codes over $\mathfrak R$ of length $n$.