There are a variety of synonyms one can use to describe the phrase "rivets one's eyes". This phrase is often used to describe a person's intense focus or concentration on something. One alternative expression is "fixes one's gaze", which implies a similar level of attention and dedication. Another option might be "engages one's sight", which emphasizes the act of actively using one's vision to study or absorb something. Some other possible synonyms could include "captivates", "mesmerizes", or "transfixes", all of which suggest a deep and sustained level of interest or attention. Ultimately, the specific synonym chosen will depend on the context and tone of the writing in question.