Listen to tonight's class, "Why Can't We Use Umbrellas on Shabbat?" here.
Follow along with the source sheet here.
For Further study:
Read the full text from Shu"t Hatam Sofer here.
See, as well, R. Yisshak Yosef's teshuvah regarding a person who attends synagogue on Shabbat with an umbrella, here.