The rule of thumb in the Talmud is that if two people are situated in a place where both are entitled to be at that time, and both are moving in the normal manner for that place, if one injures the other in the course of this normal movement (i.e. by bumping into him, stepping on his toe, etc.), he is liable to pay damages.
However, the Mishnah discusses a case in which two people are walking toward each other in the street, Moshe carrying a beam and Aaron carrying a jug. If Moshe’s beam collides with Aaron’s jug and breaks it, the Mishnah teaches, Moshe is exempt from paying damages. Why should this be so? Both Moshe and Aaron were using the street in the normal manner; according to the rule stated above, Moshe should be liable for damages done to Aaron!
The Talmud explains that when the jug and beam collided, the force that caused the damage was supplied by both Moshe and Aaron, who were, after all, walking toward each other. The breaking of the jug was thus not solely the doing of Moshe; it was a joint action of Moshe and Aaron.
The authorities of Jewish law have applied the same concept to that of an umbrella. Since the two of them collided with one of them sustaining damage in the process, nobody is obligated to pay. However, if the one who sustained the damage was standing still, or if the one who caused the damage was running or otherwise acting in a manner abnormal in the public place, he would be obligated to pay.
This is a generalization. In your specific case the two of you should together discuss it with a rabbi who is proficient in the laws of monetary manners.
Rabbi Yerachmiel Fried