Personally I'd limit it to bludgeoning damage. I've only ever seen one used as a weapon in Ranma 1/2 (which isn't exactly the most accurate of martial-arts shows) but the top of Ryoga's umbrella, which is the same style as the photo in the OP, is flat rather than a point.

You could probably have it function as either a weapon or a shield and use a swift action to switch between the two 'modes' but that's just off the top of my head.

What if it was a Spiked Shield that was described as having the shape you want and the blades on the edge? Then, you go with the shield fighting chain of feats. I could see a +1 enhancement to make a magic concealable version that folded outwards for use but stayed in umbrella form most of the time, or something.

If you wanted to describe it as something real world, that's another story. A spiked or bladed ring that size would be heavy (bad idea), would have blades pointed back at the wielder some of the time (bad idea), or if it was just a spikey thing, would have all sorts of stuff to get caught on or trapped (bad idea).