Another fixed pergola roof option is fabric in tension stretched over the rafters. This protects from both sun and rain if you use a water resistant fabric but it comes with inherent problems that are significant enough to limit its application onto pergolas.
The roof deck should be properly flashed with a highly durable material that is compatible with the deck material where it adjoins the house. Proper drainage at the edge of the deck must be provided, usually with gutters. Special attention must be paid to railing posts and any other penetrations of the roofing surface. If possible, avoid any penetrations by attaching posts to the eaves structure at the edge of the deck.
Build the deck roof above and extending out over the house roof. The rain would flow from the deck roof onto the house roof and into the existing gutters. This idea accomplished a couple of things. The deck roof was higher, giving the deck an open airy feel, like it didn't have a roof.