In a heavy saucepan, bring the sugar, vinegar and water to a boil. Stir constantly until reduced to about 1/2 cup, about 5 minutes. Transfer to a large bowl and place in the refrigerator until cool. Add the pineapple, cucumber, carrot and red onion to the mixture. Toss well.
To serve, divide the salad greens among individual plates. Top with the pineapple mixture and sprinkle with toasted sesame seeds. Serve immediately.