A few clarifications on Utility equations and indifference curves?
From a mathematical point of view, the indifference curve is an equation

$$I_{\bar u}:a^{2}b^{3} = \bar u$$ 

for every fixed $\bar u$. Put $b$ on the vertical axis and $a$ on the horizontal axis and write this as a function of $a$:

$$b^3 = \bar ua^{-2} \implies b = (\bar u)^{1/3} a^{-2/3}$$

This fully characterizes the indifference curve for $\bar u$. Now you have a one-dimensional function, and for it to be convex, its second derivative with respect to $a$ must be non-negative. In fact it is strictly positive (we assume goods are measured in positive quantities, and we ignore the uninteresting corner solution for $\bar u=0$ which is a single point).