  The geometry of electronic bands in a solid can drastically alter
single-particle charge and spin transport. We show here that collective optical
excitations arising from Coulomb interactions also exhibit unique signatures of
Berry curvature and quantum geometric tensor. A non-zero Berry curvature mixes
and lifts the degeneracy of $l \neq 0$ states, leading to a
time-reversal-symmetric analog of the orbital Zeeman effect. The quantum
geometric tensor, on the other hand, leads to $l$-dependent shifts of exciton
states that is analogous to the Lamb shift. Our results provide an explanation
of the non-hydrogenic exciton spectrum recently calculated for transition metal
dichalcogenides. Numerically, we find a Berry curvature induced splitting of
$\sim 10$ meV between the $2p_x \pm i2p_y$ states of WSe$_2$.
