  The fluctuation-induced, Casimir-like interaction between two parallel rods
of length L adsorbed on a fluid membrane is calculated analytically at short
separations d<<L. The rods are modeled as constraints imposed on the membrane
curvature along a straight line. This allows to define four types of rods,
according to whether the membrane can twist along the rod and/or curve across
it. For stiff constraints, all the interaction potentials between the different
types of rods are attractive and proportional to L/d. Two of the four types of
rods are then equivalent, which yields six universal Casimir amplitudes.
Repulsion can occur between different rods for soft constraints. Numerical
results obtained for all ranges of d/L show that the attraction potential
reaches kT for d/L\simeq0.2. At separations smaller than d_c \approx
L(L/l_p)^(1/3), where l_p is the rod persistence length, two rods with fixed
ends will bend toward each other and finally come into contact because of the
Casimir interaction.
