  We present a series of models of three-dimensional rotation-symmetric fragile
topological insulators in class AI (time-reversal symmetric and spin-orbit-free
systems), which have gapless surface states protected by time-reversal ($T$)
and $n$-fold rotation ($C_n$) symmetries ($n=2,4,6$). Our models are
generalizations of Fu's model of a spinless topological crystalline insulator,
in which orbital degrees of freedom play the role of pseudo-spins. We consider
minimal surface Hamiltonian with $C_n$ symmetry in class AI and discuss
possible symmetry-protected gapless surface states, i.e., a quadratic band
touching and multiple Dirac cones with linear dispersion. We characterize
topological structure of bulk wave functions in terms of two kinds of
topological invariants obtained from Wilson loops: $\mathbb{Z}_2$ invariants
protected by $C_n$ ($n=4,6$) and time-reversal symmetries, and
$C_2T$-symmetry-protected $\mathbb{Z}$ invariants (the Euler class) when the
number of occupied bands is two. Accordingly, our models realize two kinds of
fragile topological insulators. One is a fragile $\mathbb{Z}$ topological
insulator whose only nontrivial topological index is the Euler class that
specifies the number of surface Dirac cones. The other is a fragile
$\mathbb{Z}_2$ topological insulator having gapless surface states with either
a quadratic band touching or four (six) Dirac cones, which are protected by
time-reversal and $C_4$ ($C_6$) symmetries. Finally, we discuss the instability
of gapless surface states against the addition of $s$-orbital bands and
demonstrate that surface states are gapped out through hybridization with
surface-localized $s$-orbital bands.
