  The triaxiality in nuclear low-lying states has attracted great interests for
many years. Recently, the reduced transition probabilities for levels near the
ground state in $^{110}$Ru have been measured and provided strong evidences for
a triaxial shape of this nucleus. The aim of this work is to provide a
microscopic study of low-lying states for the Ru isotopes with $A\sim100$ and
to examine in detail the role of triaxiality, and the evolution of quadrupole
shapes with the isospin and spin degrees of freedom. The low-lying excitation
spectra and transition probabilities of even-even Ru isotopes are described at
the beyond mean-field level by solving a five-dimensional collective
Hamiltonian with parameters determined by constrained self-consistent
mean-field calculations based on the relativistic energy density functional
PC-PK1. The calculated energy surfaces, low-energy spectra, intraband and
interband transition rates, as well as some characteristic collective
observables, such as $E(4_{\rm g.s.}^+)/E(2^+_{\rm g.s.})$,
$E(2^+_\gamma)/E(4^+_{\rm g.s.})$, $B(E2; 2^+_{\rm g.s.}\to 0^+_{\rm g.s.})$,
and $\gamma$ band staggerings are in a good agreement with the available
experimental data. The main features of the experimental low-lying excitation
spectra and electric transition rates are well reproduced, and thus strongly
support the onset of triaxiality in the low-lying excited states of the Ru
isotopes around $^{110}$Ru.
