  We study geometrical responses of magnons driven by a temperature gradient in
frustrated spin systems. While Dzyaloshinskii-Moriya (DM) interactions are
usually incorporated to obtain geometrically nontrivial magnon bands, here we
investigate thermal Hall responses of magnons that do no rely on the DM
interactions. Specifically, we focus on frustrated spin systems with sublattice
degrees of freedom and show that a nonzero Berry curvature requires breaking of
an effective $PT$ symmetry. According to this symmetry consideration, we study
the $J_1$-$J_2$-$J_2^\prime$ Heisenberg models on a honeycomb lattice as a
representative example, and demonstrate that magnons in the spiral phase
support the thermal Hall effect once we introduce a magnetic field and
asymmetry between the two sublattices. We also show that driving the magnons by
a temperature gradient induces spin current generation (i.e., magnon spin
Nernst effect) in the $J_1$-$J_2$-$J_2^\prime$ Heisenberg models.
