  We predict a large in-plane polarization response to bending in a broad class
of trigonal two-dimensional crystals. We define and compute the relevant
flexoelectric coefficients from first principles as linear-response properties
of the undistorted layer, by using the primitive crystal cell. The ensuing
response (evaluated for SnS$_{2}$, silicene, phosphorene and RhI$_{3}$
monolayers and for a hexagonal BN bilayer) is up to one order of magnitude
larger than the out-of-plane components in the same material. We illustrate the
topological implications of our findings by calculating the polarization
textures that are associated with a variety of rippled and bent structures. We
also determine the longitudinal electric fields induced by a flexural phonon at
leading order in amplitude and momentum.
