  Odd elasticity encompasses active elastic systems whose stress-strain
relationship is not compatible with a potential energy. As the requirement of
energy conservation is lifted from linear elasticity, new anti-symmetric (odd)
components appear in the elastic tensor. In this work, we study the odd
elasticity and non-Hermitian wave dynamics of active surfaces, specifically
plates of moderate thickness. We find that a free-standing moderately thick,
isotropic plate can exhibit two odd-elastic moduli, both of which are related
to shear deformations of the plate. These odd moduli can endow the vibrational
modes of the plate with a nonzero topological invariant known as the first
Chern number. Within continuum elastic theory, we show that the Chern number is
related to the presence of unidirectional shearing waves that are hosted at the
plate's boundary. We show that the existence of these chiral edge waves hinges
on a distinctive two-step mechanism: the finite thickness of the sample gaps
the shear modes and the odd elasticity endows them with chirality.
