  We propose a basic theory of nonrelativistic spinful electrons on curves and
surfaces. In particular, we discuss the presence and effects of spin
connections, which describe how spinors and vectors couple to the geometry of
curves and surfaces. We derive explicit expressions of spin connections by
performing simple dimensional reduction from the three-dimensional flat space.
The spin connections act on electrons as spin-dependent magnetic fields, which
have been known as `pseudomagnetic fields' in the context of, e.g., graphenes
and Dirac/Weyl semimetals. We propose that these spin-dependent magnetic fields
are present universally on curves and surfaces, acting on electrons regardless
of the kinds of their spinorial degrees of freedom and their dispersion
relations. We discuss that the curvature effects via spin connections will
induce the spin Hall effect and induce the Dzyaloshinskii--Moriya interactions
between magnetic moments on curved surfaces, without relying on the
relativistic spin-orbit couplings. We also mention the importance of spin
connections on orbital physics of electrons on curved geometry.
