  The discovery of correlated phases in twisted moir\'e superlattices
accelerated the search for low-dimensional materials with exotic properties. A
promising approach uses engineered substrates to strain the material. However,
designing substrates for tailored properties is hindered by the incomplete
understanding of the relationship between substrate's shapes and electronic
properties of the deposited materials. By analyzing effective models of
graphene under periodic deformations with generic crystalline profiles, we
identify strong $C_{2z}$ symmetry breaking as the critical substrate geometric
feature for emerging energy gaps and quasi-flat bands. We find continuous
strain profiles producing connected pseudo-magnetic field landscapes are
important for band topology. We show that the resultant electronic and
topological properties from a substrate can be controlled with circularly
polarized light, which also offers unique signatures for identifying the band
topology imprinted by strain. Our results can guide experiments on strain
engineering for exploring interesting transport and topological phenomena.
