Electronic band structures underlie the physical properties of crystalline
materials, their geometrical exploration renovates the conventional cognition
and brings about novel applications. Inspired by geometry phases, we introduce
a geometric amplitude named as the geometric density of states (GDOS) dictated
by the differential curvature of the constant-energy contour. The GDOS
determines the amplitude of the real-space Green's function making it attain
the ultimate expression with transparent physics. The local responses of
crystalline materials are usually formulated by the real-space Green's
function, so the relevant physics should be refreshed by GDOS. As an example of
local responses, we suggest using scanning tunneling microscopy (STM) to
characterize the surface states of three-dimensional topological insulator
under an in-plane magnetic field. The GDOS favors the straightforward
simulation of STM measurement without resorting to Fourier transform of the
real-space measurement, and also excavates the unexplored potential of STM
measurement to extract the phase information of wavefunction through its
amplitude, i.e., the spin and curvature textures. Therefore, the proposed GDOS
deepens the understanding of electronic band structures and is indispensable in
local responses, and it should be universal for any periodic systems.Comment: 6 pages, 2 figure