  In the present paper, we study the defocusing complex short pulse (CSP)
equations both geometrically and algebraically. From the geometric point of
view, we establish a link of the complex coupled dispersionless (CCD) system
with the motion of space curves in Minkowski space $\mathbf{R}^{2,1}$, then
with the defocusing CSP equation via a hodograph (reciprocal) transformation,
the Lax pair is constructed naturally for the defocusing CSP equation. We also
show that the CCD system of both the focusing and defocusing types can be
derived from the fundamental forms of surfaces such that their curve flows are
formulated. In the second part of the paper, we derive the the defocusing CSP
equation from the single-component extended KP hierarchy by the reduction
method. As a by-product, the $N$-dark soliton solution for the defocusing CSP
equation in the form of determinants for these equations is provided.
