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

Geometric formulation and multi-dark soliton solution to the defocusinig complex short pulse equation