Since the proposal of monopole Cooper pairing in Ref. [1], considerable research efforts have been dedicated to the study of Copper pair order parameters constrained (or obstructed) by the nontrivial normal-state band topology at Fermi surfaces. In the current work, we propose a new type of topologically obstructed Cooper pairing, which we call Euler obstructed Cooper pairing. The Euler obstructed Cooper pairing widely exists between two Fermi surfaces with nontrivial band topology characterized by nonzero Euler numbers; such Fermi surfaces can exist in the $PT$-protected spinless-Dirac/nodal-line semimetals with negligible spin-orbit coupling, where $PT$ is the space-time inversion symmetry. An Euler obstructed pairing channel must have pairing nodes on the pairing-relevant Fermi surfaces, and the total winding number of the pairing nodes is determined by the sum or difference of the Euler numbers on the Fermi surfaces. In particular, we find that when the normal state is nonmagnetic and the pairing is weak, a sufficiently-dominant Euler obstructed pairing channel with zero total momentum leads to nodal superconductivity. If the Fermi surface splitting is small, the resultant nodal superconductor hosts hinge Majorana zero modes, featuring the first class of higher-order nodal superconductivity originating from the topologically obstructed Cooper pairing. The possible dominance of the Euler obstructed pairing channel near the superconducting transition and the robustness of the hinge Majorana zero modes against disorder are explicitly demonstrated using effective or tight-binding models.
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