  We develop a frame-invariant theory of material spike formation during flow
separation over a no-slip boundary in three-dimensional flows with arbitrary
time dependence. Based on the exact evolution of the largest principal
curvature on near-wall material surfaces, our theory identifies fixed and
moving separation. Our approach is effective over short time intervals and
admits an instantaneous limit. As a byproduct, we derive explicit formulas for
the evolution of the Weingarten map and the principal curvatures of any surface
advected by general three-dimensional flows. The material backbone we identify
acts first as a precursor and later as the centerpiece of Lagrangian flow
separation. We discover previously undetected spiking points and curves where
the separation backbones connect to the boundary and provide wall-based
analytical formulas for their locations. We illustrate our results on several
steady and unsteady flows.
