On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains

Consider a Gaussian random field with a finite Karhunen--Loève expansion of the form Z(u)=.ni=1uizi, where zi, i=1,.,n, are independent standard normal variables and u=(u1,.,un). ranges over an index set M, which is a subset of the unit sphere Sn.1 in Rn. Under a very general assumption that M is a manifold with a piecewise smooth boundary, we prove the validity and the equivalence of two currently available methods for obtaining the asymptotic expansion of the tail probability of the maximum of Z(u). One is the tube method, where the volume of the tube around the index set M is evaluated. The other is the Euler characteristic method, where the expectation for the Euler characteristic of the excursion set is evaluated. General discussion on this equivalence was given in a recent paper by R. J. Adler. In order to show the equivalence we prove a version of the Morse theorem for a manifold with a piecewise smooth boundary.