I’m studying Kleinert theory and Delta functions of surfaces and curves, defined as

$\boldsymbol{\delta}_S(x)=\int_S \delta^{(3)}(x-y) dy$

Do you know some references about the extension of the Dirac Delta for generic submanifold? Specifically, I would need references related to the delta product.

$\int_V\boldsymbol{\delta}_S(x)\cdot \boldsymbol{\delta}_M(x) =\int_M \ \boldsymbol{\delta}_S(x) dy$

I found this work where there is a brief introduction

https://core.ac.uk/download/pdf/52922262.pdf

but it is not clear to me how they get to the result about the crossing and whether it is extendable to submanifolds that have not only a single common point