Several products of distributions on manifolds

C. K. Li

The problem of defining products of distributions on manifolds, particularly un the ones of lower dimension, has been a serious challenge since Gel'fand introduced special types of generalized functions, which are needed in quantum field. In this paper, we start with Pizetti's formula and an introduction on differential forms and distributions defined on manifolds, and then apply Pizetti's formula and a recursive structure of $\triangle^j(X^l\phi(x))$ to compute the asymptotic product $X^l\delta(r-1)$. Furthermore, we study the product \[ f(P_1,\dots,P_k)\frac{ tial^{|lpha|}ẹlta(P_1,\dots,P_k)}{ tial P_1^{lpha_1}\dots tial P_k^{lpha_k}} \] on smooth manifolds of lower dimension, which extends a few results obtained earlier. Several generalized functions, such as $\delta(QP_1,\dots,QP_k)$ and $\delta(Q_1P_1,\dots,Q_kP_k)$, are derived based on the transformation of differential form $\omega$.