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$D$-Dimensional Gravity from $(D+1)$ Dimensions

We generalise Wesson's procedure, whereby vacuum $(4+1)-$dimensional field equations give rise to $(3+1)-$dimensional equations with sources, to arbitrary dimensions. We then employ this generalisation to relate the usual $(3+1)-$dimensional vacuum field equations to $(2+1)-$dimensional field equations with sources and derive the analogues of the classes of solutions obtained by Ponce de Leon. This way of viewing lower dimensional gravity theories can be of importance in establishing a relationship between such theories and the usual 4-dimensional general relativity, as well as giving a way of producing exact solutions in $(2+1)$ dimensions that are naturally related to the vacuum $(3+1)-$dimensional solutions. An outcome of this correspondence, regarding the nature of lower dimensional gravity, is that the intuitions obtained in $(3+1)$ dimensions may not be automatically transportable to lower dimensions. We also extend a number of physically motivated solutions studied by Wesson and Ponce de Leon to $(D+1)$ dimensions and employ the equivalence between the $(D+1)$ Kaluza-Klein theories with empty $D-$dimensional Brans-Dicke theories (with $ω=0$) to throw some light on the solutions derived by these authors.