Theoretical Necessity of an External Scalar Field in the Kaluza-Klein   Theory (I)
  We show that the principle of least action is generally inconsistent with the usual Kaluza-Klein program, the higher dimensional Einstein-Hilbert action being unbounded from below. This inconsistency is also present in other theories with higher dimensions like supergravity. Hence, we conclude to the necessity of an external scalar field to compensate this flaw. 
J.P. Mbelek and M. Lachieze-Rey