The isometric immersion of strongly parabolic metrics in the class of strongly parabolic surfaces

Abstract

It is known that a k-strongly parabolic surface in Euclidean space (index of relative nullity is at least k) inherits a k-strongly parabolic metric (index of nullity is at least k). For the definitions of the index of relative nullity υ(x) and the index of nullity μ(x) see Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York (1969), Vol. II, pp. 347–348. The converse is not true. In particular, there exists an analytic, three-dimensional, 1-strongly parabolic metric that does not admit a local isometric immersion into Euclidean space of any dimension in the class of three-dimensional 1-strongly parabolic surfaces. The proof is carried out with the help of the method of exterior Cartan forms.