Codimension one isometric immersions between Lorentz spaces

The theorem of Hartman and Nirenberg classifies codimension one isometric immersions between Euclidean spaces as cylinders over plane curves. Corresponding results are given here for Lorentz spaces, which are Euclidean spaces with one negative-definite direction (also known as Minkowski spaces). The pivotal result involves the completeness of the relative nullity foliation of such an immersion. When this foliation carries a nondegenerate metric, results analogous to the Hartman-Nirenberg theorem obtain. Otherwise, a new description, based on particular surfaces in the three-dimensional Lorentz space, is required. The theorem of Hartman and Nirenberg [HN] says that, up to a proper motion of E"+', all isometric immersions of E" into E"+ ' have the form id X c: E"-1 xEUr'x E2 (0.1) where c: E1 ->E2 is a unit speed plane curve and the factors in the product are orthogonal. A proof of the Hartman-Nirenberg result also appears in [N,]. The major step in proving the theorem is showing that the relative nullity foliation, which is spanned by those tangent directions in which a local unit normal field is parallel, has complete leaves. These leaves yield the E"_1 factors in (0.1). The one-dimensional complement of the relative nullity foliation gives rise to the curve c. This paper studies isometric immersions of L" into Ln+1, where L" denotes the «-dimensional Lorentz (or Minkowski) space. Its chief goal is the classification, up to a proper motion of L"+1, of all such immersions. Part I introduces the terminology and states the elementary results required to achieve the desired classification. In particular, null curves-curves in V whose tangent vectors, although nonzero, have zero "length"-and appropriate frames for them are discussed, the theory of Lorentz hypersurfaces in Ln+1 is described, and examples are exhibited. Received by the editors February 22, 1977 and, in revised form, August 6, 1978. AMS {MOS) subject classifications (1970). Primary 53C50, 53C40; Secondary 83A05.