The standard

isometric immersions of [M.sup.3.sub.r] ([rho]) into [E.sup.4.sub.v] ([8] p.

3 and 1), then Kowalczyk [5] and De Lira-Tojeiro-Vitorio [8] proved independently that there exists a local isometric immersion from (M, g) into [M.sup.2] (c) x [R.sup.2] (resp.

There exists a local isometric immersion of ([M.sup.2], g) into P = [M.sup.2] (c) x [R.sup.2] (resp.

Then

isometric immersion x : g [right arrow]] [R.sup.m] is of finite type if and only if the Fourier series expansion of each coordinate function of [gamma],

In general, the 'induced' connection [Nabla] in (41) does not coincide with the 'intrinsic' Webster connection of M [Mathematical Expression Omitted], nor is [Alpha](f) (the CR analogue of the second fundamental form of an isometric immersion between Riemannian manifolds) symmetric.

This follows from (41) as f is an isometric immersion. By Tanaka's theorem it remains to show that:

The mapping [beta], [beta](x,y) = ([c.sub.1](x),[c.sub.2](y)) is an isometric immersion of [R.sup.2] into [R.sup.4], where, [c.sub.1](x) (resp.

An element f of F is said to be a Riemannian product of two curves in [R.sup.4] if f is equivalent to an isometric immersion a in Example 1, or to an isometric immersion [beta] in Example 2.

Let [psi]: [M.sup.n.sub.s] [right arrow] [Q.sup.n+p.sub.t](c) be an isometric immersion of a connected indefinite Riemannian manifold into a s ace form.

[20] O.Omori, Isometric immersions of Riemannian manifolds, J.Math.Sci.Japan,19 (1967) 205-214.

Tojeiro:

Isometric immersions in codimension two of warped products into space forms, Illinois J.

Honda,

Isometric immersions with singularities between space forms of the same positive curvature, J.