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Examples of isometric immersions of [R.sup.2] into [R.sup.4] with vanishing normal curvature: Dedicated to the memory of Prof. Takashi OKAYASU.

1. Introduction and result. Hartman [2] showed that, for each pair of integers (n,p) with 1 [less than or equal to] p < n, an isometric immersion f of [R.sup.n] into [R.sup.n+p] is reduced to an isometric immersion h of [R.sup.p] into [R.sup.2p], f = B o(1 x h)o A, where A is an isometry of [R.sup.n], B is an isometry of [R.sup.n+p], and 1 is the identity mapping of [R.sup.n-p]. For p = 1, every h is completely charaterized by a real-valued function of a single variable (see Dajczer et al. [1]). For p [greater than or equal to] 2, the problem of describing all h remains elusive, even for p = 2.

Few isometric immersions of [R.sup.2] into [R.sup.4] are known. In this paper, we construct a family of new isometric immersions with vanishing normal curvature by getting solutions of a system of second order partial differential equations of hyperbolic type. The definition of the normal curvature [R.sub.n] is given in [3], p. 526.

We are in the [C.sup.[omega]]-category, unless otherwise is stated.

Proposition 1. There exists a family of isometric immersions of [R.sup.2] into [R.sup.4] with vanishing normal curvature, each of which depends on four real parameters s, a, b, c and an analytic function w on [R.sup.2].

Corollary. Except for one, every immersion f in the family is not a Riemannian product of two curves in [R.sup.4] (see Remark 1 below). As [R.sup.4]-valued functions, every such f is an analytic function on [R.sup.2] everywhere.

2. Preliminaries. We recall basic results (see [3]). Let (x,y) be a standard coordinate system of [R.sup.2] and D a domain in [R.sup.4]. For real-valued functions [u.sub.1] and [u.sub.2] defined on [R.sup.2] x D, let us consider a system of total differential equations for [R.sup.4]-valued functions f, [e.sub.1], [e.sub.2], [e.sub.3] and [e.sub.4].

(1) [mathematical expression not reproducible].

The integrability condition of equations (1) is interpreted as a system of the following partial differential equations of hyperbolic type.

(2) [mathematical expression not reproducible].

Getting solutions [u.sub.1] and [u.sub.2] of (2) and applying Proposition 1.1 of [3], we shall prove our main results in the next section.

3. Proof of the result.

3.1. Solutions of partial differential equations (2). Let a, b, c, (c < b) be positive constants and s a constant. We define the real numbers [alpha], [beta], e by

(3) [mathematical expression not reproducible]

Let us define a function w(x, y) on [R.sup.2] by F(x + ey), where F(t) is a real analytic function defined on whole R with F(0) = F'(0) = F"(0) = 0.

Lemma 1. For each function w(x,y) as above, the functions

(4) [mathematical expression not reproducible]

are solutions of the equations (2).

Proof. Denote by h the partial derivative [[partial derivative].sup.2.sub.x]w(x,y). Then, we have the identities [[partial derivative].sub.x][[partial derivative].sub.y]w(x, y)= eh and [[partial derivative].sup.2.sub.y]w(x,y) = [e.sup.2]h from which the identities

[mathematical expression not reproducible]

follows, and hence [u.sub.1],[u.sub.2] are solutions of the first equation of (2).

Since the function h has no constant terms, the left-hand side of the second equation (2) is of the form.

(5) [s.sup.2] {[lambda] + [mu]h + v[h.sup.2]},

where

[lambda] = 4([[alpha].sup.2] + [[beta].sup.2]) - ([a.sup.2] + 1)[b.sup.2],

[mu] = ([a.sup.2] + 1)b([e.sup.2] - 1) + 2([alpha] + [alpha][beta])([e.sup.2] + 1),

v = ([a.sup.2] + 1)[e.sup.2].

The right-hand side of the second equation (2) is of the form.

(6) ([a.sup.2] + 1)[s.sup.2](1 + 2eh + [e.sup.2] [h.sup.2]).

From (5) and (6) together with (3) it follows that [u.sub.1], [u.sub.2] are solutions of the second equation of (2).

3.2. Equivalence relation. We recall here classical isometric immersions of [R.sup.2] into [R.sup.4], and introduce an equivalence relation.

Example 1. The mapping [alpha], [alpha](x,y) = (c(x),y), is an isometric immersion of [R.sup.2] into [R.sup.4], where c(x) is a curve in [R.sup.3] [congruent to] [R.sup.3] x{0}, x being the arc length parameter.

Example 2. 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. [c.sub.2](y)) is a curve in [R.sup.2] [congruent to] [R.sup.2] x{(0, 0)} (resp. [R.sup.2] [congruent] {(0, 0)} x [R.sup.2]), x and y being the arc length parameters. In Examples 1 and 2, we mean [congruent] by a congruent under the action of O(4) on [R.sup.4].

An isometry [??] of [R.sup.n] onto itself is given by [??]([x.sup.1], ..., [x.sup.n]) = ([x.sup.1], ..., [x.sup.n])[tau] +([b.sup.1], ..., [b.sup.n]),

where [b.sup.i] are constants, [tau] = ([a.sup.i.sub.j]) is in O(n), the orthogonal group.

Denote by F the space of all isometric immersions of [R.sup.2] into [R.sup.4], and introduce a relation ~ in F. Two elements f and h of F are said to be equivalent if and only if h o [??] = [??] o f with an isometry [??] of [R.sup.2] and an isometry [??] of [R.sup.4]. The relation ~ is an equivalence relation.

Definition. 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.

Lemma 2. Let f (a,b, c, s,w(x,y)) (denote by f*) be an element of F, which is constructed by functions u1 and u2 as in Lemma 1. If s [not equal to] 0, then the immersion f* is not a Riemannian product of two curves in [R.sup.4].

Proof. We prove that f* is not related to one in Example 1. Similarly, f* will not be related to an isometric immersion in Example 2.

By using (1), it can be easily shown that an isometric immersion f given in Proposition 2 is related to one in Example 1 if and only if for a constant [theta], (--[pi] < 2[theta] [less than or equal to] [pi]), the following equations hold identically.

(7) [mathematical expression not reproducible].

We now prove Lemma 2 by reduction to absurdity. Suppose that for a constant [theta], all the equations (7) (with [u.sub.j] in (3)) hold identically. The constant terms in (7) are of the form.

(8) [mathematical expression not reproducible].

Multiplying a on the first equation of (8) and subtracting the second one of (8), we have an equality s(a[alpha] - [beta]) = 0. The equality is inconsistent with the condition s(b - c) [not equal to] 0 by virtue of (3).

Lemmas 1 and 2 imply that Proposition 1 is valid.

Remark 1. If s = 0 in Lemma 1, the functions [u.sub.1] and [u.sub.2] are identically zero, and hence f is a standard isometric imbedding of [R.sup.2] into [R.sup.4] with a standard basis {[e.sub.1],[e.sub.2],[e.sub.3],[e.sub.4]} along the isometric immersion f.

By using Lemma 3 below, f(a,b, c, s,w(x,y)) depends only on parameters a, b, c and s, and an analytic function w(x,y).

3.3. Solutions of partial differential equations (1). Next lemma is given by Prof. N. Shimakura.

Lemma 3. Let {[e.sub.1];[e.sub.2],[e.sub.3],[e.sub.4]} be functions of class [C.sup.2] of (x, y) defined in an open subset [OMEGA] in [R.sup.2] with values in [R.sup.4] which satisfy the equations

(9) [mathematical expression not reproducible].

If [s.sub.jk] and [t.sub.jk] are real-analytic functions of (x,y) in [OMEGA], then [e.sub.1], [e.sub.2], [e.sub.3], [e.sub.4] are real-analytic functions of (x, y) in [OMEGA].

Proof. If [omega] is an open subset of [OMEGA] whose closure is a compact subset of [OMEGA], there exist positive number [[mu].sub.1] and [[rho].sub.1] independent of x, y such that

(10) [mathematical expression not reproducible]

for all integers p [greater than or equal to] 0 and q [greater than or equal to] 0 if (x, y) [member of] [omega]. Let us show that there exist positive numbers [[mu].sub.2] and [[rho].sub.2] independent of x, y such that

(11) [mathematical expression not reproducible]

for all integers p [greater than or equal to] 0 and q [greater than or equal to] 0 if (x, y) [member of] [omega].

(11) is true for p = q = 0 with [[rho].sub.2] = 1 and a [[mu].sub.2] > 0. Given n, assume (11) for all p, q satisfying p + q [less than or equal to] n with a [[mu].sub.2] > 0 and a [[rho].sub.2]. The Leibniz formula

[mathematical expression not reproducible]

and (10), (11) with p + q [less than or equal to] n yield

[mathematical expression not reproducible].

If we choose a [[rho].sub.2] greater than [[rho].sub.1], the right-hand side is smaller than

[mathematical expression not reproducible].

If we choose again a [[rho].sub.2] greater than [[rho].sub.1] + 4[[mu].sub.1], we have

[mathematical expression not reproducible].

We can prove for this choice of [[mu].sub.2] and [[rho].sub.2] also that

[mathematical expression not reproducible]

starting from [[partial derivative].sub.y][e.sub.j] = [[SIGMA].sub.k][t.sub.jk][e.sub.k]. So, (11) is true for all p and q satisfying p + q [less than or equal to] n + 1, and hence for all p [greater than or equal to] 0 and q [greater than or equal to] 0 if (x, y) [member of] [omega].

Proof of the corollary. Let [u.sub.j], (j = 1,2) be functions given in (3). By applying Lemma 3 in case where

[mathematical expression not reproducible],

the solutions [e.sub.1], [e.sub.2], [e.sub.3], [e.sub.4] of (1) are real-analytic functions on [R.sup.2], so is the solution f of (1).

Acknowledgements. The present author would like to thank to Prof. Norio Shimakura for his constant encouragement and for his useful comments. He would like to thank to the referee for his useful comments.

References

[1] M. Dajczer, M. Antonucci, G. Oliveira, P. Lima-Filho, et al., Submanifolds and isometric immersions, Mathematics Lecture Series, 13, Publish or Perish, Inc., Houston, TX, 1990.

[2] P. Hartman, On the isometric immersions in Euclidean space of manifolds with nonnegative sectional curvatures. II, Trans. Amer. Math. Soc. 147 (1970), 529-540.

[3] H. Mori and N. Shimakura, Isometric immersions of Euclidean plane into Euclidean 4-space with vanishing normal curvature, Tohoku Math. J. (2) 61 (2009), no. 4, 523-550.

By Hiroshi MORI *

Professor Emeritus, Joetsu University of Education

(Communicated by Kenji FUKAYA, M.J.A., Feb. 12, 2019)

2010 Mathematics Subject Classification. Primary 53C42.

* Present address: 2-5-45 Kitashiro-cho, Joetsu, Niigata 943-0824, Japan.

doi: 10.3792/pjaa.95.21
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Author:Mori, Hiroshi
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Date:Mar 1, 2019
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