On C1-isometric imbeddings. II

MATHEMATICS

ON C1-ISOMETRIC IMBEDDINGS. II BY

NICOLAAS R KUIPER (Communicated by Prof. H. FREUDENTHAL at the meeting of September 24, 1955)

Dedicated to Prof. dr. W. van der Woude on the occasion of his 80th birthday

§ 11. Introduction In our first paper ([2] § 1-10) we considered the isometric C1 -imbedding of a Riemannian n-manifold xn in Euclidean N-space EN, in case xn is compact ( = every infinite set of points in xn has a point of accumulation). Now we will present some theorems concerning the case in which xn is non-compact. Our solution of this problem is obtained in two parts. First we obtain, starting from a C00-imbedding of X .. in E .. , a short (in the sense of NAsH) imbedding in EN+I (Theorem 2, § 12). Next we prove with much reference to § 1-10, that the existence of a short imbedding of xn in EN, N>n, implies the existence of a C1-isometric imbedding of xn in EN (theorem 3, § 13). Theorem 4 is the immediate consequence of theorems 2 and 3. In § 14 we apply our results and obtain a C1-isometric imbedding of the hyperbolic plane H 2 in E 3 , which is closed in E 3 1 ). The following definition is needed in the sequel: If i : X -+ i(X) is an imbedding of the manifold X in EN, and x, EX, i= 1, 2, ... , is a divergent sequence of points in X, whereas f(x1) i = 1, 2, ... is convergent in EN with limitpoint u E EN, then u is called a limit point of the imbedding. We will avoid that the image f(X) meets the limit set, as is necessary for any C1 -imbedding=C1 -homeomorphic map with non-vanishing Jacobian. § 12.

Short imbeddings Definition: The C00 -imbedding I : X -+ f(X) of a C00 -Riemannian n-manifold X .. in E .. is called shortable if the ratio ds' fds of the differential of the arclength ds' induced in f(X) and X, and the original differential ds of the arclength in X, is uniformly bounded on the space of tangent vectors of X: (12.1)

ds'fds
If a shortable imbedding exists then also a short imbedding exists, which 1) HILBERT [3] proved that a C4 -isometric imbedding of the hyperbolic plane H 2 in E 3 is impossible. BIEBERBACH [4] n = 2 and BLANUASA [5] n:;;;:. 2 obtained isometric imbeddings of H 11 in Hilbert space.

684 can be obtained as in§ 2. Hence we may restrict ourselves to the consideration of shortable imbeddings. Before we give our final result concerning short imbeddings, we present the fairly simple proof of a weaker theorem of NASH ([1] p. 395). The reader who is in a hurry may proceed with theorem 2. Weak theorem. Every C00 -Riemannian n-manifold X" has a short C00 -imbedding in Euclidean 2n+ 1-space E 2"+1, which does not meet its limit set. This theorem follows by induction from NAsH's result, that X" has a short imbedding which does not meet its limit set in a Euclidean space of some finite dimension, and the Lemma: If X"' has a shortable C00 -imbedding in EN, N>2n+ 1, which does not meet its limit set, then it has an analogous shortable imbedding in EN-1. From now on in this paragraph "Imbedding" will mean: "C00 -imbedding which does not meet its limit-set". Proof of the lemma. If I :X-+ f(X) is a shortable C00 -imbedding of xn into EN, N> 2n + 1, then the pointset T of chords and tangents of f(X) has dimension < 2n+ 1 0 to P. We next apply the analytic stereographic projection u with centre P, of SN- 1 with the exception of P, onto the hyperplane EN-1 C EN that is equatorial with respect to the sphere SN-l and the north pole P. The image unf(X) is bounded in EN-1 because c: > 0, and so is the (variable) factor that is the quotient of the arclengths induced in unf(X) and nf(X) ds' (unf(X) )jds' (nf(X))
Then the 0 00 -imbedding X

-+

unf(X) of X into EN-1 is shortable.

Theorem 2. If a C00 -Riemannian n-manifold X has a C00 -imbedding in EN which does not meet 1:ts limit set, and it has if N = 2n, then it has a short C00 -imbedding in EN+I which also does not meet its limit set. Proof: Let f : X--+ f(X) be a given C00 -imbedding of the noncompact C00 -Riemannian n-manifold X in EN, which does not meet its limit set. 2) The symbols 1p, cp, 1), -r, a used in this paragraph have no relation with the notions for which they were used in earlier paragraphR.

685

If /(X) is not bounded then we apply the analytic transformation

z"'=arctg z"'

(I2.2)

to /(X). Hence we may and will assume that /(X) is bounded in EN. \Ve also assume that f is not shortable. We want a transformation of the imbedding under which ds'Jds, defined as in the definition (I2.I), is diminished considerably in those points x EX, in which it assumes high values. To obtain this new imbedding we put EN into EN+2 as the linear space (with respect to preferred orthogonal coordinates z1 , •.. zN+2): zN+l =

Let the given C00-imbedding XEX:

I

o,

zN+2 = 1.

be given by the N

+2

real functions of

{zl(x), ... ,zN(x), 0, I}.

We replace this imbedding first by the imbedding g: {z1(x), ... ,zN(x),?J(X),

1}

in which ?](X) will be a positive C00 -function of X EX, which assumes very small values in points x EX in which max ds' Jds has high values. Next we project the image g(X) centrally from (0, 0, ... 0) onto the unit hypersphere (12.3)

N

1: (z"')2+ (zN+1_1)2+ (zN+2)2 =

1

D<-1

which has at (0, 0, ... 0) the tangenthyperplane zN+l=O! The arclength of any arc that is close to this hyperplane zN+l= 0, and in zN+2= 1, is diminished very much under this projection, which is an analytic transformation of the pointset zN+l > 0, zN+2 = I onto the part zN+2 > 0 of the hypersphere SN+l given by (12.3). The result of the projection is the C00-imbedding h which does not meet its limit set: (12.4}

{r(x) z1 (x), ... , r(x) zN(x), r(x) ?J(X}, r(x)} 21)(X)

) T ( X=

N

I+(1J(x))2+

l: (z"'(x))s

<¥-1

Now we may hope that the imbedding his shortable for a suitable choice of the function ?J(X). This choice follows now! The limit set R of f(X) in EN is not void according to the assumption following (12.2). Let e(x) be the distance from the point f(x) to R, measured in EN: (l(X)=g.l.b!IE.R [distance (/(X}, y)]. \Ve define a real function

~X(t)

of the real variable t by

~X(t) =max {max11<~>;;.tds'Jds,

1}.

The maximum between brackets is taken over the closed ( !) set of all

686

tangent unit vectors at all points x for which e(x):;;. t. cx(e(x)) is a continuous function of x E X which satisfies: cx(e(x)) :;;.max ds' fds (at x). We observe that lim 1_ 0 cx(t) = oo; the level pointsets cx(e(x)) =constant are closed and bounded subsets of EN. We chose a C00 -function C(x) such that 1 < cx(e(x)) < C(x) < cx(e(x)) + 1 C(x);;;.max ds'fds (at x).

(12.5) (12.6)

Then also the level pointsets of C(x) are closed subsets of EN and dCfds has a maximum on the set of tangent unitvectors with respect to f(X) at those points x EX for which C(x) =C. . 00 We next introduce a C -function '1p(C0 ) of one real variable such that: (12.7)

Finally we define

f dC 00

(12.8)

X (C)=

tp(C), 17 (x) =X (C(x)).

'

Then the metric (ds") 2 induced by the C00 -imbedding h : so obtained, is seen to be shortable as follows:

X~-?-

h(X[C SN+I

(ds0 ) 2 = (dTz1)2-t ,,, -t (dTzN) 2-t (dT1]) 2-t (dT) 2 = [(z1)2+ ... +
c(::alr + ... + (~Y + (~zrJ ds2 ds 2

d-r: [ 1 dzl N dzN dTJJ + 2Tds z ds+ ... +z Ts+1Jds

=0(1)·ds 2,

because (see 12.5, 6, 7 and 8): ~ is bounded on f(X);

! T(x) ~ 1J(X) =

X (C (x)) ~ (C(x))- 1 ~ 1;

I:ZI =~~~ ·:1 =I~ :1 ~ oo

l ds' ds ~ ds ~ C and: ldz"' d-r: = ds

1JC = X (C) · C ~ C ~

0 (1). dTJ- 21] [1 +172+ ds

dC

cs =

1;

1;

f. (z"')2] -2!:... .,=I (z~)2

.,= 1

ds

1

= 0(1)+0(1)·1J·C = 0(1). From the shortable imbedding h : X --?- h(X) C SN+I we obtain a shortable imbedding in EN+I as follows: Choose a point P in SN+I such that

687

the distance from P to h(X) is e > 0. Then apply the stereographic projection 2& with centre P, of SN+I minus P onto the hyperplane EN+I which is equatorial with respect to SN+I and the northpole P·2&h(X) is bounded in EN-l and so is the (variable) factor that is the quotient of the arclengths induced in 2th(X) and h(X). Then the 0""-imbedding 2th : X -+ 2th(X)

of X into EN+I is shortable. It does not meet its limit set either, because f : X -+ f(X) did not. The imbedding of open Riemannian n-manifolds Theorem 3. If an open n-manifold with 0""-Riemannian metric X" has a short 0""-imbedding in EN, N > n, which does not meet its limit set, then it has a 0 1 -isometric imbedding in EN which does not meet its limit set. (In our construction the two mentioned limit sets are identical). § 13.

Proof: In order to prove this theorem we start from a 0""-imbedding f1 of X in EN, which does not meet its limit set, and for which the original and the induced metrics ds 2 and (ds')2 satisfy (13.1)

This is possible because a short imbedding exists (see § 2). We will now cover X, which is assumed to be non-compact, by an increasing sequence of subsets XT, T=T0 , T 0 + 1, ... , and we will apply the construction given in § 1-10 to these compact subsets, such that in successive stages successive subsets XT will enter the process. This must be done carefully, however, in particular because (ds'fds) 2 is not known to be uniformly bounded away from zero on X. The choice of T 0 and XT is as follows: T 0 is an integer which obeys: T 0 > (10nP)32.

(13.2)

Let X;
X~

be the set of all x EX for which (in all directions at x) (ds')2fds2

(13.3)

~

T-'t•.

The sequence X~ has the properties (which) we mentioned for the sequence except the fact that may be open. However: XT=X; n x;, T;;.To, is an infinite sequence of increasing closed submanifolds of X, covering X, neither of them meeting the boundary of the next, for which (13.3) also holds. We call XT, also ZT,, and we call the closure in X of XT+l-XT : ZT+1· ZT+l is a closed set. In each stage or step a system of coordinate systems will be used which apart from other properties mentioned in § 1-10, will have the property that the coordinate neighborhood in X of any system meets at most two

x;

x;

688

of the sets ZT, and if so, then only two successive ones ZT and ZT+1· In the first stages nothing is changed: IT.= i 1 • In stage T, in which the C00 -imbedding IT is replaced by I T+l we first consider the neighborhoods that meet XT+l" We give a C00 -partition of unity on XT+l and we construct the change of the imbedding of XT+l as in paper I, however, omitting all changes with respect to coordinate neighborhoods that do not meet XT! This has the following consequences: In stage T nothing is altered in the imbedding of X- XT+1; the change of the imbedding of XT is as in § 5-10; and the new imbedding of X is coo again! The construction in § 5-10 cannot be followed literally, however. We have to meet the following extra requirements: In stage T we must avoid selfintersections, not only in the image of XT, but also in the image of X; Intersections with the (constant) limitset of the imbedding of X must be avoided; This last condition has to be fulfilled also in thelimitimbeddingloo· These requirements are, however, easily met (see § 8-10) and we omit the details. The construction is now completely defined. The proof of the validity of the construction is also taken from paper I, but with the following modifications. We study what happens with the image of ZT under the process, and we define with respect to ZT the numbers c1• 1, c2, 1, e1 and w 1 by equations (5.1) (5.2') and (7.11). In the sequel of this paragraph all considerations concern ZT unless stated otherwise. The following inequalities follow from the definitions and (13.1,3): cl,t

(13.4) (13.5)

~

c2, 1 ~!

p-'1,

for t < T,

and as we may assume that the change in c2, 1 is small in stage t=T-1-;;.T0 -1

(5.3 only gives an upper bound in this stage, which is exceptional as far as ZT is concerned, because only part of the "wanted" change in the metric is obtained): ( 13.6)

Substituting (13.4) in the definition of et(5.2') we get for t;;.T -1:

et =

(13.7)

Bt

3 (c:i:}-1} t-'1. < 3-T'I,-'1•.

< 3-T-·1•• ~ 3-Tr;"l.. < (3P)-l

uniformly in t;;.T-1-;;.T0 -1 (compare 13.2). This takes care of the inequality (5.4') for any point x EX, because x is contained in at least one of the sets ZT. With these modifications we follow the proof of paper I, applied to ZT• until the number w; appears, which is defined by (7.11). We substitute

689

the expression (5.2') for Ot in (7.ll) and we obtain, when applying (13.4, 5) and then (13.3). (compare the deduction of 7.14): WT-1

< 0,1,

Wp

< O,l.

In stages following fp, ZT is handled according to § 5, 6, 7 and the inequality Wt+lfwt < l is obtained as before. This takes care of the essential inequalities (7.14). For the rest the proof is as in paper I and theorem 3 is proved. As a consequence of theorems 2 and 3 we have: Theorem 4. If an open n-manifold with C00 -Riemannian metric xn has a C00 -imbedding in EN, N;;.n, which does not meet its limit set, and it has for N = 2n (WHITNEY), then it has a C 1 -isometric imbedding in EN+l which does not meet its limit set.

The imbedding of the hyperbolic plane tn euclidean three space Theorem 5. The hyperbolic space Hn (e.g. the hyperbolic plane H2) has a C 1 -isometric imbedding in euclidean n+ !-space En+l, which is closed in En+ 1 ( = limit set is void). § 14.

Proof: If da 2 is the metric on a unit n-l sphere in En with centre M, and e is the distance between M and a point with polar "coordinates" (e, a) where a is a unit vector, then the Euclidean metric in En is (13.1)

A hyperbolic n-space Hn of constant negative curvature K = - 3c 2 can be imbedded on En by introducing in En the Riemannian metric with

ds2 = ~ [de2 + cp(e) da2]

e) _-

cp (

(sinh 2ce)2 ·"""' ::>-: 2 _2_c_ (! .

Thus we have a C00-imbedding of Hn in En C En+l which is short because

(ds') 2 /ds 2 < !. Theorem 5 follows from the construction in § 13 because the limit set of the given initial imbedding is void, hence the same is true for the result of the construction: the isometric C 1 -imbedding foo•

W ageningen, Landbouwhogeschool. REFERENCES l. NASH, JoHN, 0 1 -isometric imbeddings. Annals of Math. 60, 383-396 (1954). 2. KuiPER, NICOLAAS H., On 0 1 -isometric imbeddings I, Proceedings Kon. Ned. Akad. v. Wetensch. Amsterdam ASS = Indagationes Math, 17, 545-556 (1955). 3. Hn,BERT, D., Ueber Flachen von konstanter Gausscher Kri.i.mmung. Trans. Am. Math. Soc. 2 (1901). Also: Grundlagen der Geometrie Anhang V. 4. BrEBERBACH, L., Comm. Math. Helv. 4, 248-255 (1932). 5. BLANUSA, D., Monatsch. f . .:\lath., Wien 57, 102-108 (1953).