Allen Shields*

Differentiable manifolds: Weyl and Whitney

H e r m a n n Weyl ' s book (Weyl [1913]) on R iemann sur- faces a p p e a r e d 75 years ago. And just over 50 years ago Hass l e r W h i t n e y [1936] p r o v e d his f a m o u s em- b e d d i n g theorem. We quote f rom the in t roduct ion to his p a p e r (p. 645).

A differentiable manifold is generally defined in one of two ways; as a point set with neighborhoods homeomor- phic with Euclidean space En, coordinates in overlapping neighborhoods being related by a differentiable transfor- mation, or as a subset of E n, defined near each point by expressing some of the coordinates in terms of the others by differentiable functions.

The first fundamental theorem is that the first definition is no more general than the second; any differentiable manifold may be imbedded in Euclidean space. In fact, it may be made into an analytic manifold in some En. As a corollary, it may be given an analytic Riemannian metric. The second fundamental theorem (when combined with the first) deals with the smoothing out of a manifold. Let f be a map of any character (continuous or differentiable, without an inverse) of a differentiable manifold M of di- mension m into another, N, of dimension n . . . . Then if n

2m, we may alter f as little as we please, forming a regular map F. ( A map is regular if, near each point, it is differentiable and has a differentiable inverse.) Moreover, if n I> 2m+1 , F may be made (1 - 1). We show in Theorem 6 that if n t> 2m + 2, then any two regular maps f0, fl of M into E, are equivalent, in the following sense. fo(M) may be deformed into h(M) by maps ft (0 ~< t ~ 1) so that the path crossed by the manifold is the regular map of an (m + 1)-dimensional manifold. Moreover, if n / 2m + 3, and fo(M) and f1(M) are non-singular, so is the (m + 1)- manifold.

A fundamental unsolved problem is the following: Can any analytic manifold be mapped in an analytic manner into Eu- clidean space?

Theorem I shows only that there is a differentiable map (with all derivatives), such that the resulting point set forms an analytic manifold.

Many portions of the proofs are based on the Weier- strass approximation theorem, if the manifolds are closed; if they are open, this theorem must be replaced by a cor- responding theorem on functions defined on open sets. This and other theorems which will be useful may be found in a previous paper.

Two c o m m e n t s come to mind. First, Whi tney ' s re- sult was p robab ly mot iva ted by earlier results in gen- eral topology. G. N6bel ing [1930] first p roved the re- sult, s ta ted wi thou t p roof in K. Menger [1926], that an n-d imens iona l separable metric space can be homeo- morphica l ly e m b e d d e d in R2n+l. W. Hurewicz [1933] gave a proof based on the Baire ca tegory theorem. He s h o w e d that in the space of all con t inuous maps f rom a compac t n-d imens iona l metric space into ~2n+1, the h o m e o m o r p h i s m s form a dense G s set. Thus, an arbi- t rary con t inuous m a p can be changed into a homeo- m o r p h i s m b y an arbi t rar i ly small change . K. Kura- towski [1937] modif ied Hurewicz ' s m e t h o d to obtain an ana logous result for separable metr ic spaces.

O u r second c o m m e n t is that in a footnote on the first page, Whi tney [1934] p roves a quant i ta t ive ver- sion of the Tietze extension theorem. Let E be a subset of a metric space M, let f be a real -valued funct ion on E, and a s s u m e that ~(x)-f(y) I ~ oo(d(x,y)) for all x, y e E. Then f can be ex tended to all of M, preserv ing this

* C o l u m n ed i tor ' s address : Depar tment of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003 USA

THE MATHEMATICAL INTELL|GENCER VOL. I0, NO. 2 �9 1988 Springer-Verlag New York 5

inequality. Here 00 is a cont inuous, strictly increasing func t ion def ined on the nonnega t ive real numbers , wi th 00(0) = 0, and 00(x + y) ~ co(x) + ~0(y).

We quote now from the Introduction to Weyl [1913].

The present work presents the contents of a course I gave at the University of G6ttingen in the winter semester 1911/12, whose purpose was to develop the fundamental ideas of Riemann's function theory in a form that would fully satisfy all modern requirements of rigor. Such a rig- orous presentation, including set-theoretically precise proofs of all the fundamental concepts and theorems of Analysis situs that enter into function theory, rather than merely invoking intuitive plausibility, has not yet been given . . . . I have benefitted in a much higher measure than the citations would indicate from the fundamental topological investigations of Brouwer that have recently appeared . . . and it is my secret hope that some of the spirit that animates his works remains alive in the present book.

�9 . . it is usual in all presentations of the theory of Rie- mann surfaces to take the concept of curve as given by our intuition, without a precise definition, and to make naive use of those properties that seem intuitively evident (for example, that a curve has two sides). But today there can no longer be any doubt that precisely these "intuitively evident" truths require proofs which in the final analysis must rest on the Axioms of Arithmetic . . . . It is all the more necessary to give a strict set-theoretic foundation for the topological concepts and theorems used in Rieman- nian function theory, since the "points" of which the basic objects (curves and surfaces) consist, are not spatial points in the ordinary sense, but can be mathematical objects of quite another sort (for example, function elements) . . . .

H e r m a n n W e y l .

It cannot be denied that the discovery of the vast gener- ality of concepts such as "function" and "curve", going far beyond what one had imagined, along with the need for logical precision, although necessary and fruitful for our science, have called forth unhealthy symptoms in the development of present day mathematics. Part of the mathematical activity that probes these concepts into their finest details and pa thologies . . , has vaporized to noth- ingness or oozed away in side channels, and lost the con- nection to the living stream of science. The concept of Rie- mann Surface requires for its presentation, if we are to meet the rigorous modern requirements for precision, a profusion of abstract and subtle concepts and consider- ations. But one has only to look a bit more sharply to re- alize that this elaborate logical structure (in which the be- ginner may perhaps become confused) is not the essential thing: it is only the net with which we haul up the essential concept, which is simple and great and godly, from the zo~o~ cLzo~o~, as Plato would say, like a pearl from the sea, to the surface of our understanding. But to grasp the kernel that is enclosed by this fine mesh of exact concept, this kernel that gives life, true content and value to the theory, each person must struggle anew for themselves to obtain understanding; a book (or even a teacher) can at best give only some sketchy hints and indications.

One still encounters here and there the conception that Riemann surfaces are merely "pictures" (admittedly, very valuable, very suggestive), a sort of means to visualize the multiple-valuedness of functions. This conception is fun- damentally wrong. Riemann surfaces . . . are not some- thing more or less artificially distilled a posteriori from an- alytic functions, but must be regarded as the prius, as the mother soil on which the functions first grow and flourish. It is true that Riemann himself partially con- cealed the true relation of functions to Riemann surfaces by the form of his presentation--perhaps because he didn't want to overwhelm his contemporaries with too unfamiliar a presentation; this relation was also hidden in that he considered only those multi-sheeted surfaces with individual winding points that spread out over the plane as covering surfaces . . . . instead of taking the more gen- eral viewpoint (developed later by Klein with penetrating clarity) . . . in which the relation to the complex plane as well as to three-dimensional space is completely dis- solved. And yet there can be no doubt that only with Klein's interpretation do Riemann's ideas come to full re- alization in their natural simplicity and penetrating power. The present book is based on this conviction.

The book is dedicated to Felix Klein, and Weyl states that he and Klein wen t over all the material thor- oughly dur ing m a n y conversations. In w 4 he gives the m o d e m definition of an abstract (Hausdorff) two-di- mensional manifold, perhaps the first place where this was wri t ten down. I have looked th rough Klein [1892] and I did not find a real definition of a manifold. I quote f rom p. 19: " . . . a Riemannian manifold, that is, a two-dimensional closed manifold on which there is given a differential expression ds 2 = @dp 2 + 2~dpdq + (~dq 2. Whether this manifold lies in a space of 3 or more di- mensions , or is thought of as being independent of any external space, is quite immaterial ."

The fo l lowing mater ia l (on Klein and Friedrich Prym) comes f rom L. Sario's fascinat ing review of

6 THE MATHEMATICAL INTELLIGENCER VOL. 10, NO. 2, 1988

A page from Felix Klein's lecture notes on Riemann surfaces. Can anyone recognize the hand- writing?

W e y l [1955], w h i c h w e s t r o n g l y r e c o m m e n d . Klein [1882] wri tes in the s e c o n d p a r a g r a p h of the in t roduc- t ion:

A presentation of the kind attempted is necessarily very subjective and the more so in the case of Riemann ' s theory, since but scanty material for the purpose is to be found explicitly given in Riemann's papers. I am not sure

that I should ever have reached a well-defined conception of the whole subject, had not Herr Prym, many years ago (1874), in the course of an opportune conversation, made to me a communicat ion which has increased in impor- tance to me the longer I have thought over the matter. He told me that Riemann's surfaces originally are not necessarily many-sheeted surfaces over the plane, but that, on the contrary, complex functions of position can be studied on arbitrarily given

THE MATHEMATICAL INTELLIGENCER VOL. 10, NO. 2, 1988 7

curved surfaces in exactly the same way as on the surfaces over the plane. The following presentation will sufficiently show how valuable this remark has been to me.

O n the o ther hand Klein's collected works [1923] contain numerous historical notes and commentar ies w r i t t e n by Kle in h i m s e l f , i n c l u d i n g a f ive p ag e (477-481) article entitled: Vorbemerkungen zu den Ar- beiten iiber Riemannsche Funktionentheorie. On p. 479 he wri tes as follows.

There is the historical question, whether Riemann's ideas really developed as I have presented them. One will prob- ably not be able to clarify this question by going back to Riemann's original manuscripts . . . . . I once questioned students of Riemann about this, without obtaining a de- finitive answer. Even the reference to a conversation with

The more general idea of Riemann surface was developed in part because of a comment by Prym that was misunderstood by Klein.

handwritten notes. I do not know if the handwriting is Klein's.]

- - [ 1 9 2 3 ] , Gesammelte Mathematische Abhandlungen, III, Julius Springer-Verlag, Berlin. JFM 49, 11-12.

K. Kuratowski [1937], Sur les thdor6mes de "plongement" dans la thdorie de la dimension, Fundam. Math. 28, 336-342. JFM 63, 1168.

K. Menger [19.26], Allgemeine R/iume und Cartesische R/iume. II: Uber umfassendste n-dimensionale Mengen, Proc. Akad. Wetensch. Amsterdam 29, 1125-1128. JFM 52, 595.

G. NObeling [1930], Ober eine n-dimensionale Universal- menge im R2n+l, Math. Ann. 104, 71-80. JFM 56, 506.

H. Weyl [1913], Die ldee der Riemannschen FMche, B. G. Teubner, Leipzig; [1923] 2d Ed.; [1955] 3d Ed.(Stuttgart). MR 16, 1097. English translation: [1964] The concept of a Riemann surface, Addison Wesley, Reading, Mass. MR 29 #3628.

H. Whitney [1934], Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36, 63-89. JFM 60, 217.

- - [ 1 9 3 6 ] , Differentiable manifolds, Ann. of Math. 37, 645-680. JFM 62, 1454.

Prym [see above] . . . must be modified. When I sent Prym the book (Klein [1882]) he wrote back on 8 April 1882. He thoroughly approved of my point of view in the book, but he also wrote ' . . . when I made my remark to you I was probably thinking of the mapping problem; un- fortunately I no longer remember the connection in which I made it. The only thing I am sure of is that I never har- bored the thought that with Riemann the investigation of functions on arbitrary surfaces preceded the investigation of functions on multi-sheeted surfaces covering the plane; as to my use of the word "originally" in my conversation with you, in the context it must have had a different meaning from the one you have attributed to it . . . .

Thus we have the curious situation that the more general idea of Riemann surface was deve loped in part because of a commen t by Prym that was misunder- s tood by Klein. In any case Klein deserves m u c h credit for emphas i z ing and ex tend ing the concep t of Rie- m a n n surface, and for popularizing Riemann 's geo- metric v iewpoint in general .

References

Abbreviations: MR = Mathematical Reviews, JFM = Jahrbuch iiber die Fortschritte der Mathematik, ZBL = Zentralblatt far Mathematik.

W. Hurewicz [1933], Uber Abbildungen yon endlichdimen- sionalen R~umen auf Teilmengen Cartesischer R~iume, Sitzb. Preuss. Akad. Wiss., phys. math. Klasse, 754-768. JFM 59, 1267.

F. Klein [1882], Ueber Riemann' s Theorie der algebraischen Func- tionen und ihrer Integrale, B. G. Teubner Verlag, Leipzig, JFM 14, 358. English translation: [1893], Macmillan and Bowes, Cambridge. Reprinted [1963] Dover Publica- tions, New York. MR 28 #1295.

- - [ 1 8 9 2 ] , Riemannsche F1/ichen I., Lecture notes for the winter semester 1891-92, G6ttingen. [These notes are written in the first person and are a reproduction of

8 THE MATHEMATICAL INTELLIGENCER VOL. 10, NO. 2, 1988

Remarks

Vol. 9, no. 2. We ment ioned that P. S. Alexandrov, in his in t roduct ion to the Russian edition of Kelley's book General Topology stated that "convergence along a d i r ec t ed s e t " was d i s co v e red t wo decades before M o o re a n d Smi th by S. O. ~ a tu n o v sk i of Odessa . However , he did not give any reference and I was un- able to find one in the reviewing journal of the time (Jahrbuch). Since then I have hea rd from a former stu- den t of V. I. Arno l 'd that the material is discussed in a book by Arno l ' d ' s father, I. V. Arnol 'd , who was also a mathemat ic ian. I will report on this w h e n I have fur- ther information.

Vol. 9, no. 3. T . J . Rivlin of the Watson Research Center, IBM, informs me that the reference for Henri Lebesgue 's proof of the Weierstrass theorem on ap- proximating cont inuous functions by polynomials is Lebesgue [1898]. Rivlin [1987] is a pleasant survey of the subject, wi th reference to cur ren t work.

Ben Fitzpatrick, Jr., of Auburn University, has called our a t tent ion to the amusing little book by Philip J. Davis [1983], in which the au thor recounts his search for the origin of Ceby~ev's given name, Pafnuti.

Finally, we ment ioned in the article that a modern reader is surpr ised that Ceby~ev did not discover the Weierstrass polynomial approximat ion theorem, since he cons idered more detailed quest ions. But that is a mo d e rn view. Probably the principal reason that Ce- by~ev did not discover the theorem is that at that t ime (1853, 1859) mathematicians were not accustomed to thinking of general cont inuous functions. Indeed, Ce- by~ev always assumes the funct ions are differentiable, and he uses this in his proofs.

Vol. 9, no 4. Correction: D. E. Egorov was born in 1869. This is given correctly under the photograph on p. 26, but due to a misprint, 1859 was written at the beginning of the second paragraph of the article.

As we stated in the article, Egorov proved the theorem that bears his name after reading Hermann Weyl [1909]. This paper of Weyl was influential in an- other way, as well. Weyl's principal result states that if {~n} is a normalized system of orthogonal functions on the interval [0,1], and if "d, c2n ~a < % then ~,c,,~,(x) con- verges almost everywhere. In particular, if {a,}, {b,} are the Fourier cosine and sine coefficients of a periodic function, and if "d,(a2+b2)n v2 < % then the Fourier series converges to the function almost everywhere. He must have been one of the first to emphasize al- most everywhere convergence of orthogonal series. Later authors took this up and replaced the sequence {n 1"~} by other, smaller, "Weyl multipliers," which would give the same conclusion. For general orthog- onal series this development culminated in H. Rade- macher [1922] and D. E. Men~ov [1923] who showed that {n w} could be replaced by {(log y/)2}, and that this result could not be improved.

For trigonometric series N. N. Lusin [1915] conjec- tured in his dissertation that no multipliers were needed, that is, he conjectured that for functions in L 2 the Fourier series converges to the function almost everywhere. However, this remained unproved, and the culmination of this early period was the famous result of A. N. Kolmogorov and G. A. Seliverstov [1926], who showed that {n v2} could be replaced by {log n}. There the matter stood for just over 40 years, until L. Carleson [1966] proved Lusin's conjecture.

T. W. K6rner of Cambridge has supplied additional references for the problem of the convergence set on the boundary for a power series with convergence radius one. We had ment ioned the result of Fritz Herzog and George Piranian [1949], [1953] that each Fr subset of the boundary is a convergence set for power series (that is, there is a power series converging at each point of the set and diverging at all other boundary points). On the other hand, one shows that the convergence set must be of type FCs, and we stated that it was an open question whether every such set really was the convergence set for some power series. Kbrner pointed out to us that S. Yu. Luka~enko [1978] constructs a G8 set (of positive measure) such that no trigonometric series (in particular, no power series) can converge on the set and diverge off it. Earlier KOrner [1971] had done this for L 1 Fourier series. KOrner [1983] gives an excellent presentation of all these results, with full details of the proofs. Unfortu- nately, he gives no references to the literature, merely stating in each case the name of the person to whom the result is due. At least one question is still open: is every FCa set of measure zero the convergence set of some power series?

Bibliography

L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 33 #7774.

P. J. Davis, The thread, a mathematical yarn. Birkh/iuser: Boston (1983).

F. Herzog and G. Piranian [1949], [1953], Sets of conver- gence of Taylor series I., II., Duke Math. J. 16, 529-534; ibid 20, 41-54. MR 11, 91; MR 14, 738.

A. N. Kolmogorov and G. A. Seliverstov, Sur la convergence des s6ries de Fourier, Rendiconti Accad. Lincei, Roma, 3, 307-310. JFM 52, 269-270.

T. W. Kbrner, Sets of divergence for Fourier series, Bull. Lond. Math. Soc. 3 (1971), 152-154. MR 44, 7207.

- - T h e behavior of power series on their circle of conver- gence. Lecture Notes Math. 995 (1983): Banach spaces, Harmonic analysis, and Probability theory. Proceedings, Univ. of Conn. 1980-81. Berlin, Heidelberg, New York: Springer Verlag (1985). MR 84j:30005.

H. Lebesgue, Sur l'approximation des fonctions, Bull. Soc. Math. 22 (1898), 278-287

S. Yu. Luka~enko, Sets of divergence and nonsummability for trigonometric series, Vestnik Moskov. Univ. Ser. I Mat. Mekh . (1978), no. 2 65-70. MR 84d42006; ZBL 386-42002. English translation: Moscow Univ. Math. Bull. 33 (1978), no. 2, 53-57.

N. N. Lusin [1915], Integral i trigonomet, ryad; [1951], 2d Ed. with commentaries by N. K. Bari and D. E. Men~ov, Gos. izdat, tekh.-teor, lit., Moscow-Leningrad (Rus- sian). MR 14, 2.

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H. Rademacher, Einige S/itze fiber Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 112-138. JFM 48, 485.

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