All the way with Gauss-Bonnet
and the sociology of Mathematics
Daniel Henry Gottlieb
Introduction
My reasons for writing this story were stimulated by the discussion in The Am*
*er-
ican Mathematical Monthly between Peter Hilton and Jean Pederson on the one
hand and Branko Grunbaum and G. C. Shephard on the other hand [HP] [GS]. The
discussion as well as my story involves the Euler-Poincare Number, alias the Eu-
ler Characteristic. The discussion centers on whether the Euler-Poincare Number
should be discussed in a historical way without mentioning the vast and dramatic
generalization and depth of understanding that this most interesting invariant *
*has
acquired in this century. My story spans the history of mathematics. It concerns
the, perhaps, most widely known non-obvious theorem of mathematics and it con-
tains the same stunning generalization that characterizes the recent history of*
* the
Euler-Poincare number. In fact it concerns one of the most important and ear-
liest of the applications of the Euler-Poincare number. It shows the reticence*
* of
mid century topologists to explain their work to the wider mathematical public.*
* It
shows the fickleness of mathematical fame, and it shows the unreasonable power
of unreasonable points of view, and it shows how easy it is for mathematicians *
*to
miss and forget beautiful and important theorems.
This is a history of the Gauss-Bonnet theorem as I see it. I am not a mathe-
matical historian. I am quoting only secondary sources or first hand papers whi*
*ch
I quickly scanned and I did not conduct any thorough interviews. Nontheless, I
am writing this history because I have contributed the last sentence to it (for*
* the
moment).
The Normal Map
What is the most widely known, not immediately obvious mathematical theo-
rem? I contend that is the following. The sum of the interior angles of a trian*
*gle
equals ss. The ordinary person might admit lightly that he doesn't quite rememb*
*er
the Pythagorian theorem, but if he does not know the sum of the angles equals 1*
*80
____________
Mathematics Subject Classification. 55.
Key words and phrases. Gauss map, normal degree, total curvature.
1
2 Daniel Henry Gottlieb
degree, he brands himself as uneducated. I will call this theorem the 180 degr*
*ee
theorem.
This 180 degree theorem was proven in the time of Thales. It has undergone
a remarkable generalization through the ages culminating in the Gauss-Bonnet
Theorem as I give it here. The first generalization involves the concept of ext*
*erior
angle. Exterior angles contain the same mathematical information as interior an*
*gles
because they are related by a simple equation: ff + fi = ss where ff is an inte*
*rior
angle and fi is the corresponding exterior angle. Now the sum of the exterior a*
*ngles
of a polygon sum to 2ss. This immediately implies the 180 degree theorem by the
above equation.
At this point I digress to give an analogy of the issues in the Hilton et al *
*contro-
versy. For a closed polygon, the interior angles are the best from the visual p*
*oint
of view. But from the mathematical point of view the exterior angles are supe-
rior because they can be defined in more generality than for polygons and more
importantly they are the form in which the angular information is presented in a
far reaching theorem. Thus the exterior angles of a square and a triangle and a
hexagon sum in each case to 2ss whereas the interior angles of a triangle sum t*
*o ss
and of a square sum to 2ss and so on. Now it may be interesting that the interi*
*or
angles of a square add up to 360 degrees and it may be very important in some
problems; but it is also very true that we can completely understand that fact *
*and
a host of similar ones through the medium of exterior angles and their constant
sum. So it is with Euler's Formula. We may get all kinds of formulas using gene*
*ral
shapes for the faces, but we can instantly understand those formulas by means of
faces which are contractible cells.
To continue, we approximate the polygon by a smooth closed curve. Then the
rate of change of the tangent, which is called the curvature of the curve corre*
*sponds
to the exterior angle, and the total turning of the tangent, or equivalently th*
*e total
curvature of the closed curve, corresponds to the sum of the exterior angles. T*
*he
tangent turns through 2ss as it completes a tour of the closed non-self interse*
*cting
curve. That is, the total curvature is 2ss. This then implies the exterior angl*
*es sum
to 2ss by continuity, so we have greatly generalized the original theorem about*
* the
triangle.
Again we change the basic concept by an equivalent but more general concept.
Instead of the turning of the tangent, we consider the turning of the normal to*
* the
tangent. We have the same curvature as the rate of change of the normal as from
the rate of change of the tangent. As in the change from interior to exterior a*
*ngles,
we go from a visually better point of view to a mathematically more generalizab*
*le
point of view. For on surfaces in three-space, the normals are defined whereas *
*there
is no unique tangent direction.
We formalize this concept by introducing the idea of the Gauss map, also call*
*ed
the normal map. To each point on the smooth surface in three-space there is a
unique unit normal vector pointing outside. The mapping maps the surface to the
unit sphere. It is given by sending the point to its normal vector and then par*
*allel
transporting the unit vector through space so that the beginning of the vector *
*is
on the center of the unit sphere and then taking the point on the unit sphere w*
*hich
corresponds to the tip of the transported unit vector.
This same idea gives the normal map in dimension two from the closed curve to
the unit circle; and from a smooth closed n - 1 dimensional manifold M imbedded
Gauss-Bonnet 3
in n-dimensional Euclidean space Rn to the unit sphere Sn-1. So fl : M ! Sn-1
denotes the normal map.
Curvature
Just as the infinitesimal change of length at x on a curve compared to the le*
*ngth
at the image fl(x) is the definition of curvature of a curve in the plane at x,*
* so is the
change in area from x to fl(x) the curvature of a surface at x in space. One wo*
*uld
think that the same name would hold for the higher dimensional examples of the
infinitesimal change of volume, but for historical reasons this did not happen.*
* For
the purposes of this paper I will call this number the normal curvature of M at*
* x
in Rn.
Let us pause and consider the reason that normal curvature, the natural gener-
alization of angle, is not called curvature in dimensions higher than 2. It is *
*because
in dimension two, the normal curvature depends not on how the surface sits in
R3, but on the intrinsic geometry of the surface. That is the curvature can be
calculated by considering only the surface and not the ambient space. This is t*
*he
famous Theorema Egregium of Gauss. So for higher dimension, curvature means
the Riemann curvature tensor. This is based on the two dimensional curvature and
does not agree at all with the normal curvature in higher dimensions and does n*
*ot
even make sense for dimension 1 curves. This curvature tensor plays an important
role in differential geometry and physics, but it does not replace the normal c*
*urva-
ture the way interior angles are replaced by exterior angles. Outside of dimens*
*ion
2 they are very different concepts. This intrinsic vs. extrinsic will play a ke*
*y role
in my story.
Now consider an (n - 1)-dimensional manifold M with no boundary in Rn. Now
M divides Rn into two pieces, the interior and the exterior. The interior of M
is a manifold with boundaryRM denoted by N. Now if we integrate the normal
curvature K over M we get KdM, the analogue of the sum of the exterior angles.
Call this the Curvatura Integrala of M in Rn. Now we can state our version of t*
*he
Gauss-Bonnet theorem. Here the Euler-Poincare number of N is O(N).
R
Gauss-Bonnet Theorem. KdM = O(N) x (the volume ofSn-1)
Normal degree
The unit volume of the (n - 1)-sphere is 2ss for the 1-sphere and 4ss for the*
* two
sphere and it changes form for each dimension. Thus we define the degree of fl *
*by
the Curvatura Integrala divided by the volume of the unit sphere of the dimensi*
*on
of M. The degree of fl is denoted by deg(fl) and is called the normal degree. T*
*here
is a much more general homotopy invariant definition of degree which gives the
formula we have used to define the degree here. In this notation we can write t*
*he
Gauss-Bonnet theorem as the
Topological Gauss-Bonnet Theorem. deg(fl) = O(N).
Now observe that a simple closed curve M in a plane bounds a contractible
setRN so that O(N) = 1 . Hence deg(fl) = 1 by the Gauss-Bonnet theorem so
KdM = 2ss where K denotes the curvature of the curve in the plane. As we have
said, this gives the 180 degree theorem.
4 Daniel Henry Gottlieb
Thus we have a tremendous generalization of the sum of angles concept valid f*
*or
every dimension and given by a simple formula. We continue with the remarkable
history of this result.
The Nineteenth Century
The Gauss-Bonnet Theorem is such an interesting result that various authors
could not resist including parts of its history in their textbooks. Especially *
*Spivak
[Sp] and Stillwell [St] give accounts of its early history.
In 1827 C. F. Gauss published the following result. Let T be a geodesic trian*
*gle
on a surface. That is the three sides of T are geodesic curves in the surface. *
*Then
the sum of the interior angles minus ss is the integral of the curvature K over*
* the
triangle T .
R
Gauss-Bonnet Formula. KdT = ff + fi + fl - ss
Note that if the surface is a plane, then the geodesics are straight lines an*
*d K
is identically equal to zero, so the Gauss-Bonnet Formula implies the 180 degree
theorem.
In 1848 O. Bonnet extended this formula to smooth closed curves on the surface
by a limiting argument which is like the extension from polygons to curves men-
tioned above. Here the sum of the angles is replaced by the integral of the geo*
*desic
curvature. This generalized formula acquired the name Gauss-Bonnet sometime
later.
If the geodesic triangles triangulate a closed surface S, then using Euler's *
*For-
mula, the number of vertices minus the number of edges plusRthe number of trian*
*gles
equals 2, gives the first global Gauss-Bonnet theorem: KdS = 2ss.
A lost manuscript of Descartes copied in Leibnez' hand was discovered some
years later and published in the Comptes Rendus in 1860. A note by Bertrand
immediately following Descartes' article points out its relationship to the glo*
*bal
theorem. Bertrand notes that Descartes seems to get the polyhedral version of t*
*he
global Gauss-Bonnet Theorem. He attributes the global theorem to Gauss.
However we know that nobody understood the Euler-Poincare Number at that
time, and the result really only held for a surface diffeomorphic to a sphere. *
* A
good account of the difficulty involved with the development of the Euler-Poinc*
*are
Number is found in [La]. Indeed, the Hilton et al discussion would fit right in*
*to the
dialogue method that Lakatos used to present his thesis.
Walter von Dyck seems to be the first who realized that the Gauss-Bonnet The-
orem should hold for more that just spherical surfaces. He did this in 1888. Ac-
cording to Hirsch [Hi], von Dyck was the first one to connect the degree with t*
*he
Euler-Poincare number and thus prove "what is wrongly called the Gauss-Bonnet
Theorem".
Actually, part of this story shows that the name of a theorem is not really f*
*or
attribution. It is very convenient to have a name for important theorems and the
main point is that people should know approximately what theorem is meant by
the name rather than who gets the credit. Still, one can reflect that Bonnet's *
*name
is famous and von Dyck's is virtually unknown these days.
Gauss-Bonnet 5
Hopf to Chern
Von Dyck worked at a time when the ideas of degree of a map and the Euler-
Poincare number were not clearly understood. By 1925, these concepts were well-
defined and found to be useful. This was due in no small measure to Heinz Hopf.
Hopf in [H1] made the biggest advance. He proved that deg(fl) = O(M)=2 for
closed hypersurfaces of even dimension. The factor 1=2 is explained by the fact
that O(N) = O(M)=2 whenever N is a compact odd-dimensional manifold with
boundary M. Since O(M) = 0 for closed odd-dimensional manifolds, the theorem
as stated by Hopf did not seem to generalize to the odd dimensional case, and in
particular did not generalize the 180 degree theorem, which as we saw is genera*
*lized
by the Gauss-Bonnet Formula.
Since the curvature of a surface is intrinsic in dimension 2, Hopf asked for *
*intrinsic
proofs and generalizations of his result [H2]. He did this repeatedly and inter*
*ested
several mathematicians in the question. The story is told in [Gr].
Using Hermann Weyl's theory of tubes, two mathematicians independently an-
swered Hopf's question in 1940. Allendoerfer [Al] and Fenchel [Fe] discovered t*
*hat
deg(fl) of the boundary of a tubular neighborhood of a closed 2n dimensional
manifold imbedded in a 2r dimensional Euclidian Space is equal to the integral *
*of
a 2n form constructed out of the components of the Riemannian curvature ten-
sor and combined together as a Pfaffian. Since the tubular neighborhood has the
same Euler-Poincare Number as the embedded manifold, they got a formula for the
Euler-Poincare Number in terms of the Reimannian curvature of an imbedded even
dimensional manifold. This remarkable formula held for every Riemannian man-
ifold because every Riemannian manifold can be isometrically imbedded in some
Euclidian space. However this last result was not known until the 1950's when it
was proved by Nash.
Aside from the fact that the Allendoerfer-Fenchel Formula held only for an
imbedded manifold, it was obviously independent of the imbedding and begged
for an intrinsic proof. S. S. Chern provided one in 1944 [Ch]. This proof was so
well received that the Allendoerfer-Fenchel Formula is frequently called the Ga*
*uss-
Bonnet-Chern Formula and the Gauss-Bonnet-Chern theorem. In fact one of the
goals of Gray's book [Gr] was to prevent the interesting methods of the Tube pr*
*oof
from being totally submerged by the powerful ideas of Chern's proof.
Whodunnit ?
Now we come to the most interesting part of the story. In 1956, Hopf gave
lectures at Stanford University on global differential geometry. These lectures*
* were
honored by being published as volume number 1000 of Springer-Verlag's Lecture
Notes In Mathematics in 1983 [H3] . On pages 117-118, Hopf describes his version
of the Gauss-Bonnet theorem for even dimensions.
It is clear that at that time Hopf did not know that the Gauss-Bonnet theorem
held for all dimensions and thus was a generalization of 180 degree theorem. Or*
* else
he knew it, but was embarrassed to state it. Hopf certainly knew all the ingred*
*ients
for the proof in all dimensions for many years, and the proof is of the same or*
*der of
difficulty as his even dimensional proof. Had he known the version that held fo*
*r all
dimensions it seems likely he would not have asked for intrinsic proofs, since *
*there
6 Daniel Henry Gottlieb
are none in odd dimensions. So two very fruitful lines of research probably wou*
*ld
not have been undertaken.
Yet the topological Gauss-Bonnet theorem was known to several topologists
around the mid fifties, among them Milnor and Lashof. Nobody seems to know
who it was who first stated the theorem. At the time there were sophisticated g*
*en-
eralizations and studies of deg(fl), for example [Ke] and [Mi]. Just recently B*
*redon,
in his textbook [Br], stated and proved the result as " Theorem 12.11 (Lefschet*
*z)".
He proves it as a corollary of the Lefschetz fixed point theorem.
Finally, in 1960, the topological Gauss-Bonnet theorem was stated in the lite*
*r-
ature, but in an even more generalized form by Samelson [Sa] and Haefliger [Ha]:
Let N be a compact n-dimensional manifold with boundary M and let f : N ! Rn
be an immersion. Then the Gauss map fl : M ! Sn-1 can still be defined and
deg(fl) = O(N).
The unasked for answer
In hindsight we see that Hopf's question amounted to: Find a formula giving
O(M) in terms of the curvature tensor for even dimensional closed Riemannian
manifolds. The more reasonable question should have been: Find a formula giving
deg(fl) for all dimensions. Nobody asked this question. An answer has been fou*
*nd
however. It is what I will call the Gauss-Bonnet-Morse Theorem to distinguish it
from the Gauss-Bonnet-Chern Theorem:
Gauss-Bonnet-Morse Theorem. Let f : N ! Rn be a map which is an immer-
sion on a collar of the boundary M of a compact n-manifold N. Then if x is the
projection of Rn to some x-axis and r is the gradient vector field of (x O f) a*
*nd
Ind is its index, we have
deg(fl) = O(N) - Ind(r(x O f))
We can now compare the even-dimension and all-dimension proofs of the Gauss-
Bonnet Theorem. The even-dimensional proof follows from the easy fact that the
index of a vector field V changes sign if the direction of V is reversed to -V *
*on an
odd dimensional manifold. The all-dimensional proof follows from the equally ea*
*sy
fact that that the index of a vector field consisting of nonzero vectors is zer*
*o.
First note that any co-dimension one immersion of M in Rn can be extended to
an f as in the above theorem. Then if f is an immersion, the gradient r(x O f) *
*has
no zeros and so deg(fl) = O(N) for all dimensions.
On the other hand, if M is a 2n-dimensional manifold imbedded in R2n+1, it
bounds a 2n + 1 dimensional manifold N. If we reverse the sense of the x-axis, *
*we
reverse the direction of r(x O f) and hence the sign of the index. But the othe*
*r two
terms in the above displayed equation do not change. Thus again deg(fl) = O(N).
However this last paragraph is not a weaker proof than the previous paragraph
when the question of co-dimension one immersions is considered. For any immersi*
*on
extends to a continuous map f : N ! R2n+1 as in the theorem. So deg(fl) =
O(N) = O(M)=2 for all immersions for even dimensional M, whereas deg(fl) = O(N)
in general only when the immersion of M extends to an immersion for some N.
So for example, whereas any integer can be the normal degree deg(fl) of a reg*
*ular
closed path, that is the immersion of the circle in the plane, the analogous st*
*atement
for the sphere immersed in three-space is that deg(fl) = 1.
Gauss-Bonnet 7
In 1929 Marston Morse [Mo] discovered a beautiful equation involving the index
of a vector field V on a compact manifold N with boundary M which I call the
Law of Vector Fields.
The Law of Vector Fields. Ind V + Ind@- V = O(M) where @- V is a vector
field induced by V and defined on that part of the boundary M where V points
inside.
This result is literally a self contained definition of the Index of vector f*
*ields [gS]
(actually I am hoping this reference is historical in its own right as the firs*
*t citation
of a paper in an electronic journal). But it was not used much by Morse, probab*
*ly
because he was inventing Morse theory and may have thought unconsciously, as
many topologist have, that all vector fields come from gradient vector fields. *
*At any
rate this result was not used much and was virtually forgotten. When I rediscov*
*ered
it in the 1980's, it took almost a year of questioning before someone told me a*
*bout
[Mo].
I thought of a simple scheme to try to exploit the Law of Vector Fields. I lo*
*oked
at interesting vector fields and plugged them into the equation. I plugged in w*
*hat
I called pullback vector fields, which generalize gradient vector fields, and I*
* got an
equation involving the normal degree and the Euler-Poincare number [G1] [G2]. It
took me years before it occurred to me that I had generalized the Gauss-Bonnet
Theorem. A simplified version of that result is the Gauss-Bonnet-Morse Theorem
above. I named it after Morse since it follows easily from his equation and I c*
*alled
that equation the Law of Vector Fields since there are already too many Morse
theorems.
Conclusions
Mandelbrot in proposing the name "fractals" complained that mathematicians
do not give names to concepts and results. He was right. In the deepest sense t*
*his
story really revolves about the naming of theorems and of curvature.
But it also demonstrates that several of the bromides that we have grown up
with are seriously flawed. That great men do not overlook simple points. That
there are no great results found in using old methods. That you can't discover
something good unless you have asked the right questions. That mathematics
progresses mostly by the work of a few great mathematicians; this seems to need
the phrase "and anonymous mathematicians". It seems to me that what happened
with the Gauss-Bonnet Theorem happened very frequently with the best of our
mathematical ideas. Nobody seems to know who invented "Cartesian Coordinates",
or who first thought about higher dimensional spaces. Great mathematicians are
quoted denigrating ideas that blossomed and dominated mathematics. From our
present vantage point these ideas seem trivial, but our greatest forbearers had
trouble grasping them. What seems to be trivial now was once the most difficult
part of mathematics; infinity, velocity and acceleration, arbitrary axioms, abs*
*tract
groups, functions.
So what about the Euler-Poincare controversy? It is clear that if the Euler-
Poincare Number figures so simply in the correct generalization of angle, that *
*it
must be a most basic mathematical concept. Hilton speaks for topologists in his
impatience that many mathematicians do not realize the fundamental importance
8 Daniel Henry Gottlieb
of this concept. But on the other hand it should be clear that topologists have*
* not
been very good at spreading the word to their colleagues.
References
[Al]C. B. Allendoerfer, The Euler Number of a Reimannian manifold, Amer. J. Mat*
*h. 62 (1940),
243-248.
[Br]Glenn E. Bredon, Topology and Geometry, Springer-Verlag: Graduate Text in M*
*athematics,
New York (1993).
[Ch]S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet theorem for close*
*d Riemannian
manifolds, Annals of Math. (2) 45 (1944), 747-752.
[Fe]W. Fenchel, On the total curvature of Riemannian manifolds, I, J. London Ma*
*th. Soc. 15
(1940), 15-22.
[G1]Daniel H. Gottlieb, On the index of pullback vector fields, Springer-Verlag*
*, Lecture Notes
in Mathematics 1350 (1987), 167-170.
[G2]Daniel H. Gottlieb, Zeroes of pullback vector fields and fixed point theory*
* for bodies, Con-
temporary Mathematics, Amer. Math. Soc. 96 (1989), 163-179.
[Gr]Alfred Gray, Tubes, Addison -Wesley, Redwood City California (1990).
[gS]D. H. Gottlieb and G. Samaranayake, Index of discontinuous vector fields, N*
*ew York Journal
of Mathematics (to appear).
[GS]B. Gr"undbaum and G. C. Shephard, A new look at Euler's Theorem for polyhed*
*ra, Amer.
Math. Monthly 101 (1994), 109-128.
[Ha]Andre Haefliger, Quelques remarques sur les applications differentiables d'*
*une surface dans
le plan, Ann. Inst. Fourier 10 (1960), 47-60.
[HP]P. J. Hilton and J. Pederson, Euler's Theorem for polyhedra: A Topologist a*
*nd a Geometer
respond, Amer. Math. Monthly 101 (1994), 959-962.
[Hi]Morris W. Hirsch, Differential Topology, Springer-Verlag, New York (1976).
[H1]Heinz Hopf, Uber die Curvatura integrala geschlossener Hyperflachen, Math. *
*Ann. 95
(1925), 340-376.
[H2]Heinz Hopf, Differential Geometrie und Topological Gestalt, Jahresbericht d*
*er Deutcher
Math. Verein. 41 (1932), 209-229.
[H3]Heinz Hopf, Differential Geometry in the Large: Seminar Lectures NYU 1946 a*
*nd Stanford
1956, Lecture Notes in Mathematics, Springer Verlag 1000 (1983).
[Ke]M. E. Kervaire, Coubure integrale generalisee et homotopie, Math. Ann. 131 *
*(1956), 219-
252.
[La]Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, *
*Cambridge
University Press, Cambridge (1976).
[Mi]J. W. Milnor, On the immersion of n-manifolds in (n+1)-dimensional space, C*
*ommentarii
Math. Helvetici 30 (1956), 275-284.
[Mo]Marston Morse, Singular points of vector fields under general boundary cond*
*itions, Amer.
J. Math 51 (1929), 165-178.
[Sa]Hans Samelson, On the immersion of manifolds, Canadian J. Math. 12 (1960), *
*529-534.
[St]John Stillwell, Mathematics and its History, Springer-Verlag, New York (197*
*4).
[Sp]Michael Spivak, A Comprehensive Introduction to Differential Geometry, Publ*
*ish or Parish,
2nd ed, vol. 1-5, Houston (1979).