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\title All the way with Gauss-Bonnet and the Sociology of Mathematics
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\rightheadtext{Gauss-Bonnet} % Argument should contain a short version of the
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\author Daniel Henry Gottlieb
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\leftheadtext{ Daniel Henry Gottlieb} 
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  % Give separate address and email, entries for each author.
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\address Department of Mathematics
         Purdue University
         West Lafayette
         Indiana 47907
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\email     gottlieb@math.purdue.edu 
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\keywords Gauss map, normal degree, total curvature
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\head Introduction
\endhead


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
My reasons for writing this story
were stimulated by the discussion in The American Mathematical
Monthly between Peter Hilton and Jean Pederson on the one hand and Branko
Gr\"unbaum and G. C. Shephard on the other hand \cite{HP} \cite{GS}.
The  discussion as well as
my story involves the Euler-Poincar\'e Number, alias the Euler 
Characteristic.
The {\it discussion} centers on whether the Euler-Poincar\'e 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 position in this discussion is that Topology should not be viewed as
an advanced subject whose theorems and concepts should be avoided
until graduate school.  Rather it
is the study of continuity, and thus underlies the most basic 
geometric results. In this paper I show how the basic concept of angle
leads naturally to the basic topological ideas of 
{\it degree of mapping} and of the 
Euler-Poincar\'e Number. 


My {\it story} spans the history of
mathematics. It concerns the, perhaps, most widely known non-obvious
theorem of mathematics and it contains the same stunning generalization
that characterizes the recent history of the
Euler-Poincar\'e number. In fact it concerns one of the  most important
and earliest
of the applications of the Euler-Poincar\'e number. 
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 as well as simple and revealing points of view.

This is a history of the Gauss-Bonnet theorem as I see it. I am not a
mathematical historian. I am quoting only secondary sources or
first hand papers which I quickly scanned and I did not
conduct any thorough interviews. Nonetheless, I am writing this history
because I have contributed the last sentence to it (for the moment).

I especially want to acknowledge the help of Hans Samelson. His
scholarship greatly altered the thrust of earlier versions of this
paper. He discovered Satz VI. He informed me of many points in this
history; about Gauss' work, Decartes work and Hopf's work. Also he was
a student of Hopf who generalized the Gauss-Bonnet theorem.



\head The Normal Map
\endhead

What is the most widely known, not immediately obvious
mathematical theorem? 
I contend that is the following. {\it The sum of the interior angles 
of a triangle equals $\pi$.} The ordinary person might admit lightly
that he doesn't quite remember the Pythagorean theorem, but if he 
does not know the sum of the angles equals 180 degrees, he brands himself
as uneducated.
I will call this theorem the {\it 180 degree 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 exterior
angle. Exterior angles contain the same mathematical information as
interior angles because they are related by a simple equation:
$\alpha + \beta = \pi $ where $\alpha$ is an interior angle and $\beta$ is
the corresponding exterior angle. Now {\it the sum of the exterior angles of a
polygon sum to $2 \pi$.} This immediately implies the 180 degree theorem
by the above equation.

FIGURE 1.

What is the angle between two straight lines intersecting at  $O$? Let
$S^1$ be the unit circle centered at $0$. Then the length of the arc of
$S^1$
cut off by the lines is the angle between the lines. We regard angles
as a property of a subset of the unit circle rather than as a number.
This point of view is closer to the original Greek point of view.
Regarding angle as a number is a more modern point of view. 

FIGURE 2.

This Greek point of view is susceptible to immediate generalization.
Just as angle is the length, or 1-volume, of a region of the unit
circle in the one dimension case, we can think of the area, or
2-volume, of a region on the unit sphere in three space, denoted by
$S^2$, as a representation of angle in the two dimensional case. In
general, angle in $n$-dimensions can be thought of as the $n+1$-volume of a
region on the unit $n$-dimensional sphere $S^n$.

Now consider a curve $\sigma$ in the plane connecting point $A$ to
point $B$. Consider the unit vectors tangent to  $\sigma$ at $A$ and
$B$. Translate these vectors to the origin, keeping them parallel. Then
the arc on $S^1$ represents the angle the curve has turned through.

FIGURE 3.

One thing that topologists have learned in developing Topology is that
it almost always pays to convert things into functions or mappings.
This procedure has spread throughout all of mathematics in the last
half of this century. So in the case at hand, define a mapping from
$\sigma$ to the unit circle $S^1$ as follows: At each point $P$ on
$\sigma$, construct the unit tangent vector to $\sigma$ at $P$, then
parallel translate it to the origin so that its end lies on the unit
circle. Call this the {\it tangent map}.

Now let $B$ approach $A$ along $\sigma$. If we divide the angle between
the tangent at $B$ and the tangent at $A$ by the length along $\sigma$
from $A$ to $B$, we have a quantity which approaches a limit if $\sigma$
is smooth enough. This number is the {\it curvature} of $\sigma$ at $A$.
This is the same as saying that the curvature at $A$ is the 
change at $A$ of the length of an infinitesimal arc on $\sigma$ to the
length of its image on $S^1$.

Now let us approximate a polygon
by a smooth simple closed curve. Then the rate of
change of the tangent, which is called
the {\it curvature} of the curve corresponds to the exterior angle, 
and the
total turning of the tangent, or equivalently the
{\it total curvature} of the closed curve, corresponds to the sum of the
exterior angles. Now for simple closed curves, the tangent turns 
through $2 \pi$ as it  completes a
tour of the closed non-self intersecting curve. That is, the total
curvature is $2 \pi$. This then implies the exterior angles sum to
$2 \pi$ by continuity. This approximation of polygons by smooth curves
is an argument known to the Greeks.
So we have greatly generalized the original 180 degree theorem
theorem about the triangle for polygons by the theorem that the total
curvature of a simple closed curve is $2 \pi$.





Instead of the tangent, we consider the
normals to $\sigma$. Now the normal varies exactly as the tangent
does as a
point moves along $\sigma$. So we could define the curvature of 
$\sigma$ using normals instead of tangents. Thus we replace the
tangent map with the {\it normal map} from $\sigma$ to $S^1$. The
advantage of using normals instead of tangents is that we can
generalize curvature to surfaces in three space, 
for on surfaces in three-space, the normal direction is well-defined 
whereas
there is no unique tangent direction.



We formalize this concept by introducing the idea of the 
{\it Gauss map}, also
called the {\it 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 parallel 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 which corresponds to
the tip of the transported unit vector.

FIGURE 4.


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 $M$ embedded in $n$-dimensional Euclidean space $R^n$ to the
unit sphere $S^{n-1}$. So $\gamma : M \rightarrow S^{n-1}$ denotes the
normal map.


\head Curvature
\endhead

Now we can define the concept
of {\it normal curvature} at a point $m$ of 
$M$ in $R^n$. Let $R$ be a small region around $m$ in $M$. Let 
$\gamma (R)$ denote the image in $S^{n-1}$ of $R$. Then the 
{\it normal curvature} at $m$, denoted $K(m)$, is the limit as $R$
tends to $m$ of the $(n-1)$ volume of $\gamma (R)$ divided by the 
$(n - 1)$ volume of $R$. This is given a positive sign if 
$\gamma$ preserves the orientation at $m$ and a negative sign if 
$\gamma$ reverses the orientation at $m$.
In the presence of a suitable coordinate system, $K(m)$ is the Jacobian
of $\gamma$  at $m$.



Just as the infinitesimal change of length at $x$ on a curve,
compared to the length at the image $\gamma (x)$, is the definition of
curvature of  a curve in the plane at $x$, so is the change in area  from
$x$ to $\gamma (x)$ the curvature of a surface at $x$ in space. 
One would 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
{\it normal curvature} of $M$ at $x$ in $R^n$. 

Let us pause and consider the reason that normal curvature, the natural
generalization 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 $R^3$, 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 the 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 not 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 curvature the way interior angles are
replaced by exterior angles. Outside of dimension 2 they are very
different concepts. This intrinsic vs. extrinsic will play a key role in
my story.

Now consider a compact  $(n -1)$-dimensional manifold $M$ with no boundary in
$R^n$. Now $M$ divides $R^n$ into two pieces, the interior and the
exterior.
The interior of $M$ is a manifold with boundary $M$ denoted by
$N$. Now if we integrate the normal curvature $K$ over $M$ we get 
$\int KdM$, the
analogue of the sum
of the exterior angles. Call this the {\it Total Curvature} or the old
fashioned {\it Curvatura Integra} of $M$
in $R^n$. Now we can state our version of the Gauss-Bonnet theorem.
Here the Euler-Poincar\'e number of $N$ is $\chi (N)$.

\proclaim{Gauss-Bonnet Theorem}
$\int KdM = \chi (N) \times \text { (the volume of \  } S^{n-1})$
\endproclaim




\head Normal degree
\endhead
The unit volume of the $(n-1)$-sphere is $2\pi$ for the 1-sphere and $4 \pi$ for the
2-sphere and it changes form for each dimension. Thus we define the
{\it degree} of $ \gamma $ by the Curvatura Integra divided by the 
volume of the unit sphere of the dimension of $M$. The degree of 
$\gamma$ is denoted by $ \text { deg}( \gamma)$ and is called the 
{\it normal degree}. The normal degree turns out to be an integer. In
fact, this is a special case of the concept of degree of a mapping, an
integer which plays a major role in Topology.
In this notation we can write
the Gauss-Bonnet theorem as the

\proclaim{Gauss-Bonnet-Hopf Theorem}
$ \text { deg}( \gamma) = \chi (N)$.
\endproclaim

The Euler-Poincar\'e number is the earliest invariant of Algebraic
Topology. It is a vast generalization of a formula involving convex
polyhedra due to Euler. There is evidence that Descartes knew about
this formula a century before Euler, \cite{S${}_2$} or \cite{St}.

The degree of a map can be traced back to Kronecker and was well
understood by L. E. J. Brouwer around 1913. The integral definition
given here for the Gauss map can be generalized to maps between
oriented closed manifolds
of the same dimension. The most general definition of both the degree
of a mapping and of the Euler-Poincar\'e number requires Homology
Theory. But both these concepts were discovered before Homology was
well understood and they can be used very effectively without knowledge
of Homology.

For a two dimensional surface $N$ that can be divided up nicely by triangles
which fit nicely together to form what we call a triangulation,
the  Euler-Poincar\'e number satisfies
$$\chi (N)= v - e + f$$
where $v$ is the number of vertices, $e$ is the number of edges, and $f$
is the number of triangles in the triangulation.

FIGURE 5.

Given this, it is a simple matter to show that if $N$ is bounded by a
convex polygon, then $\chi (N) = 1 $.
Hence  $ \text {deg}( \gamma) = 1$ by
the Gauss-Bonnet-Hopf theorem so 
$\int KdM = 2 \pi$ where $K$ denotes the curvature of the curve in the
plane. As we have said, this gives the 180 degree theorem.

Thus we have a tremendous generalization of the sum of angles concept
valid for every dimension  and given by a simple formula. We continue
with the remarkable history of this result.



\head The Nineteenth Century
\endhead

The Gauss-Bonnet Theorem is so interesting that various
authors could not resist including 
parts of its history in their textbooks. 
Especially Spivak \cite{Sp} and  Stillwell \cite{St} give accounts of its 
early history. 

Consider a geodesic triangle $T$ 
on a surface in three space. The edges of the
triangle are geodesics. Geodesics are 
what passes for straight lines on the surface; they are paths of
shortest length on the surface. Let 
$\alpha$ , $\beta$ , $\gamma$  denote the interior angles of the
triangle. Then if we integrate the curvature  $K$ over the triangle $T$
we get the Gauss-Bonnet Formula:


\proclaim{Gauss-Bonnet Formula for the geodesic triangle}
$\int KdT = \alpha + \beta + \gamma - \pi $
\endproclaim

FIGURE 6

This formula immediately gives interesting corollaries:

If the triangle $T$ is a plane triangle, then the geodesics are straight
lines and $K$ is identically equal to zero,
so $\alpha + \beta + \gamma = \pi $
So the Gauss-Bonnet
Formula implies the 180 degree theorem, but not at all in the same way
that the Gauss-Bonnet-Hopf Theorem implies the 180 degree theorem.

If we divide the angular excess $\alpha + \beta + \gamma - \pi $ by the
area of $T$, we get a number that is calculated intrinsically on the
surface.
As we let $T$ shrink down to a point $m$, the ratio approaches the
curvature $K(m)$ at $m$. Hence $K$ is an intrinsic concept of the surface.
This is Gauss' famous Theorema Egregium, but his published proof is not the
argument just given. In a previous unpublished manuscript, he gives this 
argument right after his proof of the Theorema Egregium.

If we triangulate a closed  surface $M$ with geodesic triangles, we get a
Gauss-Bonnet formula for each triangle. If we add these equations up, we
get on the left side the {\it total curvature} also called the Curvatura
Integra: $\int KdT$. On the right side
we can rearrange the angles cleverly and end up with 
$4\pi \times \chi(M)/2 $.

This agrees with what we named the Gauss-Bonnet Theorem because for
surfaces $\chi (M) = 2 \times \chi (N)$ where recall that $N$ is the part
of space interior to the closed surface $M$. In fact, it is true that 
$\chi (M) = 2 \times \chi (N)$ for all even dimensional $M$. For odd
dimensional closed manifolds $M$ however, $\chi (M) = 0$. These 
elementary topological facts
along with `intrinsic verses non-intrinsic' play a key role in this story.

Gauss wrote down the version of the `Gauss-Bonnet Formula for the geodesic
triangle' given
above in an unpublished manuscript in 1825. In 1827 he published a
book giving a differential formula, which if he integrated would have
given the generalization that Bonnet got of the Gauss-Bonnet formula.
I am informed of this last by Samelson.

In 1848, O. Bonnet extended the Gauss-Bonnet 
formula for a triangle to smooth closed curves on the 
surface. 
Here the sum of the angles is essentially replaced by the integral of the 
{\it geodesic curvature}. This generalized formula acquired the name
Gauss-Bonnet sometime later. Probably Blaschke was the first to use the
name
in a textbook in the early 1920's.

If the  geodesic triangles triangulate a closed surface $S$ which is
topologically a sphere, 
then using Euler's Formula
$$v-e+f = 2$$
%{\it  the number of vertices minus
%the number of edges plus the number of triangles equals 2}, 
gives
the first global Gauss-Bonnet theorem:
$\int KdS = 4 \pi $. 

A lost manuscript of Descartes copied in Leibniz' 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 global theorem.
Bertrand notes that Descartes seems to get the polyhedral version of the
global Gauss-Bonnet Theorem. He attributes the global theorem to Gauss.
See \cite {S${}_2$} for an interesting account of this manuscript.
However we know that nobody understood the Euler-Poincar\'e 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-Poincar\'e Number
is found in \cite{La}. Indeed, the Hilton et al discussion would fit right
into the dialogues that Lakatos used to present his thesis.


Walter Dyck seems to be the first who realized that the
Gauss-Bonnet Theorem should hold for more that just spherical
surfaces. He did this in 1888. According to Hirsch \cite{Hi},  Dyck
was the first one to connect the degree with the Euler-Poincar\'e number
and thus prove ``what is wrongly called the Gauss-Bonnet Theorem".

An examination of Dyck's paper reveals pictures which are reminiscent of
standard figures in Morse Theory, developed 50 years later. Dyck was a
real pioneer, but he was ahead of his time, like Descartes. Samelson tells
me that he cannot find a statement of the global Gauss-Bonnet theorem in
Gauss' works. So it appears that the global Gauss-Bonnet theorem should be
called the Descartes-Dyck theorem.

Actually, part of this story shows that the name of a theorem is not
really for 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 
Dyck's is virtually unknown these days.

\head Hopf to Chern
\endhead

Dyck worked at a time when the ideas of degree of a map and the
Euler-Poincar\'e 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 \cite{H${}_1$} made the biggest advance. He essentially proved that
$ \text { deg}( \gamma) =  \chi (M)/2$ for closed hypersurfaces of
{\it even} dimension. The factor $1/2$ is explained by the fact that 
$ \chi (N) =  \chi (M)/2$ whenever $N$ is a compact odd-dimensional 
manifold with boundary $M$. Since $\chi (M) = 0$ 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 {\it is}
generalized 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
\cite{H${}_3$}. He did this repeatedly and interested several
mathematicians in the question. The story is told in \cite{Gr}.

Using Hermann Weyl's theory of tubes, two mathematicians independently
answered Hopf's question in 1940.
Allendoerfer \cite{Al} and Fenchel \cite{Fe} discovered that 
$ \text {deg}( \gamma)$ of the boundary of a tubular neighborhood of a
closed $2n$ dimensional manifold embedded in a $2r$ dimensional Euclidean Space
is equal to the integral of a $2n$ form constructed out of the
components of the Riemannian curvature tensor and combined together 
as a Pfaffian.
All this is too complicated to describe here.
Since the tubular neighborhood has the same Euler-Poincar\'e
Number as the embedded manifold, they got a formula for the 
Euler-Poincar\'e Number in terms of the Riemannian curvature of an
embedded even dimensional manifold. This remarkable formula held for
every Riemannian manifold because every Riemannian manifold can be
isometrically embedded in some Euclidean 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 embedded manifold, it was obviously independent of the
embedding and begged for an intrinsic proof. S. S. Chern provided one
in 1944 \cite{Ch}. This proof was so well received that the
Allendoerfer-Fenchel Formula is frequently called the
Gauss-Bonnet-Chern Formula or the Gauss-Bonnet-Chern Theorem.
In fact one of the goals of Gray's book \cite{Gr} was to prevent the
interesting methods of the Tube proof from being totally submerged by
the powerful ideas of Chern's proof.

\head Satz VI
\endhead

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  \cite{H${}_4$} . On pages 117--118, Hopf describes his
version of the Gauss-Bonnet theorem for even dimensions. He does not
mention the part which held for odd dimensions. Because of this and
various conversations, I wrote the following three paragraphs:

``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 ingredients for the proof in all dimensions
for many years, and the proof is of the same order of difficulty as his
even dimensional proof. Had he known the version that held for all
dimensions it seems likely he would not have asked for intrinsic
proofs, since there are none in odd dimensions. So two very fruitful
lines of research  probably would not have been undertaken."  

``Yet the Gauss-Bonnet-Hopf 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 generalizations and studies of 
$ \text { deg}(\gamma)$, for example \cite{Ke} and \cite{Mi}. Just
recently Bredon, in his textbook \cite{Br}, stated and proved the result
as `` Theorem 12.11 (Lefschetz)''. He proves it as a corollary of the
Lefschetz fixed point theorem." 

``Finally,
in 1960, the Gauss-Bonnet-Hopf theorem was stated in the
literature, but in an even more generalized form by Samelson 
\cite{S${}_1$} and Haefliger \cite{Ha}:
Let $N$ be a compact $n$-dimensional manifold with boundary $M$ and let 
$f:N \rightarrow R^n$ be an immersion. Then the Gauss map 
$ \gamma : M \rightarrow S^{n-1} $ can
still be defined and $ \text { deg}(\gamma) = \chi (N)$."


After those words were written I received a letter from Hans Samelson.
I had asked Samelson if he knew who had first discovered the
Gauss-Bonnet-Hopf Theorem. After all he had generalized it in
\cite{S${}_1$}. In addition he is a scholar about the Gauss-Bonnet Theorem
, and he was a student of Heinz Hopf!

He thought it was Morse who first stated it. He could not find
the reference, but on a hunch he looked at Hopf's 1927
paper \cite{H${}_2$}. There on page 248, Satz VI, the Gauss-Bonnet-Hopf theorem
is clearly stated for all dimensions! 

It is a testament to Hopf's
genius that even though he knew Satz VI for all dimensions, the
fact that the even dimensional case was true for immersions, instead of
merely embeddings (concepts not well understood then), must have led
him to conjecture that there was an intrinsic proof in the even
dimensional case.

A differentiable map between two manifolds of the same dimension is an
{\it immersion} if the Jacobian of the map is not zero anywhere. It is
an {\it embedding} if in addition the map is on-to-one. Thus
immersions are one-to-one in small neighborhoods of any point, whereas
embeddings are globally one-to-one. This distinction generalizes to any
mappings.

Now Satz VI, that is the Gauss-Bonnet-Hopf theorem, was only proved for
embeddings, whereas Hopf knew 
from \cite{H${}_1$}, 
that for $M$ an even dimensional manifold, 
$ \text {deg}( \gamma) = \chi (M)/2$ was true if $M$ is immersed in
Euclidean space of codimension one, ie. the dimension of the
Euclidean space in one dimension higher than the dimension of $M$. By
the way, since locally $M$ is embedded in Euclidean space, there is a
normal direction and so the Gauss map $\gamma$ is still defined.

The distinction between the odd and even dimensional cases 
can be explained to anyone.
A circle can be immersed in a plane with arbitrary normal degree, but a
two-sphere can only be immersed in three-space with normal degree equal
to one.

FIGURE 7.

So Hopf's proof in \cite{H${}_1$} was not rendered superflous by his
proof of Satz VI, and he recognized the presence of intrinsicness in
the difference. Thus he stimulated the Geometers with  
\cite{H${}_3$} to aspire towards an intrinsic proof of the even
dimensional Gauss-Bonnet-Hopf theorem. He also asked questions about
the possible normal degrees of immersion for odd dimensional $M$. This
stimulated Milnor's beautiful \cite{Mi}
and then to \cite{BK} , wherein it is shown that the
normal degree can take on the value of any odd integer.



\head The unasked for answer
\endhead

In hindsight we see that Hopf's question amounted to: Find a formula
giving $\chi (M)$ in terms of the curvature tensor for even dimensional
closed Riemannian manifolds. The more reasonable question should have
been: Find a formula giving $ \text {deg}( \gamma)$ for all
dimensions. Nobody asked this question. An answer has been found however.
It is what I will call the
Topological Gauss-Bonnet Theorem to distinguish it from the 
Gauss-Bonnet-Chern Theorem. 

This theorem immediately  gives a proof of Satz
VI as well as a proof for the immersion portion of the even dimensional
part proved in \cite{H${}_1$}. The proof of this theorem requires nothing
that was known after 1929. It is completely extrinsic. If Hopf had
discovered this proof it is unlikely he would have asked for an intrinsic
proof of \cite{H${}_1$}, and so some very important mathematics would not
have been discovered so quickly. 
For the Allendoerfer-Fenchel Formula, now
known as the Gauss-Bonnet-Chern Theorem, could not have been discovered by
accident.
It is too complicated.
 Very talented mathematicians were looking for it explicitly. On
the other hand, the Topological Gauss-Bonnet Theorem is simple enough so
that it could have been discovered by accident. And it was!


\proclaim{Topological Gauss-Bonnet Theorem}
Let $f: N \rightarrow R^n $ be a map whose Jacobian is nonzero 
on the oriented boundary $M$ of a compact $n$-manifold $N$. Then if x is the
projection of $R^n$ to some $x$-axis and $\nabla (x \circ f)$ 
is the gradient vector
field of the composition of maps $(x \circ f)$ and Ind is its index, we have
$$
\text{deg}(\gamma) = \chi (N) - \text{Ind}(\nabla (x \circ f))
$$

\endproclaim

The fact that $f$ has nonzero Jacobian on the boundary $M$ of course
means that $f$ is an immersion on $M$. Since the composition 
$(x \circ f)$ is a map from  $N$ to the real line $R$, the gradient can
be defined as in advanced calculus and gives a vector field on $N$. The
{\it index} of a vector field, which is a new term in this paper, is
in fact another topological invariant which predates the start of
Algebraic topology. It was defined for vector fields in two dimensions
by Poincar\'e in the late nineteenth century. Hopf generalized the
index of vector field for any dimensional manifold, and used the
concepts in his proofs of the Gauss-Bonnet-Hopf Theorem in 
\cite{H${}_1$} and \cite{H${}_2$}.

The index of a vector field  $V$  is an integer. It is closely related
to the degree of a map, yet it was defined historically before that
concept. In contrast to the degree of a map, the best definition of
index does not necessarily need homology theory. In fact it can be
defined by means of a simple formula which it must satisfy.


In 1929 Marston Morse \cite{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.

\proclaim{The Law of Vector Fields}
Suppose that $V$ is a vector field defined on $N$ which is not zero on
the boundary $M$. Then
$\text{Ind}\ V+\text{Ind}\ \partial_-V=\chi(N)$
where $\partial_-V$ is a vector field induced by $V$ and defined on that
part of the boundary $M$ where $V$ points inside.
\endproclaim

The vector field $ \partial_-V $ is induced by $V$ by considering the
component vector field of $V$ which is tangent to the boundary. Since
$\partial_-V$ is defined on a one dimension lower space, part of the
boundary $M$ of $N$, an inductive scheme of calculating the index
suggests itself. In fact, the Law of Vector fields 
is literally a self contained definition of the Index of
vector fields by induction \cite{gS}. This is elementary topology, but
tricky. Nontheless, the whole theory of $Ind(V)$ spins out from this
simple `A plus B equals C' equation. This equation is the key to the last
part of the story.
 
Among the facts which follow easily from the Law of Vector Fields are 
two well known properties of the index 
which combine with the Topological Gauss-Bonnet
Theorem to give all the previous global results labled Gauss-Bonnet:

If $V$ is a vector field with no zeros, then $\text{Ind}\ V = 0$.

If $V$ is a vector field on an odd dimensional manifold, then 
$\text{Ind}\ (-V) = -  \text{Ind}\ (V)$ where $-V$ is the vector field in
which every vector of $V$ is reversed.

The Gauss-Bonnet-Hopf Theorem follows immediately from the first
property, since if $f$ is an embedding, the vector field 
$\nabla (x \circ f)$ is just $\nabla x $, that is a constant vector
field parallel to the $x$-axis restricted to $N$. This has no zeros,
so applying the Topological Gauss-Bonnet with the index zero yields
the Gauss-Bonnet-Hopf Theorem. In fact, if $f$ is an immersion, the
vector field $\nabla (x \circ f)$ still has no zeros
(because $x \circ f$ has no critical points). So we get
Samelson and Haefliger's generalization of Gauss-Bonnet-Hopf from
embeddings to immersions.

On the other hand, Hopf's first version in \cite{H${}_1$}, that for
even dimensional $M$ {\it immersed} in $R^{n+1}$ we have 
$ \text { deg}( \gamma) =  \chi (M)/2$, follows from the second
property. If we choose the $x$-axis in the Topological Gauss-Bonnet
Theorem to run in the opposite direction, we reverse the direction of
the gradient. The other two terms in the Topological Gauss-Bonnet
Theorem certainly do not care which way the $x$-axis is going. So we
must have 
$\text{Ind}(\nabla (x \circ f)) = 0$. Thus 
$\text{deg}(\gamma) = \chi (N) = \chi (M) / 2$ . The last equality
follows because of the fact mentioned before, that the
Euler-Poincar\'e number for an even dimensional boundary is twice the
Euler-Poincar\'e number of its  bounded manifold. 

There is one point that remains to be clarified. Does every orientable 
$M$ which
can be immersed in a codimension 1 Euclidean space bound an $N$ so that
the immersion can be extended to an $f$? The answer is yes. But I must
admit that my way of proving this fact is immediate from a famous fact
of Thom's involving cobordism theory and Stiefel-Whitney numbers, which
was not available until the 1950's.

\head The accidental discovery
\endhead

The Law of Vector Fields was discovered by Morse in 1929
,
\cite{Mo}. In an
interesting parallel with Satz VI, Morse rarely referred to the result or
exploited its potentialities. Maybe it was 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 rediscovered it in the 1980's, it took almost a year
of questioning
before someone told me about \cite{Mo}.

Ten years ago I shared the common misconception about how mathematics
is created. I did not know the lessons of this story or of history. So
I was shocked to find what I regarded as an elementary relationship
satisfied by classical topological concepts index and Euler-Poincar\'e
number virtually unknown among topologists. So I thought, perhaps, that
there might be some interesting unknown consequences of the Law of
Vector Fields.  

I thought of a simple scheme to try to exploit the Law of Vector
Fields. I looked at interesting vector fields and plugged them into the
equation. I had some success with various choices. When 
I plugged in what I called pullback vector fields, which
generalize gradient vector fields, I got an equation involving the
normal degree and the Euler-Poincar\'e number \cite{G${}_1$}
\cite{G${}_2$}. It took a while before it
occurred to me that I had generalized the Gauss-Bonnet Theorem. 
A
simplified version of that result is the 
Topological Gauss-Bonnet Theorem as stated above. The simplification is
only that I stated the result here for gradients since it is a concept
familiar
from advanced calculus. In fact, pullback vector fields may even be
easier than gradients. 


\head Conclusions
\endhead

Mandelbrot in proposing the name ``fractals'' complained that
mathematicians do not give names to concepts and results. He was right.
In the deepest sense this 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 particular misconception is called the Matthew Effect by historians
of Science.
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
predessesors had trouble grasping them. What seems to be trivial now was
once the most difficult part of mathematics; infinity, velocity and
acceleration, arbitrary axioms, abstract groups, functions.

Finally, the story shows how mathematical challenges can have a great 
and good effect on
the development of mathematics, even if the challenges were based on
faulty points of view. 

As an application of this last lesson I will issue a
mathematical-historical challenge. Let us agree that a theorem 
generalizes a second
theorem if the second has a short proof in which the first plays the
predominate part. Then I propose the {\it Historical Fame Score} of any
Theorem: The HFS is the product of three numbers $H$, $F$, and $S$. 

$H$
will be the percent of the history of Mathematics covered between the
time the first interesting special case  was proved and its generalizing
theorem was proved. The beginning of the History of Mathematics will be
considered as 300 BC in honor of Euclid and the unavailability of
precise dates before that time.

$F$ will be the percent of mathematicians who know the most famous
special case of the generalizing theorem.

$S$ will be the percent of results closely related to the subject
matter of the generalizing theorem which receive new proofs, or new
insights, or new corollaries from the generalizing theorem.

The maximum score is one million. I estimate that the Topological
Gauss-Bonnet Theorem receives the maximum score. The challenge is,
find generalizations with comparable scores.



%================================



\Refs
\widestnumber\key {BK}

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\ref
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\key H${}_4$
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and Stanford 1956
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\key Ke
\by  M. E. Kervaire
\paper Courbure integrale generalisee et homotopie
\jour Math. Ann. 
\vol 131
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\pages 219--252
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\by  Imre Lakatos
\paper Proofs and Refutations: The Logic of Mathematical Discovery
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\by  J. W. Milnor
\paper On the immersion of n-manifolds in (n+1)-dimensional space
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\ref\key Mo \by
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\by John Stillwell
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\jour Springer-Verlag, New York 
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\by Michael Spivak 
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\endref

\end



 

