MORSE THEORY AND THE GAUSS-BONNET FORMULA Alina ...

MORSE THEORY AND THE GAUSS-BONNET FORMULA
Alina Nicoleta Badus
A THESIS
in
Mathematics
Presented to the Faculties of the University of Pennsylvania in Partial
Fulfillment of the Requirements for the Degree of Master of Arts
Supervisor of Thesis
Graduate Group Chairman
2007

A special thanks to Professor Christopher Croke for all his help.
ii

1 Introduction
The simplest version of the Gauss-Bonnet theorem was probably known in the time
of Thales and states that the sum of the interior angles of a triangle in the Euclidean
plane equals π. The nineteenth century version applies to a compact surface S with
smooth boundary ∂S and Euler characteristic χ(S):
�
S
�
KdA +
∂S
κg(s)ds = 2πχ(S)
where K is the Gauss curvature of S and κg is the geodesic curvature of the boundary.
In this paper we survey generalizations of the Gauss-Bonnet formula to higher-
dimensional manifolds, with or without boundary. Section 2 describes the work of
Hopf, the contribution of Allendoerfer and Fenchel, as well as the intrinsic proof by
Chern. In section 3 we present tools from Morse theory and the theory of isometric
immersions, which are used in section 4 to give an extrinsic proof of the Gauss-Bonnet
formula for closed manifolds embedded in Euclidean space. Section 5 generalizes these
ideas to manifolds with boundary, manifolds embedded in Euclidean spheres and non-
compact manifolds. While many of the results have already appeared in [10] and [5],
this is to our knowledge the first study of the Gauss-Bonnet formula for submanifolds
of the sphere.
1

2 Gauss-Bonnet formula in higher dimensions
The Euler characteristic generalizes easily to manifolds M of any dimension n:
n�
χ(M) = bk, where bk = dim Hk (M) is the k-th Betti number of M with coefficients
k=0
in Z, Q or Zp. An alternative definition begins with a smooth vector field V on M. A
point x ∈ M is a singular point of V if V (x) = 0. A singular point x is called isolated
if it has a neighborhood U such that U \ {x} contains no singular points of V .
The index of an isolated singular point of V is defined as follows. For ɛ > 0
small enough, the set Sɛ(x) = {y ∈ M : d(x, y) = ɛ} is diffeomorphic to S n−1 and it
encloses a ball Bɛ(x) = {y ∈ M : d(x, y) ≤ ɛ} that contains no singular points of V
except for x. For every y ∈ Sɛ(x) we can connect x and y via a unique minimizing
geodesic γ (if ɛ is small enough) and parallel translate V (y) along γ to obtain a vector
Vy ∈ TxM. As Vy points to a direction in the unit sphere S n−1 ⊂ TxM, we obtain a
map φ : Sɛ(x) → S n−1 . If we choose compatible orientations on the two spheres (for
example, via the exponential map expx) then the degree of φ is well-defined and we
say that the index of V at x is exactly the index of the map φ.
For a compact manifold M we have a theorem of Poincaré:
Theorem 2.1. The sum of the indices of a smooth vector field V with isolated sin-
gularities on M is equal to the Euler characteristic of M.
However, it is not entirely obvious how to generalize the integrands in the Gauss-
Bonnet formula. One approach is to use the fact that the Gaussian curvature of a
surface is the product of its principal curvatures. If M is a hypersurface in Euclidean
2

space, we can define principal curvatures in a canonical fashion, up to a choice of
normal vector (see section 3.2); the Gaussian curvature of M is then defined to be
the product of the principal curvatures. For an even-dimensional hypersurface, this
Gaussian curvature does not depend on the choice of normal vector, and in 1925 Hopf
proved the following:
Theorem 2.2. Let M 2n ⊂ R 2n+1 be a compact embedded hypersurface. Then
(2π) n �
χ(M) = 1 · 3 · · · (2n − 1) κ1 · · · κ2ndVM
M
where the κi’s are the principal curvatures of M and dVM is the Riemannian volume
form on M.
For an odd-dimensional hypersurface M 2n+1 , the sign of the expression κ1, ..., κ2n+1
does depend on the choice of a normal vector; more specifically, it changes sign if the
normal vector changes sign. In any case, χ(M) = 0 for odd-dimensional compact
manifolds, so we expect that any integral in a generalized Gauss-Bonnet formula
should be zero. The interesting case is the even-dimensional one, so for the rest of
this section we will assume that M has even dimension 2n.
2.1 Allendoerfer and Fenchel’s tube proof
In 1940 Allendoerfer [1] and Fenchel [7] succeeded in generalizing Hopf’s result to
all compact even-dimensional submanifolds of Euclidean space, using techniques from
Weyl’s theory of tubes. In general, if M is a submanifold of N we define the tube of
radius r around M as
3

T (M, r) = {x ∈ N : there is a geodesic γ of length l(γ) ≤ r from x meeting M
orthogonally}
Weyl [11] showed that, for a submanifold M 2n of R N :
Vol T (M, r) = (πr2 ) (N−2n)/2
((N−2n)/2)!
n�
i=0
k2i(M)r 2i
(N − 2n + 2)(N − 2n + 4)...(N − 2n + 2i)
where the k2i’s are integrals of certain rather complicated curvature functions. For
�
example, k0(M) = Vol M and k2(M) = τdV where τ is the scalar curvature of
M. For our purposes, the most interesting coefficient is the top one, k2n(M). The
connection between this coefficient and the Gauss-Bonnet theorem is provided by the
Pfaffian of a certain matrix of curvature forms, as described below.
M
Let A be a 2n × 2n antisymmetric matrix, with respect to the standard basis
x1, ..., x2n on R 2n . We can define a 2-form ω on R 2n by ω(X, Y ) = 〈AX, Y 〉 for
any vectors X, Y . If dx1, ..., dx2n are the dual 1-forms for the standard basis, then
we define the Pfaffian Pf(A) by ω n = Pf(A)dx1 ∧ ... ∧ dx2n. One can show that
Pf(A) = 1
2 n n!
�
σ∈S2n
ɛσAσ1σ2...Aσ2n−1σ2n where S2n denotes the set of permutations on
2n letters and ɛσ is 1 if σ is an even permutation, −1 otherwise. The Pfaffian also
satisfies Pf(A) 2 = det(A) and Pf(CDC T ) = Pf(D) det(C) for any antisymmetric
matrix D and arbitrary matrix C.
In our case, let {Ei, ..., E2n} be a local orthonormal frame. Define curvature forms
Ωij with respect to this frame as the 2-forms satisfying Ωij(X, Y ) = 〈R(X, Y )Ei, Ej〉
for any vector fields X, Y , where R is the curvature tensor of M. Then the matrix
4

of curvature forms Ω = (Ωij) is antisymmetric and we can define its Pfaffian as
the 2n-form Pf(Ω) = 1
2 n n!
�
σ∈S2n
ɛσΩσ1σ2...Ωσ2n−1σ2n on M. One can show (see [8] for
details) that Pf(Ω) does not depend on the choice of local orthonormal frame and
�
that k2n(M) = Pf(Ω).
M
If M 2n �
is a hypersurface, then it can be proven that 1·3···(2n−1)
�
κ1···κ2ndVM
M
equals Pf(Ω), so theorem 2.2 can be expressed as: (2π) nχ(M) = k2n(M).
M
Allendoerfer and Fenchel’s contribution is to relate the Euler characteristics and
top k coefficients of a manifold and the tube around it. Recall that M 2n was a sub-
manifold of R N . Without loss of generality we may replace N by 2N +1 and consider
the tube T (M, r) around M. Its boundary Mr = ∂T (M, r) is an even-dimensional
hypersurface if r is small enough, so (2π) N χ(Mr) = k2N(Mr). On the other hand,
Mr is a bundle over M with fiber S 2N−2n , so χ(Mr) = χ(M)χ(S 2N−2n ) = 2χ(M).
Finally, they were able to prove that k2N(Mr) = 2(2π) N−n k2n(M). Combining these
results, we obtain:
Theorem 2.3. Generalized Gauss-Bonnet theorem
(2π) n �
χ(M) = k2n(M) =
Proof. k2n(M) = k2N (Mr)
2(2π) N−n = (2π)N χ(Mr)
2(2π) N−n
M
Pf(Ω)
= (2π)N ·2χ(M)
2(2π) N−n = (2π) n χ(M) �
At the time it was not known whether every compact manifold can be embedded
in Euclidean space; an answer to this question was only given by Nash in the 1950’s.
However, the result is independent of the actual embedding of M and in 1944 Chern
5

gave an intrinsic proof of the formula. His paper [2] was so well received that
Allendoerfer and Fenchel’s formula is now known as the Gauss-Bonnet-Chern formula.
2.2 Chern’s intrinsic proof: a sketch
As above, let M 2n be a compact, even-dimensional manifold and consider its unit
tangent bundle UM. Let π be the projection map UM → M. Chern proved that
the pullback of the 2n-degree Euler form ω = 1
(2π) n/2 Pf(Ω) to UM is an exact form:
π ∗ (ω) = dΦ for some (2n − 1)-form Φ on UM.
If V is a smooth vector field on M with isolated non-degenerate zeroes then the
vector field X = V
�V �
has only isolated singularities; let U be the set of singularities
of X on M. We can view X as a section X : M \ U → UM. Chern proved that the
boundary of the image X(M \ U) is an (2n − 1)-dimensional cycle of UM. Then, by
Stokes’ theorem:
� �
ω = X ∗ �
(dΦ) =
M
M\U
X(M\U)
�
dΦ =
∂X(M\U)
With some work one can show that the last integral is exactly the sum of the
indices of the vector field X. By theorem 2.1, this is equal to χ(M).
3 Tools: Morse Theory and Integral Geometry
For the remainder of this paper we drop the requirement that M has even dimension.
Φ
6

3.1 Morse theory review
Let f : M n → R be a smooth function on a closed manifold M. We say that a
point p ∈ M is a critical point for f if the induced map f∗ : TpM → Tf(p)R is zero.
For such a critical point we define the Hessian of f at p as a symmetric bilinear
functional f∗∗ on TpM acting on vectors as follows: if v, w ∈ TpM then consider
their extensions V, W to vector fields tangent to M in a neighborhood of p and let
f∗∗(v, w) = Vp(W (f)) (where Vp = v). One can check that f∗∗ is symmetric and
independent of the extensions V, W .
The index of a bilinear functional A on a vector space V is defined to be the
maximal dimension of a subspace of V on which A is negative definite; the nullity of
A is the dimension of its nullspace, the space of all vectors v such that A(v, w) = 0
for all w ∈ V . We define the index of f at p to be the index of the bilinear map f∗∗
at p. We say that p is a degenerate critical point if the nullity of f∗∗ at p is zero and
a non-degenerate critical point otherwise.
If (x1, ..., xn) is a local coordinate system around p, then p is critical if and only
if ∂f
∂f
(p) = ... = ∂x1 ∂xn (p) = 0. If v = � ai ∂
∂xi |p and w = � bj ∂
∂xj |p are vectors in
TpM(for some constants ai, bj) then f∗∗(v, w) = �
i,j
aibj
7
∂2f (p). Therefore the
∂xi∂xj
matrix associated to f∗∗ with respect to the basis ∂
∂x1 |p, ..., ∂
∂xn |p is ( ∂2 f
∂xi∂xj
(p)) and
one can check that p is a nondegenerate critical point of f if and only if this matrix
is nonsingular. We can also prove that the index of f∗∗ is the number of negative
eigenvalues of this matrix, a number which does not depend on the coordinates chosen.

Definition 3.1. f is a Morse function if it has no degenerate critical points on M.
Morse functions are very useful for understanding the geometry and topology of
compact manifolds and the standard reference for their properties is [9]. Here are
some of the statements most relevant for our study:
Theorem 3.2. Morse Lemma. [9] Let p be a nondegenerate critical point of f and
let k be the index of f at p. Then there is a local coordinate system (x1, ..., xn) in a
neighborhood U of p with xi(p) = 0 for all i such that in the entire neighborhood U
we have f = f(p) − x 2 1 − ... − x 2 k + x2 k+1 + ... + x2 n.
Theorem 3.3. [9] If f is a Morse function on M and every M a = f −1 (−∞, a] is
compact, then M has the homotopy type of a CW-complex, with one cell of dimension
k for each critical point of index k.
Theorem 3.4. (Weak) Morse Inequalities. [9] If bk is the k-th Betti number of
M and Ck denotes the set of critical points of index k of a Morse function on M then
bk ≤ #Ck for any k and �
(−1) k #Ck = �
(−1) k bk = χ(M).
k
k
Morse theory also works on a manifold with boundary, if we count nondegenerate
critical points carefully. If M is a compact manifold with boundary ∂M and f :
M → R is a smooth function, consider f|M\∂M and f|∂M separately. A critical point
of f|M\∂M is a point p ∈ M \ ∂M where the induced map from Tp(M \ ∂M) to Tf(p)R
is zero, and degeneracy is defined in the same fashion as before. Since ∂M is an
(n − 1)-manifold, we can simply consider f|∂M : ∂M → R; a critical point is again
a point p ∈ ∂M where the map (f|∂M)∗ : Tp∂M → Tf(p)R is zero. Such a critical
8

point is degenerate if the Hessian (f|∂M)∗∗ has nullity > 0. The function f is called a
Morse function if both the maps f|M\∂M and f|∂M are Morse functions in the sense
of definition 3.1 above.
At every point x ∈ ∂M we can define a canonical unit vector νout pointing outward
with respect to M. Let f be a Morse function on M such that ∇f is never orthogonal
to νout. Consider the sets
Ck = {p ∈ M \ ∂M : the index of f|M\∂M at p is k}
Dk = {p ∈ ∂M : 〈∇f, νout〉 < 0 and the index of f|∂M at p is k}.
n�
Then (−1) k �n−1
#Ck + (−1) k #Dk = χ(M).
k=0
k=0
Example. Let M be a punctured torus T 2 − {pt}; it is not difficult to compute
χ(M) = −1. Consider two embeddings of M into R 3 and in both cases let f : M → R
be the height function corresponding to the z coordinate of the embedding.
In the left-most embedding, the entire boundary consists of critical points, all of
which are degenerate; therefore, the height function is not a Morse function for this
embedding. In the second embedding, there are 2 critical points on the boundary and
4 critical points in the interior of M, all of which are nondegenerate. However, the
point b on the boundary is not counted, as 〈∇f, νout〉 > 0 at b.
9

index(p) = 2, index(q) = index(r) = index(a) = 1, index(s) = 0, and we obtain
χ(M) = (−1) 0 · 1 + (−1) 1 · 3 + (−1) 2 · 1 = −1 as expected.
Example. Consider the annulus M ⊂ R 2 and the height function h representing
the y-coordinate. There are four critical points, all on the boundary, but only two of
them are used for computing χ(M) = (−1) 0 · 1 + (−1) 2 · 1 = 0.
3.2 Isometric immersions and the Gauss map
Let M n be a submanifold of R N . The unit normal bundle of M is a bundle with fiber
S N−n−1 and total space of dimension N − 1 defined as:
ν 1 M = {(x, u) : x ∈ M, u ∈ TxR N , u ⊥ TxM and �u� = 1}
For any (x, u) ∈ ν 1 M, the tangent space T(x,u)ν 1 M splits into a horizontal subspace
and a vertical subspace: T(x,u)ν 1 M = Hor T(x,u)ν 1 M ⊕Vert T(x,u)ν 1 M. In the canonical
metric on ν 1 M the horizontal and vertical subspaces are orthogonal to each other.
This splitting gives a canonical volume form dV ν 1 M on ν 1 M which locally coincides
with dVM ∧ dV S N−n−1.
10

Since we are working with an embedded manifold, we will also need to consider
the ”shape” of the embedding, given by the second fundamental form. If f : M → N
is an isometric embedding, we define the second fundamental form (or shape operator
in [6]) to be a map that associates to every normal vector u ⊥ TxM the unique
self-adjoint operator Au : TxM → TxM satisfying 〈Au(v), w〉 = 〈∇V W, u〉. Here V
and W are extensions of v and w, respectively, to vector fields tangent to M in a
small neighborhood of x, and ∇ is the covariant derivative in the ambient manifold
N. One can show that this definition does not depend on the extensions chosen and
that Au(v) = −(∇vU) T where U is a vector field normal to M extending u on a small
neighborhood and T denotes the tangential component of the derivative. Again, this
equality does not depend on the extension U.
As a self-adjoint operator, Au has real eigenvalues κ1, ..., κn called the principal
curvatures of the embedded submanifold M in the direction of u. The determinant
det Au = κ1 · ... · κn is a generalization of the Gaussian curvature of a surface. In the
case of a hypersurface, det Au is called the Gauss-Kronecker curvature.
Recall that M is a submanifold of R N . For some x ∈ M we can interpret TxM as
an n-subspace of R N , and view ν 1 xM as the set of all unit vectors in R N orthogonal
to TxM. Then we can define the Gauss map G : ν 1 M → S N−1 ⊂ R N by G(x, u) = u.
It is known (see [3] or [10]) that G ∗ (dVS)G(x,u) = |det Au|(dV ν 1 M)u for all u ∈ ν 1 M.
In particular, u ∈ ν 1 M is a regular point for G if and only if Au is invertible.
11

4 An Extrinsic Proof
4.1 A proof of the Gauss-Bonnet theorem
Let M n be a compact manifold embedded isometrically in R N for some N > n with
unit normal bundle ν 1 M.
Theorem 4.1. Extrinsic Gauss-Bonnet Theorem
�
ν 1 M
�
ν 1 M
det AudV ν 1 M(u) = χ(M) Vol(S N−1 )
Notice that if n is odd then det A−u = − det Au. By Fubini’s theorem,
� �
�
det AudVν1M(u) = det AudVν1 pMdp = 0dp = 0, and χ(M) = 0.
M
ν 1 pM
The proof is a beautiful application of Morse theory and integral geometry and
we will describe it in the remainder of this section.
Definition 4.2. For a unit vector v ∈ S N−1 , define the height function hv : M → R
by hv(p) = 〈p, v〉. In this context, p represents a point in M as well as the vector
connecting the origin of R N with that point.
Proposition 4.3. S0 = {v ∈ S N−1 : hv is a Morse function } has full measure in
S N−1 .
From the definition of height functions we can deduce that p ∈ M is a critical point
for the height function hv if and only if v is orthogonal to TpM. This is equivalent
to v being parallel to some normal vector u ∈ ν 1 M based at p. Moreover, there is a
close connection between height functions and the second fundamental form:
M
12

Lemma 4.4. If u ∈ ν 1 M is based at p ∈ M then the Hessian of hu at p is equal to
the second fundamental form Au at p.
Proof. Let f : R n → R N be a local parametrization of M in a neighborhood U
around p, with f(0) = p. Then X1 = ∂
∂x1 , ..., Xn = ∂
∂xn form a basis for TqM, for all
q ∈ U. As both the Hessian and the second fundamental form are linear operators, it
suffices to show that hess hu(p)(Xk, Xl) = 〈Au(Xk), Xl〉 for all k, l. This is equivalent
to proving that ∂ 2 hu
∂xk∂xj = 〈∇Xk Xj, u〉 for all k, l.
Write f(x1, ..., xn) = (f1(x1, ..., xn), ..., fN(x1, ..., xn)) and let e1, ..., eN be the stan-
dard basis of R N . Without loss of generality we may assume that u = eN and therefore
hu(f(x1, ..., xn)) = fN(x1, ..., xn). Also, Xk = � ∂fi
ei and Xl =
∂xk
� ∂fj
ej. We can
∂xl
now compute:
∇XkXl = �
∇Xk
j
(∂fj ej) =
∂xl
� ∂
j
2fj ej +
∂xl∂xk
� ∂fi
∇ei
∂xk i,j
ej = � ∂
j
2fj ej
∂xl∂xk
Finally, 〈∇XkXl, u〉 = 〈∇XkXl, eN〉 = ∂2fN ∂xl∂xk
as desired. �
Corollary 4.5. p is a nondegenerate critical point for hu iff Au is invertible.
Proof of proposition 4.3. Corollary 4.5 implies that p is a nondegenerate
critical point for hv if and only if v is a regular value of the Gauss map G. But G is
a smooth map, so by Sard’s theorem the set of its critical values has measure zero in
S N−1 . As S0 is exactly the set of regular values of G, we can conclude that S0 has
full measure in S N−1 . �
Lemma 4.6. For any v ∈ S0, G −1 (v) is a finite set. Moreover, there exists a neigh-
borhood U of v such that G| G −1 (U) : G −1 (U) → U is a finite covering map.
i
j
13

Proof. Suppose G −1 (v) was infinite. Since ν 1 M is compact, we can extract
a convergent sequence {ai} ⊂ G −1 (v) converging to some element a ∈ ν 1 M. By
continuity we must have G(a) = v as well, so a is a regular point of G. Therefore, for
any neighborhood V1 of a there exists some element ai that also maps to v. However,
if a is a regular point of G then there exists a neighborhood V2 of a such that G|V2 is
a homeomorphism; this contradicts the existence of the sequence ai.
As v is regular and has finitely many preimages, we can conclude that v has a
neighborhood U such that G−1 � �
(U) = U1 ... Uk is a finite disjoint union of open
sets Ui, each of which is homeomorphic to U. �
Corollary 4.7. For any function f : ν 1 M → R we have
�
ν 1 M
�
f(u)|det dG|dVν1M(u) =
�
SN−1 u∈G−1 (v)
f(u)dV S N−1(v)
Technically, the corollary should only allow us to compute integrals over S0. How-
ever, S0 has full measure in S N so we can replace integrals over S0 with integrals over
S N .
Proof of theorem 4.1
Define Ck(v) = {(v, p) ∈ S0 × M : p is a critical point of index k for hv} for every
n�
v ∈ S0. Then (−1) k #Ck(v) = χ(M) by the Morse inequalities.
k=0
14

Apply the previous corollary to the function f(u) = (−1) index Au :
�
ν 1 M
�
det AudVν1M(u) =
�
=
�
=
ν 1 M
(−1) index Au |det dG|dV ν 1 M(u)
�
SN−1 u∈G−1 (v)
SN−1 k=0
4.2 The Chern-Lashof results
(−1) index Au dV S N−1(v)
n�
(−1) k #Ck(v)dVSN−1(v) = χ(M) Vol(S N−1 )
In [3] and [4], Chern and Lashof study the absolute total curvature of a mani-
fold M immersed in R N . The total absolute curvature is defined as AT C(M) =
�
M
�
ν 1 p M|det dGu|dudp. They proved that:
Theorem 4.8. [3], [4] Let M n be a compact manifold immersed in RN . Then:
n�
i) AT C(M) ≥ bk · Vol(S N−1) ; in particular, AT C(M) ≥ 2 Vol(SN−1 );
k=0
ii) If AT C(M) < 3 Vol(SN−1 ) then M is homeomorphic to Sn .
iii) If AT C(M) = 2 Vol(S N−1 ) then M n is embedded as a convex hypersurface in
an (n + 1)-dimensional linear subspace of R N .
The purpose of this subsection is to prove the first two parts of this theorem using
Morse theory methods for embedded manifolds. Chern and Lashof also use Morse
theory and the Gauss map, but they use differential forms to show that almost all
the height functions are Morse functions.
15
�

Apply corollary 4.7 for the function f : ν 1 M → R, f(u) = 1; then:
�
AT C(M) =
�
=
�
=
M
�
ν 1 pM
SN−1 u∈G−1 (v)
SN−1 k=0
�
|det dGu|dudp =
�
1 · dV S N−1(v)
ν 1 M
n�
#Ck(v)dVSN−1(v) ≥
|det dGu|dV ν 1 M
n�
bk · Vol(S N−1 )
But b0 ≥ 1 for any manifold and bn = 1 if M is compact. Therefore,
desired.
k=0
16
n�
bk ≥ 2 as
If AT C(M) < 3 Vol(S N−1 ) then there exists a set A ⊂ S0 with positive measure
such that for any v ∈ A the height function hv has less than 3 critical points on M.
But if M is compact then every such hv has at least two critical points: an absolute
minimum and an absolute maximum. We can conclude that all height functions hv
with v ∈ A have exactly two critical points. By a theorem of Reeb’s (see [9]), if M
is a compact manifold and f is a differential function on M with only two critical
points, both of which are non-degenerate, then M is homeomorphic to a sphere. This
completes the proof of (ii).
The proof of part (iii) is more complicated. If AT C(M) = 2 Vol(S N−1 ) then hv
must have exactly two critical points on M for almost all v ∈ S0. Chern and Lashof
prove that if M is not contained in a linear subspace of dimension (n+1) then there is
a neighborhood U ∈ S N−1 such that hv has at least three critical points for all v ∈ U.
This is a contradiction and shows that M must be embedded in an (n + 1)-linear
subspace. The proof that M is actually a convex hypersurface is also very interesting
n=0

ut falls outside the scope of this discussion.
5 Generalizations
5.1 Manifolds with boundary
Let M n be a compact submanifold of R N with boundary ∂M. Then ∂M is an
embedded (n − 1)-submanifold of R N and at every point of ∂M we can define a
unique outward pointing unit vector νout(x) tangent to M.
The discussion in section 3.1 suggests that we should consider the bundle
NM = ν 1 (M \ ∂M) � ν 1 +∂M where
ν 1 (M \ ∂M) = {(x, v) : x ∈ M \ ∂M, v ∈ TxR N , v ⊥ TxM and �v� = 1} and
ν 1 +∂M = {(x, v) : x ∈ ∂M, v ∈ TxR N , v ⊥ Tx∂M, �v� = 1 and 〈v, νout(x)〉 < 0}
Notice that ν 1 (M \ ∂M) and ν 1 +∂M are bundles with the same total dimension
N − 1. For ν 1 (M \ ∂M), the base space M \ ∂M has dimension n and the fiber has
dimension N − n − 1; for ν 1 +∂M the base space ∂M has dimension n − 1, but the fiber
has dimension N −n. The volume form on ν 1 (M \∂M) is locally a product dVM\∂M ∧
dV S N−n−1 and the volume form on ν 1 +∂M is locally the product dV∂M ∧dV H N−n, where
H N−n denotes the round unit hemisphere of dimension N − n.
We can still define a smooth Gauss map G : NM → S N−1 by G(x, u) = u as
a map between manifolds of the same dimension N − 1. As in section 3, one can
prove that G ∗ (dVS)G(u) = |det Au|(dVNM)u for all u ∈ NM. In this formula, Au is
17

the second fundamental form of M \ ∂M if u is based at some x ∈ M \ ∂M or the
second fundamental form of ∂M if u is based on the boundary.
Theorem 5.1. Gauss-Bonnet for manifolds with boundary
�
�
det AudVν1M\∂M(u) +
u∈ν 1 (M\∂M)
u∈ν1 + ∂M
det AudVν1 + ∂M(u) = χ(M) Vol(S N−1 )
Proof. For every v ∈ S N−1 we consider the height function hv : M → R,
hv(p) = 〈v, p〉. The result follows via essentially the same proof as for theorem 4.1.�
Dillen and Kühnel offer an alternative proof of the theorem via the tube method,
using the relation χ(NM) = (1 + (−1) N )χ(M); the details can be found in [5].
5.2 Submanifolds of the sphere
Let M n be a compact manifold without boundary isometrically embedded in a unit
sphere S N . Applying Morse theory to distance functions on M will allow us to
obtain yet another version of the Gauss-Bonnet formula, with the advantage that the
integrand has a very interesting geometric interpretation.
Consider the tube of radius r around M as defined in section 2.1. If r is small
enough, the tube is an embedded submanifold of S N but for r = π the tube has self-
intersections. However, its signed volume contains important topological information
about M. More precisely, consider the normal disk bundle:
ν π M = {(x, w) : x ∈ M, w ∈ TxS N , w ⊥ Tx and �w� ≤ π}
18

and the map F : ν π M → S N , F (x, w) = expxw. Then the tube of radius π around M
is T (M, π) = F (ν π M) and its signed volume is defined as
�
SVol T (M, π) =
ν π M
det dFudVν π M(u)
where dV olν π M is the canonical volume form on ν π M.
Example. Consider M = S 1 embedded in S 2 as a great circle. Then ν π S 1 is a
cylinder S 1 × [−π, π] and the tube T (S 1 , π) = S 2 has volume 4π.
However, the signed volume SVol T (S 1 , π) is zero, by the following theorem:
Theorem 5.2. Gauss-Bonnet for submanifolds of the sphere.
SVol T (M, π) = χ(M) Vol(S N )
In order to prove this theorem, consider the distance function dp : M → R, x ↦→
d(p, x) for some fixed p ∈ S N . Notice that x is a critical point of dp if and only if
p = F (x, w) for some (x, w) ∈ ν π M.
19

Lemma 5.3. S0 = {p ∈ S N : dp is a Morse function } has full measure in S N .
Proof. From [9] we know that x ∈ M is a degenerate critical point of dp for
some p = expxw if and only if p is a focal point of M. Therefore, dp is a Morse
function whenever p is not a focal point of M. But focal points are critical values of
the normal exponential map (see [6]), which is a smooth map; therefore, by Sard’s
theorem, the set of focal points of M has measure zero in S N . �
The Morse index theorem for distance functions (see [9]) states that the index of
x ∈ M as a nondegenerate critical point of dp is exactly the number of focal points of
M with base x situated on the segment from x to p, counted with multiplicities. As
focal points are critical points of F , we can conclude that for every point q = F (x, v)
on the segment from x to p the index of dF(x,v) equals the number of focal points
between q and x. The inverse image of the set of focal points thus divides ν π M into
regions with indices that increase as we move further from M.
In the following proof we essentially compute the volume of all these regions, with
the corresponding indices. It might be possible to study the regions individually, as
they contain information about the number of critical points of distance functions
and, implicitly, about the topology of M.
Proof of theorem 5.2. For any p ∈ S0, let Ck(p) be the set of critical points of
20
�

dp of index k. By theorem 3.2,
�
ν π M
n�
(−1) k #Ck(v) = χ(M). Then:
k=0
det dFudVνπ �
M(u) =
�
=
�
=
ν π M
(−1) index dFu |det dFu|dVν π M(u)
�
S0 u∈F −1 (p)
S0 k=0
(−1) index dFu dV S N (p)
n�
(−1) k #Ck(p)dVSN (p) = χ(M) Vol(S N )
Remark. We can also compute explicitly the signed volume of a tube of radius
r around M. The formula is
�
SVol T (M, r) =
νM
� r
(sin t) N−n−1 (cos t + λ1 sin t)...(cos t + λn sin t)dtdVνM
0
where νM is the entire normal bundle of M inside S N , dVνM is the canonical volume
form on νM and λ1, ..., λn are the eigenvalues of a second fundamental form. When
n is odd the integral equals zero, as expected, since χ(M) = 0 for compact, odd-
dimensional manifolds. However, attempts to compute the integral when M is even-
dimensional have not been fruitful.
Remark. We can also ask whether every compact manifold can be isometrically
embedded in some S N , perhaps after a suitable scaling. Notice that a flat torus T n
can be isometrically embedded (with scaling) in S 2n−1 via the map:
(x1, ..., xn) ↦→ 1
n (cos x1, sin x1, ..., cos xn, sin xn)
By the Nash embedding theorem, any compact manifold can be isometrically
embedded in Euclidean space, which can be embedded in a flat torus, perhaps after
21
�

some scaling. Therefore, any compact manifold can be isometrically embedded in an
Euclidean sphere if we allow scaling.
5.3 Noncompact complete manifolds
Another possible generalization is to drop the assumption of compactness. Consider a
complete noncompact surface S. We should require that χ(S) is finite, which implies
that S must have finite topological type: it must have finite genus and only finitely
many ends. Also, we should require that S has an absolutely integrable Gaussian
�
�
curvature: |K|dA < ∞ in order to guarantee that the integral KdA converges
S
and is finite. In this context we have the following:
�
Theorem 5.4. Cohn-Vossen. 2πχ(S) −
for any complete, noncompact surface.
S
�
KdA ≥ 0. In particular,
S
S
22
KdA ≤ 2π
The Gauss-Bonnet formula might still hold in some cases, or one can study the
�
curvature defect 2πχ(S) − KdA. For a recent study of this area and higher dimen-
sional generalizations, see [5].
S

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