A Semiclassical Heat Kernel Proof of

the Poincaré-Hopf Theorem

arXiv:1302.6895v3 [math.DG] 5 Sep 2014

Matthias Ludewig
November 12, 2018

Universität Potsdam / Institut für Mathematik

Am Neuen Palais 10 / 14469 Potsdam, Germany
matthias.ludewig@uni-potsdam.de
Tel. +49-331-977-1248

Abstract
We consider the semiclassical asymptotic expansion of the heat kernel coming from
Witten’s perturbation of the de Rham complex by a given function. For the index,
one obtains a time-dependent integral formula which is evaluated by the method of
stationary phase to derive the Poincaré-Hopf theorem. We show how this method is
related to approaches using the Thom Form of Mathai and Quillen. Afterwards, we
use a more general version of the stationary phase approximation in the case that
the perturbing function has critical submanifolds to derive a degenerate version of
the Poincaré-Hopf theorem.

1 Introduction
Wenn man 2 Wege hat, so muss man nicht bloss diese Wege gehen oder
neue suchen, sondern dann das ganze zwischen den beiden Wegen liegende
Gebiet erforschen.
Given two routes, it is not right to take either of these two or to look for a
third; it is necessary to investigate the area lying between the two routes. 1

(David Hilbert)

The connection between semiclassical Analysis and Morse theory was discovered by
Witten [Wit82] about thirty years ago and got a lot of attention ever since. The main
1
Cited in [Hug06]

1
idea is inspired by heuristics coming from physics and goes as follows: The Laplace part
of a Schrödinger operator ~2 ∆ + V is responsible for diffusion, while the potential term V
causes concentration at its cavities. Therefore, when simultaneously taking limits ~ ↓ 0
and t ↑ ∞ in some appropriate way (where t is the time parameter in the heat equation
corresponding to the operator ~2 ∆ + V ), one expects the particles to concentrate near the
minima of the potential. With Morse theory on the other hand, one can obtain topological
information of the underlying space by investigating the critical points of a given function.
Witten intertwines the Euler operator acting on differential forms with a vector field X,
this perturbation depending on a small parameter ~. That way he obtains a Schrödinger
type operator whose semiclassical eigenfunction approximations he uses to construct the
Morse complex.
The Morse inequalities directly imply the Poincaré-Hopf theorem, which states that the
Euler characteristic of M is determined by the critical points of X, more precisely
X
χ(M) = (−1)ν(p) ,
{X(p)=0}

where the index ν(p) of a critical point p is equal to the number of negative eigenvalues
of the linearization ∇X|p .
Another way to get this theorem uses the Thom form U of Mathai and Quillen: By
the transgression formula for the Thom form, the pullbacks of U along any two vector
fields are cohomologous; on the other hand, the Euler class of M is the pullback of U
along the zero vector field. The pullback of U along the vector field Xt := t1/2 X gives a
differential form on M and the Poincaré-Hopf theorem follows then by evaluating Xt∗ U
with the method of stationary phase [BGV96, Thm. 1.56].
These proofs seem conceptually very different at first: Witten uses the low-lying eigen-
functions of his operator to construct a complex chain homotopic to the de Rham complex
whence the theorem follows by an argument of homological algebra, while the other proof
derives an integral formula that interpolates between the Gauss-Bonnet-Chern theorem
and Poincaré-Hopf. In this article, we show that in fact one can use the semiclassical
asymptotics [BP10] of the heat kernel of Witten’s operator to recover the interpolation
formula appearing in the Thom form proof.
More precisely, from the semiclassical heat kernel asymptotics, we derive the integral
formula Z
2
χ(M) = α(t)e−t|X| for all t > 0,
M

where α(t) is some function depending polynomially on t. We then use ideas from Getzler’s
proof of the local index theorem [Get85] to explicitly calculate α(t) in terms of the curva-
ture of M and Taylor coefficients of X. It turns out that the integrand above is nothing
but the Thom form, pulled back via Xt .
We show that in the limit t ↓ 0, the integral formula above yields the Gauss-Bonnet-
Chern theorem and in the limit t ↑ ∞, the integral can be evaluated with the method
of stationary phase to obtain the Poincaré-Hopf theorem. In this sense, the t-dependent
integral formula above interpolates between two classical theorems.

2
 More generally, by allowing the critical set to be a disjoint union of submanifolds of
M, we obtain a degenerate version of the Poincaré-Hopf theorem (Thm. 6.3). Let us
remark that the degenerate Morse inequalities (Morse-Bott inequalities) that imply the
degenerate Poincaré-Hopf theorem have been proved with heat kernel methods by Bismut
[Bis86], but the integrands are not explicitly calculated in terms of curvature. For other
treatments of the Morse-Bott inequalities, see [Bot54] or [AB95], for example.
This article is organized as follows: At first, we briefly review some basic notions
regarding the Clifford algebra and the exterior algebra of a Euclidean vector space V .
In Section 3, we state the needed results about the semiclassical expansion of the heat
kernel of Witten’s operator. Afterwards, we introduce Getzler symbols and use them to
calculate the relevant integrands. In Section 5, we put these results together to prove
the stated integral formula for the index and the consequences of it. The last section is
dedicated to the degenerate case, in which we derive the Morse-Bott Theorem 6.3. This
needs a longer calculation because of the more complicated nature of the corresponding
stationary phase expansion.
Acknowledgments. It is a pleasure to thank Christian Bär and Florian Hanisch for
helpful discussion as well as Potsdam Graduate School and the Fulbright Commission for
financial support.

2 The Clifford Symbol on the Exterior Algebra

We begin with a short recap of the filtrations, gradings and symbol maps associated to
the exterior algebra.
In this section, let V be a Euclidean vector space of dimension n. We consider two
different gradings on the exterior algebra ΛV : the Z-grading induced by the degree of
forms and the Z2 -grading, where the even (odd) part is the space of even-degree (odd-
degree) forms. The latter gives ΛV the structure of a superalgebra; the associated grading
operator Ξ is by definition the operator that is the identity on even-degree forms and minus
the identity on odd-degree forms.
For v ∈ V , we define the Clifford multiplications ΛV −→ ΛV by

c(v) = ε(v) − ι(v), b(v) = ε(v) + ι(v). (2.1)

Here, ε and ι denote exterior and interior multiplication, where to define the latter, we
use the Euclidean structure on V . For v, w ∈ V , we have the Clifford relations

[c(v), c(w)]s = −2hv, wi, [b(v), b(w)]s = 2hv, wi, [c(v), b(w)]s = 0,

where [ · , · ]s denotes the super commutator. Under the isomorphism (of vector spaces)
ΛV ∼
= Cℓ(V ) given by
v1 ∧ · · · ∧ vk 7−→ v1 · · · vn ,

3
c(v) acts on a homogeneous element a ∈ Cℓ(V ) by Clifford multiplication with v from
the left, while b(v) acts by Clifford multiplication with v from the right, followed by
multiplication with (−1)|a| , where |a| is the Clifford order of a. The endomorphism space
End(ΛV ) is generated as an algebra by the elements c(v), b(v) with v ∈ V .

Lemma 2.1. The grading operator Ξ can be represented by

n(n+1)
Ξ = (−1) 2 c1 · · · cn b1 · · · bn

where we wrote cj = c(ej ), bj = b(ej ) for an orthonormal basis e1 , . . . , en of V . Ξ does

not depend on the choice of this orthonormal basis.

Proof. Let I = (i1 , . . . , il ), i1 < · · · < il be a multi-index. Writing eI = ei1 · · · eil , we have

b1 · · · bn eI = (−1)l b1 · · · bn−1 eI · en = (−1)l+l+1 b1 · · · bn−2 eI · en · en−1

n(n−1)
= · · · = (−1)nl+1+2+···+n−1 eI · en · · · e1 = (−1)nl+ 2 eI · en · · · e1 .

Now (
(−1)l+1 eI if j ∈
/I
ej · eI · ej = ,
(−1)l eI if j ∈ I
so that
n(n−1) n(n−1) P
c1 · · · cn b1 · · · bn eI = (−1)nl+ 2 e1 · · · en · eI · en · · · e1 = (−1) 2
+ j ∈I
/ 1 I
e
P
Now we have j ∈I / 1 = n − l ≡ −n + l mod 2 so that

n(n+1)
(−1) 2 c1 · · · cn b1 · · · bn eI = (−1)l eI ,

which was the claim. 

The algebra End(ΛV ) has a filtration and a bi-filtration, which we will both use: An
element A ∈ End(ΛV ) has (Clifford-) bi-order (k, l) or lower if it can be written as
XX
A= AIJ cI bJ , (2.2)
|I|≤k |J|≤l

where we wrote cI := ci1 · · · cim for a multiindex I = (i1 < · · · < im ) and similarly for bJ .
Here |I| := m is the length of the multi-index. We say that A has (Clifford-) order k or
lower if it has bi-order (k, n) or lower.
For such an element A of order k, we define its k-th Clifford symbol by
X X
σ
/ k (A) := σ
/ k,•(A) := AIJ eI ⊗
b bJ ∈ ΛV ⊗ b Cℓ(−V ),
|I|=k |J|≤n

4
where ⊗b denotes the super tensor product and Cℓ(−V ) denotes subalgebra2 of endomor-
phisms generated by the bj , which super-commutes with the Clifford action induced by
c. The Clifford bi-symbol of an element A of bi-order (k, l) is defined as
XX
σ
/ k,l (A) := AIJ eI ⊗b eJ ∈ ΛV ⊗ b ΛV.
|I|=k |J|=l

It is straightforward to check that all these definitions do not depend on the choice of
orthonormal basis. Furthermore, both symbols just defined are compatible with the multi-
plication in the following sense: If A, B ∈ End(V ) have Clifford order k and u (or bi-order
(k, l), (u, v)), then AB is of Clifford order k + u (respectively bi-order (k + u, l + v)) and
we have

σ
/ k+u (AB) = σ
/ k (A)/
σ u (B) and σ
/ k+u,l+v (AB) = σ
/ k,l (A)/
σ u,v (B) (2.3)

This identity would be wrong if we hadn’t used the super tensor product in the target
spaces.

Definition 2.2. The supertrace on End(ΛV ) is defined by str(A) := tr(ΞA), where Ξ is

the grading operator.

Proposition 2.3. If A has bi-order (k, l) with k < n or l < n, then str (A) = 0. On the
other hand, str(Ξ) = 2n .

Proof. Let A = cj1 · · · cjk bi1 · · · bil ∈ End(ΛV ) with k < n and let u ∈
/ {j1 , . . . jk }. Then
straightforward calculation shows

[cu cj1 · · · cjk bi1 · · · bil , cu ]s = (−1)k+l 2 cj1 · · · cjk bi1 · · · bil ,

so that A is a super-commutator. However, str vanishes on super-commutators [BGV96,

Prop. 1.31]. In the case that k = n and l < n, replace cu by bu for some u ∈
/ {i1 , . . . , il }.
2 n
Finally, by definition, str(Ξ) = tr(Ξ ) = tr(id) = 2 . 

Corollary 2.4. We have

n(n+1)
str(A) = (−1) 2 2n σ b vol
/ n,n (A), vol ⊗

where vol = e1 ∧ · · · ∧ en for an (oriented) orthonormal basis e1 , . . . , en .

Remark 2.5. Notice that this does not depend on an orientation of M. Changing the
orientation turns vol into −vol, but the two signs cancel each other.

2
By (2.1), this is a Clifford algebra associated to the negative of the scalar product on V , hence the
notation.

5
Proof. For A of the form (2.2) (with k = l = n), we have
X  n(n+1)
str(A) = str AIJ cI bJ = str(A[n][n] c1 · · · cn b1 · · · bn ) = A[n][n](−1) 2 2n ,
I,J

by Lemma 2.1 and Prop. 2.3, where we wrote [n] for the multi-index (1, 2, . . . , n). On the
other hand,
b vol,
σ/ n,n (A) = A[n][n]vol ⊗

which gives the result. 

3 The Semiclassical Heat Kernel Expansion

Definition 3.1. Let (M, g) be a Riemannian manifold of dimension n and let E be a real
or complex vector bundle over M equipped with a scalar product (or positive definite
Hermitian form respectively). We say that an operator of the form

H~ = ~2 L + ~W + V, (3.1)

is of Schrödinger type, if L is a formally self-adjoint operator of Laplace type, i.e. locally

it has the form n
X ∂2
ij
L = −idE g i ∂xj
+ lower order terms,
ij=1
∂x
and V and W are symmetric endomorphism fields (we adopt the sign convention such
that the eigenvalues of Laplace type operators tend to +∞).

Example 3.2 (The Witten Operator). Let E = ΛT ∗M. For a one-form ξ ∈ Ω1 (M),
define
D~ = ~D + b(ξ), where D = d + d∗ .

Then D~2 is of Schrödinger type, as straightforward calculation shows that in local coor-
dinates, we have
n
X
W = [D, b(ξ)] = c(dxi ) b(∇i ξ), and V = |ξ|2 . (3.2)
i=1

Witten [Wit82] uses this operator, but with a different normalization (~ 7→ t−1 ). Also see
[Roe98, p. 125] and [Zha01, Chapter 5].

Suppose that M is compact. Then it is well-known that for each ~ > 0 fixed, a
Schrödinger type operator H~ is an unbounded operator in L2 (M, E) which is self-adjoint
on the Sobolev space H 2 (M, E) and has eigenvalues tending to +∞ (see e.g. [Gil84, 1.6]).
For t > 0, the operator e−tH~ (defined by functional calculus) has a smooth integral kernel


k~ ∈ Γ∞ M × M × (0, ∞), E ∗ ⊠ E . (3.3)

6
that depends smoothly on ~ [BGV96, Thm. 2.48]. Here E ∗ ⊠ E denotes the bundle
pr∗1 E ⊗ pr∗2 E over M × M, where pr1,2 : M × M −→ M are projections on the first and
second factor respectively. Its fiber over (p, q) ∈ M ×M is given by Ep∗ ⊗Eq ∼
= Hom(Ep , Eq ).
On any complete Riemannian manifold M, we call
 
2 −n/2 1 2
e~ (p, q, t) = (4πt~ ) exp − 2 d(p, q)
4t~
the Euclidean heat kernel, because k~ = e~ if L is the usual Laplace operator on functions
in Euclidean space and W = V = 0. Here, d(p, q) is the Riemannian distance between p
and q, so that e~ is smooth on the set

M ⊲⊳ M = (p, q) ∈ M × M | p is not a cut-point of q ,

which is a dense open neighborhood of the diagonal. (Two points (p, q) are cut-points of
each other if either there are several geodesics of minimal length joining p and q or there
is a Jacobi field along the unique shortest geodesic connecting p and q that vanishes at
both p and q [BP10].)
The ~-asymptotics of the heat kernel k~ were developed in [BP10]. We briefly re-
call the relevant ideas: Because k~ is the solution kernel to the heat equation, we have
(∂t + H~ ) k~ ( · , q) = 0 for each q ∈ M and ~ > 0. If on M ⊲⊳ M we make the ansatz
X∞
k~ ∼ e~ ~j Φj (3.4)
j=0

for an asymptotic expansion of k~ in ~, we can formally (i.e. termwise) apply the heat
operator (∂t + H~ ) to this expression and straightforward calculation shows that the result
is again of the form e~ times some power series in ~. One obtains that in order to have
this power series vanish, the Φj have to fulfill the recursive transport equations
 
∂
t + ∇V + G + tV Φj ( · , q) = −tW Φj−1 ( · , q) − tLΦj−2 ( · , q), (3.5)
∂t
for each q ∈ M, where
 
1 n 1
V = grad[d( · , q)2], G=− 2
+ ∆[d( · , q) ] . (3.6)
2 2 4
Therefore one makes the following definition.
P
Definition 3.3. Let ∞ j
j=0 ~
Φj be a formal power series with coefficients Φj in the space
Γ∞ M ⊲⊳ M × [0, ∞), E ∗ ⊠ E . Then the formal expression
X∞
b
k~ := e~ ~j Φj
j=0

is called semiclassical heat kernel expansion if for all q ∈ M, the coefficients Φj fulfill the
recursive transport equations (3.5) with the initial condition Φ0 (q, q, 0) = idEq .

7
 Regarding this, we have the following theorem.
Theorem 3.4 (Bär, Pfäffle). [BP10, Lemma 3.1] Let M be a complete Riemannian
manifold, E a vector bundle with scalar product over M and H~ be of Schrödinger type.
Then there exists a unique semiclassical heat kernel expansion.
Furthermore, for each q ∈ M, the first transport equation (3.5) has a unique solution
for each prescribed value for Φ0 (q, q, 0), while for j > 0, the j-th transport equation has a
unique solution for each right hand side.
For the solution theory of equations of the form (3.5), also see [Lud14].
The relation between the formal heat kernel and the true heat kernel is the following.
Theorem 3.5 (Bär, Pfäffle). [BP10, Thm. 3.3] Let M be compact. Then for all T > 0,
m, k ∈ N0 and N > 2n + 2m + 2k, there exist constants C > 0, ~0 > 0 such that
∂k  XN
j

sup k
k ~ − χe~ ~ Φj ≤ C~N −2n−2m−2k+1
t∈(0,T ] ∂t j=0
C m (M ×M )

for all ~ < ~0 . Here, χ is a smooth cutoff function that is compactly supported in M ⊲⊳ M
such that χ ≡ 1 on a neighborhood of the diagonal in M × M.
Remark 3.6. In fact, Bär and Pfäffle [BP10] show Thm. 3.4 only for W = 0. In this
case, Φj = 0 whenever j is odd. While this is not true for non-zero W , the transport
equations (3.5) can be solved just as well and the proof of Thm. 3.5 carries over basically
without changes.

4 Getzler Symbols
This section is dedicated to the proof of the following theorem.
TheoremP 4.1. Let H~ be the Witten operator of Example 3.2 acting on sections of ΛT ∗M
and let e~ ∞ j
j=0 ~ Φj be the corresponding formal heat kernel. Then for each j, the coeffi-
cient Φj has Clifford bi-order at most (j, j) and
n
X

~j σ
/ j,j (Φj ) = exp −t σ / 1,1 (W ) − t~2 σ
/ 0,0 (V ) − t~ σ / 2,2 (F) . (4.1)
j=0

Here,
1
F = − Rijkl ci cj bk bl . (4.2)
8
To establish a proof, we follow along the lines of Getzler’s proof [Get85] of the local
Atiyah-Singer index theorem in the form explained in [Roe98]: We define a symbol calculus
on the space D(M, ΛT ∗M) of differential operators acting on sections of ΛT ∗M that takes
into account the Clifford order. Then we show how to use it in the ~-dependent situation
discussed here.
Following Roe, a symbol map is a homomorphism-like mapping from a filtered algebra
to a graded algebra:

8
Definition 4.2. [Roe98, Def. 12.5] Let A be a filtered algebra and let G be a graded
algebra. A symbol map σ• : A −→ G is a family of linear maps Ak −→ Gk such that
(i) If a ∈ Al with l < k, then σk (a) = 0

(ii) If a ∈ Ak and b ∈ Al , then σk (a)σl (b) = σk+l (ab).

We already introduced two symbol maps in Section 2. The Clifford symbol

σ b Cℓ(−V )
/ • : End(ΛV ) −→ ΛV ⊗

and the Clifford bi-symbol. Here, A = End(ΛV ) for a finite-dimensional vector space
V considered with either its Z-filtration or its Z2 -filtration. The corresponding graded
algebras are ΛV ⊗ b Cℓ(−V ) and ΛV ⊗ b ΛV , respectively (where the first is equipped with
the grading that only takes into account the differential form degree).
Now we introduce Getzler’s symbol. Let (M, g) be a Riemannian manifold of dimension
n. The algebra D(M, ΛT ∗M) of differential operators acting on sections of ΛT ∗M has a
natural filtration by order, and comes with the map that assigns each operator its principal
symbol. This is indeed a symbol map in the sense of Def. 4.2: The corresponding graded
algebra is the algebra of sections of the bundle3
∞
M
S k T ∗M ⊗ ΛT ∗M ∼
= ΛT ∗M[T M].
k=0

Here, S T M denotes the k-th symmetric power of T ∗M; this space is canonically isomor-
k ∗

phic to the space of homogeneous polynomials of degree k on T M and the direct sum
over these spaces for all k ≥ 0 can be identified with a space of polynomials. In our
case, we obtain the space of polynomials with coefficients in ΛT ∗M, which we denote by
ΛT ∗M[T M].
Now let us use all this data to define a new filtration on D(M, ΛT ∗M) that takes into
account the Clifford order.
For q ∈ M, choose a chart x around q with x(q) = 0. If we additionally trivialize
the bundle ΛT ∗M by identifying fibers Λk Tp∗M with Λk Tq∗M for p near q, any differential
operator P ∈ D(M, ΛT ∗M) has a Taylor series with respect to these choices,
X ∂ |β|
P ∼ pαβ xα , with pαβ ∈ End(ΛTq∗M). (4.3)
αβ
∂xβ

Here, α and β are multi-indices and we employ the usual conventions for these (as ex-
plained e.g. in [Shu87, p. 1]). Of course, this Taylor series depends heavily on the two
choices made. Its order and the principal term, however, do not, allowing us to define the
following.
3
Strictly speaking, this is an infinite-dimensional bundle. However, the homogeneous parts of this bundle
are finite-dimensional, and we only consider homogeneous sections or finite linear combinations of
these.

9
Definition 4.3 (q-Symbols). Let q ∈ M. We say that P is of q-order k or less, if for
each α and β, pαβ has order less or equal to k + |α| − |β| in the Clifford filtration of
End(ΛTq∗M). In this case, its k-th q-symbol is
X ∂ |β|
σkq (P ) = σ/ j (pαβ )X α . (4.4)
∂X β
j=k+|α|−|β|−|α|

Here, Xj are the Euclidean coordinate functions on Tq M induced by the chart x. σkq (P )
is a differential operator on Tq M with coefficients in the algebra Aq [Tq M], the space of
Aq -valued polynomials on Tq M, where
Aq := ΛTq∗M ⊗
b Cℓ(−Tq M).

We denote the space of such operators by P(Tq M, Aq ). If an operator P ∈ D(M, ΛT ∗M)

is of q-order k or less for every q ∈ M, we can take its q-symbol at every point. This
associates to P its (global) Getzler symbol which we denote by σk (P ). This is a section
of the bundle P(T M, A)k .
Under Euclidean coordinates X1 , . . . , Xn on Tq M, we understand coordinates such that
for all v = v j ej ∈ Tq M, we have X j (v) = v j (here e1 , . . . , en is an arbitrary but fixed
∂
orthonormal basis of Tq M). This implies ∂X j
|v = ej for every v ∈ Tq M under the
∼
canonical isomorphism Tv Tq M = Tq M. Each Xj can be interpreted as a polynomial of
degree one on Tq M.
The space P(Tq M, Aq ) introduced in Def. (4.3) is a graded algebra: We assigns degree
∂ ∗ ∗
one to differential operators ∂X j and one-forms η ∈ Tq M ⊆ ΛTq M, and degree minus one

to polynomials Xj ∈ R[Tq M]. Elements of Cℓ(−Tq M) are assigned degree zero.

Remark 4.4. The q-symbol is a refinement (and extension at the same time) of the
Clifford filtration: If P is an endomorphism (i.e. a differential operator of order zero in
the usual filtration) and it has q-order k or less (i.e. σkq (P ) ∈ Aq [Tq M]), then the Clifford
q
order of P |q is k or less as well, and σ
/ k (P |q ) is given by evaluating the polynomial σk (P )
at X = 0 ∈ Tq M.
Proposition 4.5. The q-symbol is well-defined, i.e. its definition above is independent of
the choices made. Furthermore, we have
q
σj+k (P ◦ Q) = σjq (P ) ◦ σkq (Q) (4.5)
whenever P is of q-order ≤ j and Q is of order ≤ k.
Proof. If we have the well-definedness of the q-symbol map, (4.5) follows directly from the
the composition rules of differential operators in P(Tq M, Aq ). Let us temporarily denote
by σkq,x the k-th q-symbol with respect to a chart x. If we change from a chart x to a
chart y, then the charts have Taylor expansions with respect to each other, namely
x ∼ Ay + . . . and
y ∼ Bx + . . .

10
for some A ∈ GL(n) and B = A−1 , where the dots indicate terms of lower order in the
q-filtration. Therefore
q,x α q,y q,y α
σ|α| (x ) = X α = (AY )α = σ|α| ((Ay)α) = σ|α| (x ),

because the order of (Ay)α − xα is less than |α|. A similar computation can be done for
the differential operator ∂ |β| /∂xβ , with the matrix A replaced by B.
Now under a trivialization, we understood a smooth identification of near-by fibers
Λkp T ∗M with Λk Tq∗M. A transformation T from one such identification to another is of
course the identity at q, so the q-order of T −id is negative which means that the q-symbol
is independent of the choice of trivialization. 

Lemma 4.6. Let ∇ be the Levi-Civita connection on ΛT ∗M and choose Euclidean coor-
dinates X 1 , . . . , X n on Tq M. Then the operator ∇i is of q-order 1 and
∂ 1 Xn
σ1q (∇i ) = − Rij X j (4.6)
∂X i 4 j=1

where X
Rij = Rijkl dX k ∧ dX l (4.7)
k<l
are the 2-form entries of the curvature tensor of T M with respect to this basis.

Proof. Let x be Riemannian normal coordinates about the point q ∈ M. From [BGV96,
Lemma 4.14], we obtain that
∂ 1
∇i = i
− Rijkl xj ck cl + . . . (4.8)
∂x 8
where the dots indicate terms that are of order zero or less in the q-filtration. Hence
∂ 1
σ1q (∇i ) = i
− Rijkl X j dX k ∧ dX l ,
∂X 8
which is the claimed formula. 

Remark 4.7. We adopt the convention Rijkl = hR(∂i , ∂j )∂k , ∂l i which different from the
one used in [BGV96], also leading to a different sign of Rij (compare Lemma 4.14). Roe
[Roe98] uses the same sign convention for Rijkl as we do (compare Example 12.14) but
chooses to define his Rij forms to be the negative of those defined in (4.7). Our convention
makes formula (4.8) coincide with [Get85].

Remark 4.8. By [Roe98, Thm. 12.13], there is a unique symbol map

σ• : D(M, ΛT ∗M) −→ Γ∞ (M, P(T M, A)

that satisfies formula (4.6) and restricts to the Clifford symbol defined in Section 2 when
restricted to an operator of order zero (i.e. a section of End(ΛT ∗M)). Therefore, our

11
global Getzler symbol coincides with the one defined in [Roe98, Thm. 12.13], restricted
to the special case of the Hodge Laplacian considered here (the general definition works
on any Clifford bundle). In Getzler’s original paper [Get85], a similar calculus is implicit,
but the notion of a symbol map is not formalized.

Getzler’s main observation was that the q-symbol of a Dirac operator is a harmonic
oscillator on Tq M, meaning the following:

Proposition 4.9. [Roe98, Prop. 12.17] Let D = d+d∗ . For every q ∈ M, D 2 has q-order
2 and its q-symbol with respect to orthogonal coordinates on Tq M is
Xn  ∂ 1 Xn
2
σ2q (D 2 ) =− − Rij X j
+ σ/ 2 (F) ∈ P(Tq M, Aq ), (4.9)
i=1 ∂X i 4 j=1

where F was defined in (4.2).

Proof. This follows at once from the Weizenböck formula for the Euler operator D =
d + d∗ , which is
1
dd∗ + d∗ d = D 2 = ∇∗ ∇ + scal + F, (4.10)
4
(compare (3.16) and p. 149 in [BGV96]) and formula (4.6)). 

The q-symbols of the other terms in the Witten operator H~ = ~2 D 2 + ~W + V are

straightforward to calculate; they are given by
n
X
σ1q (W ) =σ
/ 1,• (W ) = dxi ⊗
b b(∇i ξ) (4.11)
i=1
σ0q (V ) = σ
/ 0,• (V ) = |ξ| .2
(4.12)

We now need to extend the symbol calculus to the coefficients Φj of the formal heat
kernel of H~ . We abbreviate E := ΛT ∗M. For each q ∈ M fixed and each t > 0, Φj ( · , q, t)
is a section of the vector bundle E ∗ ⊗ Eq over M. Therefore, with respect to a chart x
around q and a trivialization that identifies fibers of Ep with Eq for p near q, it has a Taylor
series
X
Φj ( · , q, t) ∼ Φjα (q, t) xα , where Φjα (q, t) ∈ Eq∗ ⊗ Eq ∼
= End(Eq ),
α

where the coefficients depend smoothly on q and t. We say that Φj has q-order k or less
if Φjα (q, t) ∈ End(Eq ) has Clifford order less or equal to k + |α| for each t, and in that
case, we define its k-th q-symbol as
X

σkq (Φj ) = / m Φjα (q, t) X α ∈ A[Tp M].
σ
m+|α|=k

Note that this also depends on t in addition to the usual dependence on q.

12
 The well-definedness can be shown as in the proof of Prop. 4.5 and a formal computation
with Taylor series shows that if the q-order of Φj is l or less, we have the multiplication
property
σkq (P )σlq (Φj ) = σk+l
q
(P Φj ) (4.13)
for any differential operator P ∈ D(M, ΛT ∗M) of order at most k, which is supposed to
act on the first entry of Φj .

Theorem 4.10. For each j = 0, 1, 2, .P . . , and each q ∈ M, Φj is of q-order at most j

and for the "heat symbol" σ (k~ ) := e~ nj=0 ~j σjq (Φj ), we have the formula
q

σ q (k~ ) = u(X, R, t~2 ) exp −t σ / 1 (W ) − t~2 σ
/ 0 (V ) − t~ σ / 2 (F) (4.14)

where     
−n/2 b 1 tR tR
u(X, R, t) = (4πt) A(tR) exp − X, coth X
4t 2 2
b
is the Mehler kernel with the A-form
 
b 1/2 R/2
A(R) = det .
sinh(R/2)

Let us first see why this implies Thm. 4.1.

Proof (of Thm. 4.1). With a view on Remark 4.4, we have

n
X

~j σ b 2R) exp −t σ/ (V )−t~ σ
/ j,• (Φj )= (4πt~2 )n/2 σ q (k~ )|X=0 = A(t~ / (W )−t~ 2
σ
/ (F) .
0,• 1,• 2,•
j=0
P
b 2 R) = 1 + n ~j A
Furthermore A(t~ bj (t) for some j-form4 Abj (t) ∈ Ωj (M) and the
Pn j=1
j
exp-part is of the form j=0 ~ σ / j,•(Ej (t)) for some Ej (t) ∈ End(ΛTq∗M) of order (j, j)
in the Clifford bi-filtration because the exponents have bi-order (0, 0), (1, 1) and (2, 2),
respectively and come with appropriate powers of ~. Therefore
n
X n
X X
j
~σ
/ j,•(Φj ) = ~j bk (t)/
A σ l,• (El (t))
j=0 j=0 k+l=j

Because A bk (t) has order (k, 0) in the Clifford bi-filtration, we now find that σ
/ j,j (Φj ) =
b σ j,j (Ej (t)) = σ/ j,j (Ej (t)). This gives the theorem.
A0 (t)/ 

We now proceed with the proof of Thm. 4.10. For each q ∈ M, the "symbolic operator"

σ q (H~ ) := ~2 σ2q (D 2 ) + ~σ1q (W ) + σ0q (V ) (4.15)

4 bj (t) = 0 unless j is divisible by four, as A

In fact, A b is an even function of R.

13
is of Schrödinger type as an operator on Tq M, acting on C ∞ -functions with values in Aq
(i.e. sections of a trivial vector bundle). Therefore, we can consider the "symbolic heat
equation"

∂t + σ q (H~ ) k~ = 0 (4.16)
on Tq M. We use bold letters for all objects associated to this symbolic equation to
distinguish them from those associated to the equation down on M. By Thm. 3.4, there
exists a unique formal heat kernel
∞
X
b~ (X, Y, t) = e~ (X − Y, t)
k ~j Φj (X, Y, t),
j=0

where the Φj solve the transport equations corresponding to the heat equation (4.16) and
 
2 −n/2 1 2
e~ (X, t) = (4πt~ ) exp − 2 |X| .
4t~
is the Euclidean fundamental solution on Tq M.
Looking at formula (4.9), the unique connection ∇ on Tq M such that σ2q (D 2 ) − ∇∗ ∇
is of order zero (compare [BGV96, Prop. 2.5]) is clearly given by
∂ Xn
∇i = − Rij X j = σ1q (∇i ).
∂X i j=1
P ∂
Because Tq M is flat as a Riemannian manifold, G ≡ 0 and V = j X j ∂X j , hence

Xn ∂ Xn
∇V = Xj − Rij X i X j = σ0q (∇V )
j=1 ∂X j ij=1

The recursive transport equations (3.5) for the operator σ q (H) therefore can be written
as
 
∂
t + σ0 (∇V ) + tσ0 (V ) Φj ( · , Y ) = −tσ1q (W )Φj−1 ( · , Y )−tσ2q (D 2 )Φj−2 ( · , Y ). (4.17)
q q
∂t
Lemma 4.11. For each j = 0, 1, 2, . . . and each q ∈ M, the coefficient Φj is of q-order
at most j.

Proof. By definition, for each q ∈ M, the Φj satisfy the transport equations

 
∂
t + ∇V + G + tV Φj ( · , q) = −tW Φj−1 ( · , q) − tLΦj−2 ( ·, q) (4.18)
∂t
with initial condition Φ0 (q, q, 0) = idEq . Straightforward calculation shows that G( · , q)
vanishes at q, hence σ0q (G) = 0. Let Φ0 be of q-order k. Taking the k-th symbol on both
sides shows (by multiplicativity (4.13)) that σkq (Φ0 ) solves
 
∂
t + σ0 (∇V ) + tσ0 (V ) σkq (Φ0 ) = 0,
q q
∂t

14
which is just the first transport equation on Tq M (4.17). By Thm. 3.4, there is a unique
solution for each initial value. Because Φ0 (q, q, 0) = idEp , we have

σkq (Φ0 )|(X,t)=0 = σ / k (idEq ) = 0

/ k (Φ0 (q, q, 0)) = σ if k > 0,

so the initial value is zero and σkq (Φ0 ) vanishes for all X and t. This shows that Φ0 has
q-order zero for all t. Now by induction and multiplicativity of symbols, the right hand
side of the j-th transport equation (4.18) has q-order ≤ j. Suppose that Φj has q order
k > j. Taking the k-th q-symbol on both sides shows that σkq (Φj ) solves
 
∂
t + σ0 (∇V ) + tσ0 (V ) σkq (Φj ) = 0.
q q
∂t
Again, this is a transport equation. By Thm. 3.4, there is a unique solution for each right
hand side, and zero is a solution, hence σkq (Φj ) = 0 whenever k > j. But this means that
Φj is of q-order ≤ j. 

This gives the following corollary.

Corollary 4.12. The terms σjq (Φj ) solve the recursive transport equations (4.17) for Y =
0 and we have
Φj |Y =0 = σjq (Φj ). (4.19)
b~ is actually a
Furthermore, because σjq (Φj ) = 0 whenever j > n, the formal heat kernel k
finite sum and we have
b~ |Y =0 .
σ q (k~ ) = k

Proof. Taking the j-th q-symbol on both sides of the j-th equation (4.18) gives exactly
the transport equations (4.17) for σ q (H~ ). Equation (4.19) follows from the uniqueness
statement of Thm. 3.4. 

Let us now finish the proof of the Thm. 4.10.

Proof (of Thm. 4.10). First, one verifies that the right hand side of (4.14) is a solution
to the heat equation (4.16). The result that u(X, R, t) satisfies

Xn  ∂ 2 !
∂ 1 Xn
− − Rij X j u(X, R, t) = 0, u0 = δ0 · idEq
∂t i=1 ∂X i 4 j=1

is usually called Mehler’s formula, see [BGV96, chapter 4.2]. Substitution t 7→ ~2 t and
straightforward calculation then shows that the right hand side of (4.14) solves (4.16).
Now explicitly expanding the Taylor series, one verifies that


u(X, R, t~2 ) exp −t σ / 1 (W ) − t~2 σ
/ 0 (V ) − t~ σ / 2 (F) = e~ (X, t)Φ~ (X, t),

15
 P∞ j
for some power series Φ~ = j=0 ~ Φj with coefficients in A[Tq M], each Φj being a
polynomial in both X and t. As seen in section 3, the Φj have to fulfill the recursive
transport equations (4.17), and by uniqueness (Thm. 3.4), we get
n
X
Φ~ = ~j σjq (Φj )
j=0

as by Corollary 4.12, the σjq (Φj ) as well solve these transport equations. 

5 The McKean-Singer Formula and its Consequences

From now on, let (M, g) be an n-dimensional closed Riemannian manifold. Again, let
D~ be Witten’s perturbed Euler operator of Example 3.2. The McKean-Singer formula
[BGV96, Thm. 3.50] states that for all t > 0, we have
Z
ind(D~ ) = str k~ (q, q, t) dq.
M

where we integrate over the volume density associated to the Riemannian metric of M.
The index of the Euler operator D = d + d∗ on the exterior algebra equipped with the
grading given by Ξ is well-known to be equal to the Euler characteristic χ(M). On the
other hand, the index is a topological invariant, i.e. it is the same for every Dirac operator
on a given vector bundle [BGV96, Thm. 3.51]. Therefore, the index of D~ is also equal
to χ(M). Expanding k~ in its semiclassical expansion and using that the asyptotics are
uniform over M by Thm. 3.5, we get
∞
X Z
2 −n/2 j
χ(M) ∼~ց0 (4πt~ ) ~ str Φj (q, q, t) dq
j=0 M

for any t > 0 fixed. The left hand side of this equation is independent of ~, so by
uniqueness of asymptotic expansions, all coefficients on the right-hand-side except the
term constant in ~ must vanish. Hence "∼" must in fact be an equality and we get
Z
−n/2
χ(M) = (4πt) str Φn (q, q, t) dq for all t > 0. (5.1)
M

By Corollary 2.4, the supertrace of Φn (t) can be calculated in terms of the Clifford bi-
symbol via the formula
n(n+1)

str Φn (q, q, t) = (−1) 2 2n σ/ n,n Φn (q, q, t) , vol ⊗ b vol
n
X
n(n+1)
n


= (−1) 2 2 b vol
σ/ j,j Φn (q, q, t) , vol ⊗
j=0

16
where the last equality holds because the pairing of σ b vol vanishes if
/ j,j (Φn ) with vol ⊗
j < n. On the other hand, by Thm. 4.1 we have
n
X


/ j,j Φj (q, q, t) = exp −t|ξ|2 − t σ
σ / 1,1 (W ) − t/
σ 2,2 (F) .
j=0

/ 0,0 (V ) = |ξ|2. Therefore, we get that for all t > 0, we have

where we used σ
Z D E
n(n+1)
−n/2


χ(M) = (−1) 2 (πt) exp −t|ξ|2 − t σ
/ 1,1 (W ) − t/ b vol .
σ 2,2 (F) , vol ⊗ (5.2)
M

Remark 5.1. The integrand above is in fact the pullback of Mathai and Quillen’s Thom
form [MQ86]

U = (2π)−n/2 T exp(−|η|2 /2 + i∇η + F ) ∈ Ωn (T ∗M)

along the section ξt := (2t)1/2 ξ. Regarding the terms appearing above, η is the tautological
section in Γ∞ (T ∗M, π ∗ T ∗M) that maps η 7→ η, T is the Berezin integral on the second
component, F is the Riemann tensor considered as (2, 2)-form on T ∗M and ∇ is the Levi-
Civita connection on T ∗M, both pulled back to T ∗M via the canonical projection π (see
[BGV96, Chapter 1.6] or [Zha01, Chapter 3]).
Note that σ/ 1,1 (W ) is not quite equal to ∇η because of the appearance of a super tensor
product, whence the lack of the factor i in (5.2). By one possible definition, the Euler
form is the pullback ι∗ U along the inclusion of ι : M −→ T ∗ M as the zero section, which
corresponds to t = 0. Therefore, (5.2) is exactly the interpolation formula
Z
χ(M) = ξt∗ U,
M

which can be found in the proof of [BGV96, Thm. 1.56].

In this sense, formula (5.2) can be seen as a generalization of a result of Mathai [Mat92],
who showed how to get the Thom form on the tangent bundle from the heat kernel of the
Laplacian.

Let us now evaluate (5.2) without knowing anything about Thom forms. Note that
σ / 1,1 (W ) and |ξ|2 all commute so that
/ 2,2 (F), σ

n k

2
X X / 2,2 (F)j σ
σ / 1,1 (W )k−j
exp −t/ σ 1,1 (W ) − t|ξ|2 = e−t|ξ|
σ 2,2 (F) − t/ (−t)k
k=0 j=0
j!(k − j)!
We are interested in the (n, n)-form part, which is produced by the summands above
where n = 2j + (k − j) = k + j. Therefore, we define the functions αk by

(−1)k D E
αk := / 2,2 (F)n−k σ
σ / 1,1 (W )2k−n , vol ⊗
b vol (5.3)
(n − k)!(2k − n)!

17
whenever 2k ≥ n. Then
D
E 2
n
X
2 −t|ξ|
exp −t/
σ 2,2 (F) − t/ b vol = e
σ 1,1 (W ) − t|ξ| , vol ⊗ tk αk .
2k≥n

Using formula (5.2), we get the following formula for the index density.
Theorem 5.2. Let M be an n-dimensional closed Riemannian manifold. Then for all
t > 0, we have
Xn Z
n(n+1) 2
−n/2 k
χ(M) = (−1) 2 (πt) t αk e−t|ξ| (5.4)
2k≥n M

where the functions αk are defined in (5.3).

Proposition 5.3. If n is even, then the function αn/2 is given by

1 X
αn/2 = sgn(τ )sgn(σ)Rτ (1)τ (2)σ(1)σ(2) · · · Rτ (n−1)τ (n)σ(n−1)σ(n) .
8n/2 (n/2)! τ,σ∈S
n

Proof. By formula (5.3),

(−1)n/2 D n/2
E
αn/2 = σ b
/ 2,2 (F) , vol ⊗ vol ,
(n/2)!
so the proposition follows directly from the formula
n
1 X
σ2,2 (F) = − Rijkl ei ej ⊗
b ek el , (5.5)
8 i,j,k,l=1

compare (4.2). 

The term
(−1)n/2 X
Ω= sgn(τ )sgn(σ)Rτ (1)τ (2)σ(1)σ(2) · · · Rτ (n−1)τ (n)σ(n−1)σ(n) (5.6)
2n (n/2)! τ,σ∈S
n

is called Killing-Lipschitz curvature (or n-th order sectional curvature) and up to a factor
coincides with the Pfaffian of the curvature tensor (compare [Zha01, (3.39)]). Taking the
limit t ↓ 0 in (5.4), we obtain the following classical theorem (note that if n is odd, there
is no term of order zero in t in (5.4) so that the integrand vanishes in the limit).
Theorem 5.4 (Gauss-Bonnet-Chern). [Che55, Thm. 1] Let M be an n-dimensional
closed Riemannian manifold. If n is even, then its Euler characteristic is given by the
integral formula Z
χ(M) = (2π)−n/2 Ω, (5.7)
M
where Ω is the Lipschitz-Killing curvature as in (5.6). If n is odd, then χ(M) = 0.

18
Proposition 5.5. We have
n(n+1)

αn = (−1) 2 detg ∇ξ (5.8)
where the determinant of the (0, 2)-tensor ∇ξ is calculated with help of the metric.
Proof. By (5.3), we have
(−1)n D E
αn = / 1,1 (W )n , vol ⊗
σ b vol .
n!
We have n
X X
σ
/ 1,1 (W ) = dxi ⊗
b ∇i ξ = (∇i ξ)j dxi ⊗
b dxj
i=1 ij

By multiplication rules of the super tensor product, we generally have

b
!k b
X k(k−1) X
α b β
cαβ dx ⊗ dx = (−1) 2 cα1 β1 · · · cαk βk dxα1 ∧ · · · ∧ dxαk ⊗
b dxβ1 ∧ · · · ∧ dxβk .
α,β=a α1 ,...,αk =a
β1 ,...,βk =a
(5.9)
Hence
D E n(n−1)
n
X
n b
σ
/ 1,1 (W ) , vol ⊗ vol = (−1) 2 (∇α1 ξ)β1 · · · (∇αn ξ)βn
α1 ,...,αk =1
β1 ,...,βk =1
n(n−1) X
= (−1) 2 sgn(σ)sgn(τ ) (∇τ (1) ξ)σ(1) · · · (∇τ (n) ξ)σ(n)
τ,σ∈Sn
n(n−1)
= (−1) 2 n! det(∇ξ),
by the Leibnitz formula for determinants. The result follows 
Now we can evaluate the integral (5.4) with the method of stationary phase, where we
take the limit t ↑ ∞ in (5.4). This allows us to express the Euler characteristic as a sum
of integrals over the set of critical points of ξ. The task is to explicitly calculate certain
jets of the functions αj , which appear in the stationary phase expansion.
We always assume that the zero locus of ξ is a disjoint union of finitely many sub-
manifolds of M. The easiest situation is the case that all critical submanifolds are zero-
dimensional, which is dealt with in the following corollary. The degenerate case is dealt
with in the next section.
Theorem 5.6 (Poincaré-Hopf). Let M be a closed Riemannian manifold. Suppose
that the vector field X has only non-degenerate critical points, i.e. whenever we have
X(p) = 0, then det(∇X|p ) 6= 0. Then
X
χ(M) = (−1)ν(p) ,
{X(p)=0}

where ν(p) is the number of negative eigenvalues of ∇X|p .

19
 This is a special case of Thm. 6.3. We give a separate proof, as it is quite short and
hopefully indicates how the proof works in the general case. Note that we do not require
M to be oriented (compare e.g. [BGV96, Thm. 1.58]).
Proof. Set ξ := X ♭ , the adjoint form to X given by the metric (i.e. "lowering the indices").
Notice that the set C of points where |X|2 = |ξ|2 = 0 is just the set Critφ of critical points
of φ := |X|2 . The method of stationary phase (see e.g. [DS99, Prop. 5.2] or compare
Lemma 6.4 below) states that if φ fulfills φ > 0 on the compact support of a function α
except at the point p, where φ(p) = 0 and det ∇2 φ > 0, then we have
Z (
−1/2
k −tφ (2π)n/2 det ∇2 φ|p α(p) if k = n/2
lim t αe = .
t↑∞ M 0 if k < n/2
Here, ∇2 φ|p is the Hessian of φ at p. By a partition of unity argument, taking the limit
t ↑ ∞ in (5.4) yields
 n/2 Z X
t 2
−1/2
χ(M) = lim αn e−t|X| = 2n/2 det ∇2 |X|2|p αn (p)
t↑∞ π M {X(p)=0}

At p, we have ∇2Y,Z |X|2 = h∇Y X, ∇Z Xi so that the determinant is given by det(∇2 |X|2) =
2n det(∇X)2 . Therefore, using (5.8)

2


2 −1/2 det(∇X) (−1)ν(p)
det ∇ |X| αn (p) = n/2 = (5.10)
2 det(∇X) 2n/2

and the theorem follows. 

6 The Degenerate Case

In this section, we generalize Thm. 5.6 to the case that we have critical submanifolds
instead of critical points.
Definition 6.1. [Bot54] Let M be a manifold. A function φ is called Morse-Bott, if
(i) The set of critical points Critφ = {p ∈ M | dφ|p = 0} is a disjoint union of finitely
many submanifolds of M.
(ii) For each connected component C ⊆ Critφ , the Hessian ∇2 φ of X is non-degenerate
when restricted to the normal bundle NC.
The second condition a priori depends on the choice of a Riemannian metric, but turns
out to be independent of this choice.

Definition 6.2. The index ν(C) of a connected critical submanifold C ⊆ Critφ of M

is the dimension of the biggest subspace V of Tp M such that the bilinear form ∇2 φ|p
restricted to V is negative definite, where p is some point in C.

20
 This is independent of the point p ∈ C because ν(C) is locally constant, as follows from
the Morse-Bott Lemma (see for example [BH04]).

Theorem 6.3 (Degenerate Poincaré-Hopf). Let M be`an closed ` manifold. Suppose

that φ is Morse-Bott as defined above and let Critφ = C1 · · · Ck for connected sub-
manifolds Cj . Then
Xk
χ(M) = (−1)ν(Cj ) χ(Cj ).
j=1

Usually one proves this theorem using the degenerate Morse inequalities [Bis86, Thm.
2.14]. We will instead start from Thm. 5.2 and evaluate the integrand in the limit t ↑ ∞
with the method of stationary phase. To this end, we first make some general observations
regarding integrals of the form
Z
2
I(t, α) = e−t|dφ| α.
M

Here M is an n-dimensional Riemannian manifold, φ is a Morse-Bott function, and α ∈

Cc∞ (M). Clearly, I(t, α) is a smooth function on (0, ∞).

Lemma 6.4. Assume that Critφ consists of exactly one submanifold C, which has co-
dimension m. Then
 π m/2 Z α
m/2
lim t I(t, α) = ,
t↑∞ 2 2
C det(∇ φ|N C )

where we integrate over the canonical measure on C induced by the Riemannian metric.

Proof. If φ is Morse-Bott, then |dφ|2 is a Morse-Bott function as well. Assume that there
κ) : M ⊇ U −→ (V ⊆ Rm ) × C (where U is a
exists a diffeomorphism κ = (κ1 , . . . , κm , e
tubular neighborhood of C and V is a neighborhood of zero in Rm ) such that

|dφ|2 U
= κ21 + · · · + κ2m

Then, assuming that α is supported in U, we have

Z Z 
−m/2 −t|v|2 −1
t I(t, α) = e (α ◦ κ )(x, p) G(x, p) dv dp, (6.1)
C V

where G = | det dκ|−1 is the Jacobian determinant coming from the transformation for-
2
mula. It is well-known that tm e−t|v| converges as a distribution to π m/2 δ0 in the limit
t ↑ ∞, where δ0 is the delta distribution at zero in Rm . Straightforward calculation (e.g.
in local coordinates) yields

−1/2
−1
G(0, p) = det ∇2 |dφ|2 Np C
= 2−m/2 det ∇2 φ|Np M .

21
This proves the lemma for α compactly supported in U, and under the assumption of the
existence of a diffeomorphism κ as above.
For general α, we can use a partition of unity to split it up into a function α1 that is
compactly supported in U and a function α2 with compact support disjoint from C. For
the first, we can use the argument above, and the second part is of order t−∞ , because
|dφ|2 ≥ ε on the support of α2 , for some ε > 0.
Regarding the diffeomorphism κ, such a diffeomorphism exists at least locally on C by
[BH04], and the statement again follows by splitting up α with a partition of unity. 

Proposition 6.5. Under the assumptions of Lemma 6.4, I(t, α) has a complete asymp-
totic expansion as t ↑ ∞, namely
 π m/2 X∞ Z
−j Lj α
I(t, α) ∼ t . (6.2)
2t j=0
2
C j! det(∇ φ|N C )

Here, L is a second order differential operator defined on a neighborhood of C that for

each q ∈ C has the q-symbol
Xm 

q αβ ∂2 1 ∇2 φ|Nq C −2 0
σ2 (L) = Aq with Aq =
α,β=1
∂X α ∂X β 4 0 0

in the decomposition Tq M = Nq C ⊕ Tq C.

The asymptotic expansion (6.2) is often called stationary phase expansion, though one
usually considers imaginary emponents.
Proof. It is more convenient to consider J(s, α) := s−m/2 I(s−1 , α) instead. The idea is
then to construct an operator L defined on some neighborhood of C such that
∂
J(s, α) = J(s, Lα). (6.3)
∂s
If we make the ansatz


L∗ α := div A · grad α + v, grad α ,

with an endomorphism field A and a vector field v, calculate both sides of (6.3) and order
by powers of s, we obtain that the endomorphism A and the vector field v have to fulfill
the coefficient equations

|dφ|2 = grad |dφ|2 , A · grad |dφ|2 (6.4)

m

v, grad |dφ|2 = − div A · grad |dφ|2 (6.5)
2
We have
grad |dφ|2 = 2h∇dφ, dφi♯ = 2∇2 φ · grad φ.

22
so that (6.4) becomes
| grad φ|2 = 4 grad φ, ∇2 φ · A · ∇2 φ · grad φ

Therefore, if we find a vector field V such that

∇grad φ V = 2∇2 φ · grad φ, (6.6)
then A = (∇V )∗ ∇V solves (6.4). Equation (6.6) is a transport equation that is singular at
C. Because the right-hand side vanishes on C as well, it admits a unique smooth solution
for all given smooth initial values on C (this is explained in [Lud14, Thm. 2.3, Thm. 7.1]),
so in particular there exists a unique endomorphism field A of the form A = (∇V )∗ ∇V ,
defined on some neighborhood of C, such that the induces operator L has the claimed
principal symbol.

Having constructed such an A, we find div A · grad |dφ|2 ≡ m2 on C. This means that
the right-hand-side of the second coefficient equation (6.5) vanishes on C, and again, the
equation has a solution of the form v = grad f , defined on some neighborhood of C.
Together, this shows that on some neighborhood of C, there exists an operator L with
the claimed principal symbol such that (6.3) holds. Therefore, we can expand in a Taylor
series around any ε > 0.
XN Z s
(s − ε)j j 1
J(s, α) = J(ε, L α) + (s − r)N J(r, LN +1 α)dr
j=0
j! N! ε

By (6.4), we can take the limit ε ↓ 0 in this formula, which implies that we have the
claimed asymptotic expansion, at least if the support of α is compactly contained in a
neighborhood of C where L is defined. The general case follows with a partition of unity
argument, as explained before. 
We now apply the stationary phase expansion (6.2) to the integral formula (5.4) for
the Euler characteristic, where we set ξ := dφ. We may and will henceforth assume that
Critφ consists of exactly one connected submanifold C of co-dimension m; the general
case follows once more from a partition of unity argument. This gives
n
X
n(n+1)
χ(M) = (−1) 2 (πt)−n/2 tk I(t, αk )
2k≥n
 π m/2 X
n X
∞ Z
n(n+1)
−n/2 k−j Lj αk
∼ (−1) 2 (πt) t .
2t k≥n j=0 C j! det(∇2 φ|N C )

Now we invoke the same argument as earlier: Because the left-hand-side is independent of
t, all coefficients of the asymptotic expansion of the right-hand-side must be zero, except
the constant term, and we must have equality in fact. The zero-order terms are those
where k = n/2 + m/2 + j (k and j being integers) so that we have zero-order terms if and
only if m and n have the same parity, or equivalently, if C is even-dimensional. We have
obtained the following partial result.

23
Lemma 6.6. If the critical set of φ consists of exactly one submanifold C of co-dimension
m, we have the following integral formula
n/2−m/2 Z
n(n+1) X Lj αn/2+m/2+j
m/2−n/2
χ(M) = (−1) 2 π . (6.7)
j=0 C j! det(∇2 φ|N C )

if and only if C is even-dimensional. Otherwise, χ(M) = 0.

Remark 6.7. This implies already that for any Morse-Bott function φ on M, Critφ can
be a union of non-empty odd-dimensional manifolds only if χ(M) = 0.

We can therefore assume that C is even-dimensional, i.e. that m and n have the same
parity. The task is now to explicitly calculate the integrands Lj αn/2+m/2+j on C. It will
turn out that αn/2+m/2+j has q-order −2j for every q ∈ C. Therefore,

Lj αn/2+m/2+j (q) = σ2q (L)j σ−2j

q
(αn/2+m/2+j ) X=0
(6.8)

where σ•q contains the q-symbol of Section 4.

To make our calculations easier, we fix a point p ∈ C and work in a special chart
around the point p, constructed as follows: Choose an orthonormal basis ν1 |p , . . . , νm |p
of Np C and obtain an orthonormal frame ν1 , . . . , νm of NC over a neighborhood of p
by parallel translation along geodesics in C emanating from p. Furthermore, choose
Riemannian normal coordinates x e in C centered around the point C. Now define the
chart x : M ⊃ U −→ V ⊂ R × Rn−m by
m

Xm 
−1 α
x (v, w) := expq v να |q , e−1 (w),
q=x
α=1

where U is a suitable neighborhood of p in M. Write ∂α for the directional derivatives

with respect to this chart. Then we have the following standard result.

Lemma 6.8. For q ∈ C ∩ U, we have ∂α |q = να |q if α ≤ m and ∂α |q ∈ Tq C if α > m.

The Christoffel symbols at the point p are
(
IIγαβ |p if γ ≤ m and α, β > m
Γγαβ |p = . (6.9)
0 otherwise

Here IIγαβ are the coefficients of the second fundamental form II|p : Tp C × Tp C −→ Np C,
compare [Cha06, II.2.1]. Furthermore, in the decomposition Tp M = Np C ⊕ Tp C,
 
2 φαβ 0
∇ φ|p =b . (6.10)
0 0

with respect to the basis ∂1 |p , . . . , ∂n |p , where φαβ = ∂α ∂β φ|p .

24
Lemma 6.9. For every p ∈ C and 0 ≤ j ≤ n/2 − m/2, the function αn/2+m/2+j has
p-order −2j or less, and its p-symbol is given by
p

X
σ−2j (αn/2+m/2+j ) = cj det ∇2 φ|Np C sgn(τ ) sgn(σ) Θτ σ (X)
σ,τ ∈Sm,n

where cj is a dimensional constant and the Θτ σ (X) are homogeneous polynomials of degree
2j on Tp M that only depend on the curvature of M, the second fundamental form of C
and the Hessian of φ at p; see (6.12) and (6.13). Here, the summation is over all σ, τ in
the group Sm,n of permutations of the numbers {m + 1, . . . , n}.

Proof. By (4.11), we have

n n n
!
X X X
σ
/ 1,1 (W ) = ∇2αβ φ dxα ⊗
b dxβ = ∂α ∂β φ − Γγαβ ∂γ φ dxα ⊗
b dxβ .
α,β=1 α,β=1 γ=1

By (6.10), ∇2αβ φ has p-order zero or less with σ0 (∇2αβ φ) = φαβ if α, β ≤ m and if α, β > m,
then ∇2αβ φ has p-order −1 or less with
m
X
p

σ−1 ∇2αβ φ =− IIγαβ φγδ X δ . (6.11)
γ,δ=1

Therefore,
  m
!m n
!2j
m + 2j X X
/ 1,1 (W )m+2j
σ = ∇2αβ φ dxα ⊗
b dxβ b dxβ +O(|x|2j+1)
∇2αβ φ dxα ⊗
m α,β=1 α,β=m+1
| {z }| {z }
:=W1 :=W2

The pre-factor accommodates for the fact that we selected m factors out of m + 2j factors
/ 1,1 (W )m+2j up this way. For the first factor we then get (compare (5.9))
to split σ
m
X
m(m−1)
W1 |p = (−1) 2 φα1 β1 · · · φαm βm dxα1 ∧ · · · ∧ dxαm ⊗
b dxβ1 ∧ · · · ∧ dxβm
α1 ,...,αm =1
β1 ,...,βm =1
m(m−1) X
= (−1) 2 sgn(τ )sgn(σ) φτ (1)σ(1) · · · φτ (m)σ(m) dx1 ∧ · · · ∧ dxm ⊗
b dx1 ∧ · · · ∧ dxm
τ,σ∈Sm
m(m−1)

= (−1) 2 m! det ∇2 φ|Np C dx1 ∧ · · · ∧ dxm ⊗
b dx1 ∧ · · · ∧ dxm

and for the second factor, we have

n
!2j
X X
W2 = − ∇2αβ φ dxα ⊗
b dxβ = (−1)j ΦAB dxA ⊗
b dxB ,
α,β=m+1 A,B

25
where A = (α1 , . . . , α2j ) and B = (β1 , . . . , β2j ) run over all 2j-tuples of numbers between
m + 1 and n. By the considerations above, ΦAB is of p-order −2j and its p-symbol is
m
X
p
γ
σ−2j ΦAB = IIγα11 β1 · · · IIα2j2j β2j φγ1 δ1 · · · φγ2j δ2j X δ1 · · · X δ2j ,
γ1 ,...,γ2j =1
δ1 ,...,δ2j =1

compare (6.11). Now remember that

(−1)n/2+m/2+j D E
n/2−m/2−j m+2j
αn/2+m/2+j = σ
/ (F) σ
/ 1,1 (W ) .
(n/2 − m/2 − j)!(m + 2j)! 2,2
By the calculations above, we obtain
p

X
σ−2j (αn/2+m/2+j ) = cj det ∇2 φ|Np C sgn(τ )sgn(σ) Θτ σ (X)
σ,τ ∈Sm,n

for a dimensional constant cj , where Θτ σ (X) is the homogeneous polynomial

n/2
p
Y A=(τ (m+1),...,τ (m+2j))
Θτ σ (X) = σ−2j ΦAB Rτ (2s−1)τ (2s)σ(2s−1)σ(2s) for B=(σ(m+1),...,σ(m+2j)). (6.12)
s=m/2+j+1

/ 2,2 (F)n/2−m/2−j . In the case that n (and hence

Here we used formula (5.5) to calculate σ
also m) is odd, the above product is supposed to run over half-integer numbers s.
For the constant cj , we find
   n/2−m/2−j
(−1)n/2+m/2+j m + 2j m(m−1)
+j 1
cj = (−1) 2 m! −
(n/2 − m/2 − j)!(m + 2j)! m 8
m(m−1)
(6.13)
(−1)n+ 2 (−1)j 8j
=
8n/2−m/2 (n/2 − m/2 − j)!(2j)!

which only depends on j, m and n. 

Lemma 6.10. For each p ∈ C, we have

n/2−m/2
n(n+1) X Lj αn/2+m/2+j (p)
ν(C) n/2−m/2
(−1) ΩC |p = (−1) 2 2 (6.14)
j=0
j! det(∇2 φ|Np C )

where ΩC is the Lipschitz-Killing curvature of C, as in (5.6).

This proves Thm. 6.3 using the stationary phase formula (6.7) and the Gauss-Bonnet-
Chern Theorem 5.4.

26


Proof. Denote by φαβ the entries of the inverse matrix of φαβ αβ≤m
. Then the operator
σ2p (L) from Prop. 6.5 is given by
m
1 X αγ βγ ∂2
σ2p (L) = φ φ .
4 α,β,γ=1 ∂X α ∂X β


To calculate σ2p (L)j σ−2j
p
ΦAB , we need to split the 2j indices into groups of two, for
which there are 2−j (2j)! possibilities. Therefore
 
j m

(2j)! Y X ′
p
σ2p (L)j σ−2j σ2p (L)  IIγα2s−1 β2s−1 IIγα2s β2s φγδ φγ ′ δ′ X δ X δ 
′
ΦAB = j
2 s=1
γγ ′ δδ′ =1
 
j m m
(2j)! Y X X ′ ′
= j j  IIγα2s−1 β2s−1 IIγα2s β2s + IIγα2s−1 β2s−1 IIγα2s β2s 
2 4 s=1 γ=1 ′ γ =1
j
(2j)! j Y
= 2 IIα2s−1 β2s−1 , IIα2s β2s .
8j s=1

Put together and using (6.8) now gives

X
Lj αn/2+m/2+j = cj det ∇2 φ|Np C sgn(τ ) sgn(σ) σ2p (L)j Θτ σ (X) X=0
σ,τ ∈Sm,n
m(m−1)
(−1)n+ 2 (−1)j 8j
(2j)!
= det ∇2 φ|Np C Υj
8n/2−m/2 (n/2 − m/2 − j)!(2j)! 8j
m(m−1)
2n/2 (−1)n+ 2

= n/2−m/2 det ∇2 φ|Np C (−1)j Υj
8 (n/2 − m/2 − j)!
where
m/2+j n/2
X Y Y
j
Υj = 2 sgn(τ )sgn(σ) IIτ (2s−1)σ(2s−1) , IIτ (2s)σ(2s) Rτ (2s−1)τ (2s)σ(2s−1)σ(2s)
σ,τ ∈Sm,n s=m/2 s=m/2+j+1
m/2+j n/2
X Y Y
= sgn(τ )sgn(σ) Sτ (2s−1)τ (2s)σ(2s−1)σ(2s) Rτ (2s−1)τ (2s)σ(2s−1)σ(2s)
σ,τ ∈Sm,n s=m/2 s=m/2+j+1

with
Sαβγδ = IIαγ , IIβδ − IIβγ , IIαδ

eαβγδ of the curvature

The Gauss formula [Cha06, Thm. II.2.1] states that the entries R
tensor of C are given by the formula
eαβγδ = Rαβγδ − Sαβγδ .
R

27
Therefore,

X n/2−m/2
Y n/2−m/2
X  
eτ (2s−1)τ (2s)σ(2s−1)σ(2s) = j n/2 − m/2
sgn(τ )sgn(σ) R (−1) Υj . (6.15)
σ,τ ∈Sm,n s=1 j=0
j

In total, we obtain
n/2−m/2 j n/2−m/2
m(m−1)
X L αn/2+m/2+j (p) X det ∇2 φ|Np C 2n/2−m/2 (−1)n+ 2
−m/2
2 = n/2−m/2 (n/2 − m/2 − j)!
(−1)j Υj
j=0
j! det(∇2 φ|Np C ) j=0
j! det(∇ 2 φ|
Np C ) 8
m(m−1) n/2−m/2
X  
ν(C) (−1)n+ 2 j n/2 − m/2
= (−1) (−1) Υj
2n−m (n/2 − m/2)! j=0 j

Finally, we use (6.15) and the fact that

n(n + 1) m(m − 1) n m
n+ + = − mod 2
2 2 2 2
because n and m have the same parity to obtain the stated result. 

Remark 6.11. It is a curious observation that the coefficients αk vanish exactly to the
order such that only there top-order terms matter in the stationary phase evaluation. In
particular, this also means that only the principal symbol of the stationary phase operator
L contributes to the end result.

References
[AB95] Austin, D. M. ; Braam, P. J.: Morse-Bott Theory and Equivariant Cohomol-
ogy. In: Progr. Math. 133 (1995), S. 123–183

[BGV96] Berline, Nicole ; Getzler, Ezra ; Vergne, Michèle: Heat Kernels and Dirac
Operators. Second Edition. Springer, 1996

[BH04] Banyaga, Augustin ; Hurtubise, David E.: A proof of the Morse-Bott

Lemma. In: Expositiones Mathematicae 22 (2004), S. 365–373

[Bis86] Bismut, Jean-Michel: The Witten Complex and the Degenerate Morse In-
equalities. In: Journal of Differential Geometry 23 (1986), S. 207–240

[Bot54] Bott, Raoul: Nondegenerate Critical Manifolds. In: Annals of Mathematics

60 (1954), Nr. 2, S. 248–261

[BP10] Bär, Christian ; Pfäffle, Frank: Asymptotic heat kernel expansion in the
semiclassical limit. In: Comm. Math. Phys. (2010), Nr. 294, S. 731–744

28
[Cha06] Chavel, Isaac: Riemannian Geometry. 2nd Edition. Cambridge : Cambridge
University Press, 2006

[Che55] Chern, Shiing-Shen: On Curvature and Characteristic Classes of a Riemannian

Manifold. In: Abhandlungen aus dem Mathematischen Seminar der Universität
Hamburg 20 (1955), S. 117–126

[DS99] Dimassi, M. ; Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical

Limit. Cambridge University Press, 1999

[Get85] Getzler, Ezra: A short proof of the local Atiyah-Singer index theorem. In:
Topology 25 (1985), S. 111–117

[Gil84] Gilkey, Peter: Invariance Theory, the Heat Equation and the Atiyah-Singer
Index Theorem. CRC Press Online, 1984

[Hug06] Hughes, Dominic J. D.: Towards Hilbert’s 24 th Problem: Combinatorial

Proof Invariants. (2006)

[Lud14] Ludewig, Matthias: Vector fields with a non-degenerate source. In: Journal
of Geometry and Physics 79 (2014), S. 59–76

[Mat92] Mathai, Varghese: Heat Kernels and Thom Forms. In: Journal of Functional
Analysis 104 (1992), S. 34–46

[MQ86] Mathai, Varghese ; Quillen, Daniel: Superconnections, Thom Classes and

Equivariant Differential Forms. In: Topology 25 (1986), Nr. 1, S. 85–110

[Roe98] Roe, John: Elliptic operators, topology and asymptotic methods. Second Edi-
tion. Boca Raton London New York Washington : Chapman and Hall, 1998

[Shu87] Shubin, M. A.: Pseudodifferential Operators and Spectral Theory. Springer,

1987

[Wit82] Witten, Edward: Supersymmetry and Morse Theory. In: Journal of Differ-
ential Geometry 17 (1982), Nr. 661-692

[Zha01] Zhang, Weiping: Lectures on Chern-Weil Theory and Witten Deformations.

World Scientific, 2001

29