Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 35 (1994), No. 2, 283-314.

Applications of Spectral Geometry to Affine and Projective Geometry

Neda Bokan^{1}, Peter B. Gilkey^{2}, Udo Simon^{3}

Fachbereich Mathematik, Technische Universität Berlin Strasse des 17. Juni 135, D-10623 Berlin Mathematics Department, University of Oregon Eugene Oregon 97403, USA E-mail: gilkey@math.uoregon.edu Fachbereich Mathematik, Technische Universität Berlin Strasse des 17. Juni 135, D-10623 Berlin E-mail: simon@sfb288.math.tu-berlin.de

Abstract. We use the asymptotics of the heat equation to construct spectral invariants in affine and projective geometry. MSC 1991: 53A15, 53B05, 53C07, 53C40, 58G25.

§1 Introduction

One of the first global affine results is Blaschke’s characterization of the ellipsoid within the class of ovaloids by the constancy of its equiaffine mean curvature [2, p. 215]. Blaschke considers a Laplace type equation and determines the space of solutions, which generalizes properties of the first eigenspace of the Laplacian on a sphere. Much later Schneider [18] generalized Blaschke’s results to relative geometry proving several global uniqueness results in affine differential geometry. The operators Schneider considers and recent transformation techniques for partial differential equations in [14], [21] are closely related. In the latter context, relations between conformal and projective structures show up naturally. This suggests a systematic study of the corresponding operators, of their spectral asymptotics, and an investigation of their properties under conformal and projective changes. We perform this study in §1-§5; §6 is devoted to applications in affine and projective geometry. As an example, we would like to point out interesting results about the Pick functional which measures the deviation of an ovaloid from an ellipsoid (see [2, p. 174]) and which

1 Research partially supported by the Technische Universität of Berlin, the DFG, and the University of Oregon foundation

2 Research partially supported by the NSF, the DFG, IHES, and MPIM

3 Research partially supported by the DFG

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1994 Heldermann Verlag, Berlin

is useful in projective differential geometry, see Sasaki [16]. In dimension 2, we will prove the Pick functional is a spectral invariant of the operators under consideration.

We shall be fairly expository in certain parts of this paper as this paper deals with two quite different areas of mathematics and we shall not assume the reader is familar with all the relevant material both from spectral geometry and from affine differential geometry.

We introduce the following notational conventions. Let M be a smooth compact manifold of dimension m ≥ 2; we assume the boundary of M, if non empty, is smooth. Let T M and T^{∗}M be the tangent and cotangent bundles of M. Let S^{2}M ⊂ T^{∗}M ⊗ T^{∗}M be the bundle of symmetric 2 cotensors. Let g ∈ C^{∞}(S^{2}M) be a Riemannian metric on M. We say that g̃ ∈ S^{2}M is conformally equivalent to g if there exists ψ ∈ C^{∞}(M) so that g̃ = e^{2ψ}g; let C(g) be the set of such conformally equivalent metrics.

Let D(M) be set of torsion free connections on T M. Let

R_{∇}(X, Y) : Z → (∇_{X}∇_{Y} − ∇_{Y} ∇_{X} − ∇_{[X,Y]})Z  (1.1)

be the curvature of ∇ ∈ D(M). We define the Ricci tensor ρ_{∇}(X, Z) by:

ρ_{∇}(X, Z) = −Tr(Y → R_{∇}(X, Y)Z).  (1.2)

Let

D^{s}(M) = {∇ ∈ D(M) : ρ_{∇}(X, Z) = ρ_{∇}(Z, X)}  1.3)

be the set of all torsion free connections on T M with symmetric Ricci tensor. If ∇ ∈ D(M), then ∇ ∈ D^{s}(M) if and only if there exists a parallel volume element on M, see Pinkall et al [14] for details. If ∇^{g} is the Levi-Civita connection of the metric g, then ∇^{g} ∈ D^{s}(M). There are, however, many connections in D^{s}(M) which do not arise in this way.

We define the Hessian H_{∇} for ∇ ∈ D(M) by:

(H_{∇}f)(X, Y) := X(Y (f)) − ∇_{X}Y (f).  (1.4)

It is clear (1.4) is tensorial in X. Since ∇ is torsion free,

(H_{∇}f)(X, Y) = (H_{∇}f)(Y, X)  (1.5)

so (1.4) is tensorial in Y. Consequently, H_{∇} is a second order partial differential operator

H_{∇} : C^{∞}(M) → C^{∞}(S^{2}M).  (1.6)

We say two connections ∇, ˜∇ ∈ D(M) are projectively equivalent if there exists a smooth 1-form θ so that:

( ˜∇_{X} − ∇_{X})Y = θ(X)Y + θ(Y)X.  (1.7)

This is equivalent to assuming that their unparametrized geodesics coincide. Let ∇ ∈ D^{s}(M). Then ˜∇ ∈ D^{s}(M) if and only if dθ = 0; see Pinkall et al [14] for details. If ∇ ∈ D^{s}(M), let

P^{s}(∇) ⊆ D^{s}(M)  (1.8)

be the set of all torsion free connections with symmetric Ricci tensor which are projectively equivalent to ∇. A projective structure on M is such an equivalence class P^{s}.

Let g be a Riemannian metric on M. Let {e_{i}} be a local orthonormal frame for T M and let {e^{i}} be the dual orthonormal frame for T^{∗}M. We adopt the Einstein convention and sum over repeated indices. We expand ω ∈ C^{∞}(S^{2}(M)) in the form ω = ω_{ij}e^{i} ∘ e^{j} where ∘ denotes symmetric tensor product. Define the trace of ω with respect to g by:

Tr_{g}(ω) := ω_{ii} ∈ C^{∞}(M).  (1.9)

We say a second order partial differential operator D on C^{∞}(M) is of Laplace type if the leading symbol of D is positive definite and defines a metric g on T^{∗}M. This means that in any system of local coordinates,

D = −(g^{νμ}∂_{ν}∂_{μ} + A^{ν}∂_{ν} + B).  (1.10)

We contract the Hessian to define an operator of Laplace type. Let ∇ ∈ D^{s}(M) and let g be a Riemannian metric. We define

Df = D(g, ∇)f := −Tr_{g}{H_{∇}(f) + (m − 1)^{−1}fρ_{∇}}.  (1.11)

We remark that D need not be self-adjoint.

The normalized Hessian

H_{∇} := H_{∇}(f) + (m − 1)^{−1}fρ_{∇}  (1.12)

arises naturally in the study of Codazzi equations; see [14] for details. If ˜∇ and ∇ are related by θ as in (1.7), we may choose a local primitive ϕ so dϕ = θ. Then

H ˜∇ = e^{ϕ}H_{∇}e^{−ϕ};  (1.13)

this equation simply means the operators H ˜∇ and H_{∇} are locally conjugate; see Lemma 2.1 for details. Let g̃ = e^{2ψ}g. Then (1.13) implies

D(g̃, ˜∇) = e^{−2ψ+ϕ}D(g, ∇)e^{−ϕ};  (1.14)

this transformation law will be crucial in our studies. If [θ] = 0 in the first cohomology group H^{1}(M), then ϕ is globally defined and D(g, ˜∇) and D(g, ∇) are conjugate and hence isospectral. This can fail if [θ] ≠ 0. For example, let M = S^{1} with the usual periodic parameter x and flat metric g. Let ∇ be trivial and let θ = ϵdx. Then

D(g, ∇) = −∂_{x}^{2}, spec(D(g, ∇)) = {n^{2}}_{n∈Z},
D(g, ˜∇) = −(i∂_{x} − ϵ)^{2}, spec(D(g, ˜∇)) = {(n − ϵ)^{2}}_{n∈Z}. (1.15)

Let τ_{g} = Tr_{g}(ρ_{∇^{g}}) be the scalar curvature of g and let Δ_{0} = δ_{0}d_{0} be the Laplacian. Then

D(g, ∇^{g}) = Δ_{0} − (m − 1)^{−1}τ_{g}  (1.16)

is neither the standard Laplacian nor is it the conformal Laplacian

:= Δ_{0} + 4(m − 2)(m − 1)^{−1}τ_{g}.  (1.17)

Let D = D(g, ∇) for ∇ ∈ D^{s}(M). If the boundary of M is non-empty, we impose Dirichlet boundary conditions B and let

domain(D_{B}) = {f ∈ C^{∞}(M) : f|_{∂M} = 0};  (1.18)

we postpone a discussion of modified Neumann boundary conditions until §2 to avoid technical complications. Let {λ_{ν}} be the eigenvalues of D_{B} where each eigenvalue is repeated according to its multiplicity and where λ_{1} ≤ λ_{2} ≤ .... If t > 0, then e^{−tD_{B}} is trace class on L^{2}(M) and

Tr_{L^{2}}(e^{−tD_{B}}) = Σ_{ν}e^{−tλ_{ν}};  (1.19)

this series converges uniformly on compact subsets of (0, ∞) since the eigenvalues satisfy the growth estimate

λ_{ν} ∼ ν^{2/m}  (1.20)

for ν large. As t ↓ 0, there is an asymptotic series of the form:

Tr_{L^{2}}(e^{−tD_{B}}) ∼ Σ_{n≥0}a_{n}(D, B)t^{(n−m)/2}.  (1.21)

The a_{n}(D, B) are spectral invariants which are locally computable in the following sense. Let dx = dx(g) and dy = dy(g) be the Riemannian measures on M and ∂M. There exist local invariants a_{n}(x, D) defined for x ∈ M and a_{n}^{bd}(y, D, B) defined for y ∈ ∂M so that

a_{n}(D, B) = ∫_{M}a_{n}(x, D)dx + ∫_{∂M}a_{n}^{bd}(y, D, B)dy.  (1.22)

The interior invariants vanish for n odd; we shall discuss these invariants in more detail in §3 and §4.

These invariants depend both upon the metric g and on the symmetric connection ∇; they are projectively invariant in ∇.

Theorem 1.1: Let D = D(g, ∇) and D̃ = D(g, ˜∇) for ˜∇ ∈ P^{s}(∇).

(a) a_{n}(x, D̃) = a_{n}(x, D) and a_{n}^{bd}(y, D̃, B) = a_{n}^{bd}(y, D, B).

(b) a_{n}(D̃, B) = a_{n}(D, B).

The dependence on g is quite complicated for general n. There are, however, two very important special cases. Let m be the dimension of M.

Theorem 1.2: Let D = D(g, ∇) and D̃ = D(g̃, ∇) for g̃ = e^{2ψ}g ∈ C(g).

(a) a_{m−2}(x, D(g̃, ∇)) = e^{(2−m)ψ}a_{m−2}(x, D(g, ∇)).

(b) a_{m}(D(g̃, ∇), B) = a_{m}(D(g, ∇), B).

Remark: Theorems 1.1 and 1.2 show that the global invariant a_{m} depends only on the conformal class of the metric and the projective class of the connection. Thus (g, ∇) → a_{m}(D(g, ∇), B) defines a scalar valued invariant

a_{m} : C(g) × P^{s}(∇) → R.  (1.23)

We shall see, however, that the corresponding local invariants a_{m}(x, ·) do not satisfy a simple transformation law and in particular are not independent of the representatives chosen. Since

dx(e^{2ψ}g) = e^{mψ}dx(g),  (1.24)

the following density is invariantly defined on C(g) × P^{s}(∇) :

a_{m−2}(x, D(g, ∇))dx(g)^{(m−2)/m}.  (1.25)

In general, constructing projective invariants is quite difficult. One such example is the projective curvature tensor of H. Weyl:

(m − 1)P_{∇}(X, Y)Z := (m − 1)R_{∇}(X, Y)Z − ρ_{∇}(Y, Z)X + ρ_{∇}(X, Z)Y.  (1.26)

If ˜∇ ∈ P^{s}(∇), then P ˜∇ = P_{∇}. For any n, the a_{n}(x, D(g, ∇)) are projectively invariant in ∇; they are local renormalizable conformally invariant in g only for n = m − 2 and their integral is conformally invariant in g only for n = m.

If m is odd, and if the boundary of M is empty, the invariants of Theorem 1.2 are trivial. There is, however, a global spectral invariant called the functional determinant which can be defined in this context. For Re(s) >> 0, let

ζ(s, D) := Tr_{L^{2}}(D^{−s})  (1.27)

where we project on the complement of the kernel of D to avoid the 0-spectrum. This has a meromorphic extension to C with isolated simple poles on the real axis. The origin is a regular value and

ζ′(0) := − log(det(D))  (1.28)

is a global invariant of D.

Theorem 1.3: Let D = D(g, ∇) and D̃ = D(g̃, ˜∇) for ˜∇ ∈ P(∇) and g̃ ∈ C(g). If the boundary of M is empty and if m is odd, then ζ′(0, D) = ζ′(0, D̃).

Here is a brief outline of the paper. §2 is devoted to the proof of Theorems 1.1 and 1.2 and to a discussion of suitable boundary conditions of Neumann type. In §3, we will give local formulas for the interior invariants a_{n}(x, D) for n ≤ 4. In §4, we give the corresponding formulas for the boundary invariants a_{n}^{bd}(y, D, B). In §5, we study the operators D(g, ∇^{g}). In §6 we study these invariants in the context of affine differential geometry; this long section is divided into a number of subsections. In §6.0, we study spectral invariants of conjugate triples and Codazzi transformations. In §6.1, we establish notational conventions and derive the basic formulas we shall need. We discuss equiaffine geometry in §6.2, centroaffine geometry in §6.3, and polar geometry in §6.4. In §6.5, we use the conserved quantity a_{2} to relate Euclidean, equiaffine, and centroaffine geometry and to derive inequalities involving the Willmore functional and use in an essential fashion the conformal and projective invariance. We conclude in §6.6 by discussing briefly the case of surfaces with boundary. It is a pleasant task to acknowledge helpful suggestions by the referee.

§2 Heat equation asymptotics

In this section, we prove Theorems 1.1, 1.2, and 1.3. We begin by studying the local geometry of the Hessian. Let ∇ ∈ D^{s}(M) and let ˜∇ ∈ P^{s}(∇). Then by hypothesis there exists a closed 1-form θ so that

( ˜∇_{X} − ∇_{X})Y = θ(X)Y + θ(Y)X.  (2.1)

Locally, we can choose a smooth primitive ϕ so dϕ = θ and express:

( ˜∇_{X} − ∇_{X})Y = X(ϕ)Y + Y (ϕ)X.  (2.2)

If M is not simply connected, ϕ need not exist globally and this causes a mild amount of technical fuss later. Let g̃ = e^{2ψ}g ∈ C(g).

Lemma 2.1: Adopt the notation established above. Then:

(a) R ˜∇(U, V)W = R_{∇}(U, V)W + e^{ϕ}H_{∇}(e^{−ϕ})(V, W)U − e^{ϕ}H_{∇}(e^{−ϕ})(U, W)V.

(b) ρ ˜∇(V, W) = ρ_{∇}(V, W) + (m − 1)e^{ϕ}H_{∇}(e^{−ϕ})(V, W).

(c) e^{−ϕ}H ˜∇(e^{ϕ}f) = H_{∇}(f) − fe^{ϕ}H_{∇}(e^{−ϕ}).

(d) H ˜∇ = e^{ϕ}H_{∇}e^{−ϕ} and D(g̃, ˜∇) = e^{−2ψ+ϕ}D(g, ∇)e^{−ϕ}.

Proof: Choose coordinate vector fields {U, V, W } which are bracket free and compute:

R ˜∇(U, V)W = ( ˜∇_{U} ˜∇_{V} − ˜∇_{V} ˜∇_{U})W  (2.3)
=( ˜∇_{U}∇_{V} W + ˜∇_{U}{V (ϕ)W + W (ϕ)V }
− ˜∇_{V} ∇_{U}W − ˜∇_{V} {U(ϕ)W + W (ϕ)U}
=∇_{U}∇_{V} W + U(ϕ)∇_{V} W + (∇_{V} W)(ϕ)U
+ ∇_{U}(V (ϕ)W) + U(ϕ)V (ϕ)W + V (ϕ)W (ϕ)U
+ ∇_{U}(W (ϕ)V) + U(ϕ)W (ϕ)V + W (ϕ)V (ϕ)U
− ∇_{V} ∇_{U}W − V (ϕ)∇_{U}W − (∇_{U}W)(ϕ)V
− ∇_{V} (U(ϕ)W) − V (ϕ)U(ϕ)W − U(ϕ)W (ϕ)V
− ∇_{V} (W (ϕ)U) − V (ϕ)W (ϕ)U − W (ϕ)U(ϕ)V
=R_{∇}(U, V)W + U(ϕ)∇_{V} W − ∇_{V} (U(ϕ)W) + (∇_{V} W)(ϕ)U
+ ∇_{U}(V (ϕ)W) − V (ϕ)∇_{U}W − (∇_{U}W)(ϕ)V + V (ϕ)W (ϕ)U
+ ∇_{U}(W (ϕ)V) − ∇_{V} (W (ϕ)U) − W (ϕ)U(ϕ)V
=R_{∇}(U, V)W + (∇_{V} W)(ϕ)U − (∇_{U}W)(ϕ)V + V (ϕ)W (ϕ)U
− U(ϕ)W (ϕ)V + UW (ϕ)V − V W (ϕ)U

We complete the proof of (a) by checking:

\begin{matrix}e^{ϕ}H_{∇}(e^{−ϕ})(V, W) = & e^{ϕ}\{V W (e^{−ϕ}) − (∇_{V} W)(e^{−ϕ})\} \\ = & V (ϕ)W (ϕ) − (V W)ϕ + (∇_{V} W)ϕ. \end{matrix}  (2.4)

Let {Y_{i}} be a local frame for T M and let {Y^{i}} be the corresponding dual frame for T^{∗}M. We use (a) to prove (b) by computing:

ρ ˜∇(V, W) = −Σ_{i}(R ˜∇(V, Y_{i})W, Y^{i})  (2.5)
=ρ_{∇}(V, W) − Σ_{i}e^{ϕ}H_{∇}(e^{−ϕ})(Y_{i}, W)(V, Y^{i})
+ Σ_{i}e^{−ϕ}H_{∇}(e^{−ϕ})(V, W)(Y_{i}, Y^{i})
=ρ_{∇}(V, W) − e^{ϕ}H_{∇}(e^{−ϕ})(V, W) + me^{−ϕ}H_{∇}(e^{−ϕ})(V, W).

We prove (c) by computing:

e^{−ϕ}H ˜∇(e^{ϕ}f)(U, V) = e^{−ϕ}(UV)(e^{ϕ}f) − e^{−ϕ}( ˜∇_{U}V)(e^{ϕ}f)  (2.6)
=(UV)(f) + U(ϕ)V (f) + U(f)V (ϕ) + f(UV)(ϕ)
+ fU(ϕ)V (ϕ) − f(∇_{U}V)ϕ − (∇_{U}V)f
− 2fU(ϕ)V (ϕ) − U(f)V (ϕ) − V (f)U(ϕ)
=H_{∇}(f)(U, V) − fe^{ϕ}(UV)e^{−ϕ} + fe^{ϕ}(∇_{U}V)e^{−ϕ}
=H_{∇}(f)(U, V) − fe^{ϕ}H_{∇}(e^{−ϕ})(U, V).

We use (1.11), (b) and (c) to see that

\begin{matrix}H ˜∇ = & (H ˜∇ + (m − 1)^{−1}ρ ˜∇) \\ = & e^{ϕ}H_{∇}e^{−ϕ} − e^{ϕ}H_{∇}(e^{−ϕ}) + (m − 1)^{−1}ρ_{∇} + e^{ϕ}H_{∇}(e^{−ϕ})) \\ = H_{∇} + (m − 1)^{−1}ρ_{∇} = e^{ϕ}H_{∇}e^{−ϕ}; \end{matrix}  (2.7)

the remaining assertion of (d) now follows.

Remark: See also [14] for a proof of (a)-(c).

Operators which satisfy an intertwining law of the form given in Lemma 2.1 are said to be conformally related. We shall deduce Theorems 1.1 and 1.2 from Lemma 2.1 and from more general results which are based on work of Branson and Orsted [4] and of Parker and Rosenberg [13]; see also Rosenberg [15].

We work in a quite general context for the moment. Let D be a partial differential operator of order d > 0 on a smooth vector bundle V. We assume the leading symbol of D is self-adjoint and positive definite. If the boundary of M is non-empty, we impose boundary conditions B and let

Domain(D_{B}) = {ω ∈ C^{∞}(V) : Bω = 0}.  (2.8)

We assume the boundary conditions B are strongly elliptic; see Gilkey [8, §1.11]. We note that Dirichlet and modified Neumann boundary conditions are strongly elliptic for second order operators of Laplace type.

Let f ∈ C^{∞}(M) be an auxiliary test function. Then there is an asymptotic series as t ↓ 0^{+} of the form

Tr_{L^{2}}(fe^{−tD_{B}}) ∼ ∑ {∞ \atop n=0}a_{n}(f, D, B)t^{(n−m)/d};  (2.9)

see Gilkey [8, Theorem 1.11.4] for details; we set f = 1 to recover the invariants a_{n}(D, B) discussed in §1. The global invariants a_{n}(f, D, B) are locally computable. It is convenient to consider measure valued invariants at this stage rather than scalar invariants since there is no canonical volume element in this general context. Let ∂_{m}^{ν}f be the ν^{th} normal covariant derivative of f. Then there exist local measure valued invariants A_{n}(x, D) defined for x ∈ M and A_{n,ν}^{bd}(y, D, B) defined for y ∈ ∂M such that:

\begin{matrix}a_{n}(f, D, B) = & ∫_{M}fA_{n}(x, D) \\ + Σ_{0≤ν≤n−1}∫_{∂M}(∂_{m}^{ν}f)A_{n,ν}^{bd}(y, D, B). \end{matrix}  (2.10)

We say that B_{1} and B_{2} are equivalent boundary conditions if they define the same domains; i.e.

B_{1}ψ = 0 ⇔ B_{2}ψ = 0.  (2.11)

Lemma 2.2: Let D and D̃ be elliptic operators of order d > 0 with self-adjoint positive definite leading symbols. Let B and B̃ be strongly elliptic boundary conditions for D and D̃. Suppose on every simply connected open subset O of M there exists ϕ ∈ C^{∞}(O) so that D̃ = e^{ϕ}De^{−ϕ} and so that B̃ and e^{ϕ}Be^{−ϕ} define equivalent boundary conditions. Then (a) A_{n}(x, D̃) = A_{n}(x, D) and A_{n,ν}(y, D̃, B̃) = A_{n,ν}^{bd}(y, D, B).

(b) a_{n}(D̃, B̃) = a_{n}(D, B).

(c) If ϕ is globally defined, (D̃, B̃) and (D, B) are isospectral.

Proof: By hypothesis, e^{−ϕ} locally intertwines the operators D̃ and D and the boundary conditions B and B̃ over O. Since A_{n} and A_{n,ν} are invariant under change of frame, (a) follows; (b) is immediate from (a) and (c) is immediate. It is crucial in (a) that one does not necessarily assume ϕ is globally defined and thus (D, B) and (D̃, B̃) need not be isospectral in general.

Remark: Since Dirichlet boundary conditions are invariant, Theorem 1.1 follows from Lemmas 2.1 and 2.2. If B is a first order operator, then B̃ ≠ B and the boundary conditions are not invariant.

Lemma 2.3: Let D be an elliptic operator of order d > 0 with self-adjoint positive definite leading symbol. Let B be a strongly elliptic boundary condition for D. Let D̃ = e^{−dψ}D for ψ ∈ C^{∞}(M). Then:

(a) a_{m}(D̃, B) = a_{m}(D, B).

(b) e^{−dψ}A_{m−d}(x, D̃) = A_{m−d}(x, D).

Remark: The case of manifolds without boundary was known previously, see [4, 13]. The proof we shall give of (b) generalizes immediately to show that

Σ_{j≤m−1−i}\binom{i+j}{i}∂_{m}^{j}(e^{−dψ})A_{m−d,i+j}^{bd}(y, D̃, B) = A_{m−d,i}^{bd}(y, D, B)  (2.12)

for 0 ≤ i ≤ m − 1; we omit details as we shall not need this generalization. Theorem 1.2 follows from Lemmas 2.1 and 2.3 once we make the necessary change of notation and note dx(e^{2ψ}g) = e^{mψ}dx(g).

Proof: Let D(a) := e^{−daψ}D. We proceed formally and use the results of Gilkey [8, §1.9] to provide the necessary analytic justification. We compute:

Σ_{n}t^{(n−m)/d}∂_{a}a_{n}(D(a), B) ∼ ∂_{a}Tr_{L^{2}}(e^{−tD(a)_{B}})
= dtTr_{L^{2}}(ψD(a)e^{−tD(a)_{B}})
= −dt∂_{t}Tr_{L^{2}}(ψe^{−tD(a)_{B}})
∼ Σ_{n}(m − n)t^{(n−m)/d}a_{n}(ψ, D(a), B).

(2.13)

We equate terms in the asymptotic expansion to see:

∂_{a}a_{n}(D(a), B) = (m − n) · a_{n}(ψ, D(a), B).  (2.14)

We set n = m to see ∂_{a}a_{m}(D(a), B) = 0 which proves (a). Let f ∈ C^{∞}(M) vanish near the boundary of M. Let

D(a, c) = e^{−daψ}(D − cf).  (2.15)

We compute:

\begin{aligned}Σ_{n}t^{(n−m)/d} & ∂_{c}a_{n}(D(a, c), B) ∼ ∂_{c}Tr_{L^{2}}(e^{−tD(a,c)_{B}}) \\ & = tTr_{L^{2}}(e^{−daψ}fe^{−tD(a,c)_{B}}) \\ & ∼ tΣ_{n}t^{(n−m)/d}a_{n}(e^{−daψ}f, D(a, c), B). \end{aligned}  (2.16)

We equate coefficients in the resulting asymptotic expansion to see:

∂_{c}a_{n}(D(a, c), B) = a_{n−d}(e^{−daψ}f, D(a, c), B).  (2.17)

We set m = n and use (a) to compute:

\begin{matrix}0 = & ∂_{c}∂_{a}a_{m}(D(a, c), B) = ∂_{a}∂_{c}a_{m}(D(a, c), B) \\ = & ∂_{a}a_{m−d}(e^{−daψ}f, D(a, c), B). \end{matrix}  (2.18)

We set c = 0 to see:

\begin{matrix}0 = & ∂_{a}a_{m−d}(e^{−daψ}f, D(a), B) \\ = & ∫_{M}f · ∂_{a}\{e^{−daψ}A_{m−d}(x, D(a))\}. \end{matrix}  (2.19)

This yields the desired vanishing for x in the interior of M since f was arbitrary; continuity then shows (b) holds on the boundary as well.

Proof of Theorem 1.3: Assume ∂M = ∅, that m is odd, and that ϕ is globally defined so

D̃ = e^{−2ψ}e^{−ϕ}De^{ϕ}.  (2.20)

Multiplication by e^{ϕ} is a bounded operator on L^{2} which intertwines D and D̃. Consequently D and D̃ have the same spectrum so

ζ′(0, e^{−ϕ}De^{ϕ}) = ζ′(0, D).  (2.21)

This may fail, of course, if ϕ is not globally defined since ζ′(0, D) is not a local invariant. Thus we may assume without loss of generality that ϕ = 0. We now use the arguments of [4, 13]. Let ϵ be a parameter and let D(ϵ) = e^{−2ϵψ}D. Then

\begin{aligned}\frac{d}{dϵ}|_{ϵ=0}ζ(s, D) & = Tr_{L^{2}}(2sψD^{−s}) \\ \frac{d}{dϵ}|_{ϵ=0}ζ′(0, D) & = Tr_{L^{2}}(2ψD^{−s})|_{s=0}. \end{aligned}  (2.22)

We note Tr_{L^{2}}(2ψD^{−s})|_{s=0} is locally computable and the local formula which appears vanishes if m is odd and if ∂M = ∅; see [8] for details.

It is immediate that Dirichlet boundary conditions satisfy the covariance property required in Lemmas 2.2 and 2.3. We conclude this section by discussing boundary conditions of Neumann type which are admissible. We adopt the following notational conventions. Let {x^{1}, ..., x^{m}} be a system of local coordinates near the boundary; we assume M = {x : x^{m} ≥ 0} and ∂M = {x : x^{m} = 0}. Let ∂_{ν} and dx^{ν} be the corresponding coordinate frames for T M and T^{∗}M. Let indices α, β range from 1 through m − 1 and index the coordinate frame for the tangent and cotangent bundles of the boundary. We assume ∂_{m} is perpendicular to the boundary and that g_{αm} ≡ 0; this normalization is preserved by conformal rescaling. Let

∇_{∂_{ν}} ∂_{μ} = Γ_{∇,νμ}^{σ}∂_{σ}  (2.23)

be the Christoffel symbols of the connection ∇. Let Γ_{g,νμ}^{σ} be the Christoffel symbols of the Levi-Civita connection ∇^{g};

Γ_{g,νμ}^{σ} = \frac{1}{2}g^{ϵσ}(∂_{ν}g_{ϵμ} + ∂_{μ}g_{ϵν} − ∂_{ϵ}g_{νμ}).  (2.24)

The difference of the two connections is tensorial; let

θ_{νμ}^{σ} := Γ_{∇,νμ}^{σ} − Γ_{g,νμ}^{σ}.  (2.25)

Since ∇ and ∇^{g} are torsion free,

θ_{νμ}^{σ} = θ_{μν}^{σ}.  (2.26)

The second fundamental form is given by:

L_{αβ} = (∇_{∂_{α}}^{g}∂_{β}, ∂_{m}) = −\frac{1}{2}∂_{m}g_{αβ}.  (2.27)

Let a, b, c, and d be constants. We define

\begin{matrix}S = & S(g, ∇, ∂_{m}) \\ := & aθ_{αm}^{α} + bθ_{mm}^{m} + cg^{αβ}L_{αβ} + dg_{mm}g^{αβ}θ_{αβ}^{m}. \end{matrix}  (2.28)

We define modified Neumann boundary conditions

B(S)f = (∂_{m} + S)f|_{∂M}.  (2.29)

Since S(g, ∇, c∂_{m}) = cS(g, ∇, ∂_{m}), rescaling the normal vector yields equivalent boundary conditions so the boundary conditions are independent of the particular normal vector field chosen.

We say S defined by (2.28) is admissible if

\begin{matrix}B(S(g̃, ˜∇, ∂_{m})) = & e^{ϕ}B(S(g, ∇, ∂_{m}))e^{−ϕ}; i.e. \\ S(g̃, ˜∇, ∂_{m}) = & S(g, ∇, ∂_{m}) − ∂_{m}ϕ. \end{matrix}  (2.30)

Theorem 2.4:

(a) Let S be admissible. Then Theorems 1.1 and 1.2 hold if we replace Dirichlet boundary conditions by modified Neumann boundary conditions B(S).

(b) S is admissible if and only if

\begin{matrix}−1 = & (m − 1)a + 2b \\ 0 = & − a(m − 1) − b − c(m − 1) + d(m − 1). \end{matrix}

Proof: The first assertion is immediate from Lemmas 2.1, 2.2, 2.3 and from the definition of admissiblity. To prove (b), we first vary ∇ and keep g fixed. We compute:

\begin{matrix}θ̃_{νμ}^{σ} − θ_{νμ}^{σ} = & Γ_{˜∇,νμ}^{σ} − Γ_{∇,νμ}^{σ} \\ = & ∂_{ν}(ϕ)δ_{μ}^{σ} + ∂_{μ}(ϕ)δ_{ν}^{σ}; \end{matrix}  (2.31)

This establishes the first equation since

S(g, ˜∇, ∂_{m}) − S(g, ∇, ∂_{m}) = ((m − 1)a + 2b)∂_{m}(ϕ).  (2.32)

Next we vary g and keep ∇ fixed. Let g̃ = e^{2ψ}g and ∇ = ˜∇. Then

\begin{matrix}θ̃_{νμ}^{σ} − θ_{νμ}^{σ} = −\{Γ_{g̃,νμ}^{σ} − Γ_{g,νμ}^{σ}\} \\ = −g^{σϵ}(g_{νϵ}∂_{μ}ψ + g_{μϵ}∂_{ν}ψ − g_{νμ}∂_{ϵ}ψ) \\ g̃^{αβ}L_{αβ} − g^{αβ}L_{αβ} = −(m − 1)∂_{m}ψ. \end{matrix}  (2.33)

We complete the proof by computing:

\begin{matrix}0 & = S(g̃, ∇, ∂_{m}) − S(g, ∇, ∂_{m}) \\ = & (−a(m − 1) − b − c(m − 1) + d(m − 1))∂_{m}ψ. \end{matrix}  (2.34)

Remark: There do not seem to be any canonical choices of the constants a, b, c, and d so we shall concentrate for the most part on Dirichlet boundary conditions.

§3 Local formulas

We adopt the following notational conventions to study these local invariants. For the moment, let M be a compact smooth Riemannian manifold of dimension m without boundary; we take up the corresponding questions for manifolds with boundary in §4. Roman indices {i, j, ...} will range from 1 through m and index a local orthonormal frame {e_{1}, ..., e_{m}} for the tangent bundle T M; use the metric to provide a corresponding dual frame for the cotangent bundle T^{∗}M. Let x = (x^{1}, ..., x^{m}) be a system of local coordinates on M. Greek indices {μ, ν, ...} range from 1 through m and index the local coordinate frames {∂_{1}, ..., ∂_{m}} and {dx^{1}, ..., dx^{m}} for T M and T^{∗}M respectively. To simplify notation, we shall supress the distinction between lower and upper indices {i, j, ...} since these relate to an orthonormal frame; we shall maintain this distinction for indices {μ, ν, ...} since these index a coordinate frame.

Let

∇_{e_{i}}^{g}e_{j} = Γ_{g,ijk}e_{k}  (3.1)

define the Christoffel symbols of the Levi-Civita connection on M. Let

R_{g,ijkl} = (∇_{e_{i}}^{g}∇_{e_{j}}^{g} − ∇_{e_{j}}^{g}∇_{e_{i}}^{g} − ∇_{[e_{i},e_{j}]}^{g}e_{k}, e_{l})  (3.2)

be the curvature tensor of ∇^{g} with the sign convention R_{1212} = −1 on the standard sphere. We sum over repeated indices to define

\begin{matrix}ρ_{g,ij} = R_{g,ikkj}, & τ_{g} = ρ_{g,ii}, \\ |ρ_{g}|^{2} = ρ_{g,ij}ρ_{g,ij}, & |R_{g}|^{2} = R_{g,ijkl}R_{g,ijkl}. \end{matrix}  (3.3)

We begin by studying the local geometry of operators of Laplace type. Let

D = −(g^{νμ}∂_{ν}∂_{μ} + A^{σ}∂_{σ} + B)  (3.4)

be an operator of Laplace type on C^{∞}(M). Our first task is to get a more invariant expansion.

Lemma 3.1: There exists a unique connection ∇_{D} on C^{∞}(M) and a unique function E_{D} ∈ C^{∞}(M) so that D = −(Tr(∇_{D}^{2}) + E_{D}). If ω_{D} is the connection 1-form of ∇_{D}, then

\begin{matrix}ω_{D,δ} = & \frac{1}{2}g_{νδ}(A^{ν} + g^{μσ}Γ_{g,μσ}^{ν}), and \\ E_{D} = & B − g^{νμ}(∂_{μ}ω_{D,ν} + ω_{D,ν}ω_{D,μ} − ω_{D,σ}Γ_{g,νμ}^{σ}). \end{matrix}

Proof: Fix a local frame for V. Since

∇^{g}(dx^{ν}) = −Γ_{g,μσ}^{ν}dx^{μ} ⊗ dx^{σ}  (3.5)

we compute:

∇_{D} =dx^{μ} ⊗ (∂_{μ} + ω_{D,μ}),
(∇_{D})^{2} =dx^{μ} ⊗ dx^{σ} ⊗ (−Γ_{g,μσ}^{ν}(∂_{ν} + ω_{D,ν})
+ (∂_{μ} + ω_{D,μ})(∂_{σ} + ω_{D,σ})),  (3.6)
D(∇_{D}, E_{D}) = − g^{μσ}{∂_{μ}∂_{σ} + 2ω_{D,μ}∂_{σ} − Γ_{g,μσ}^{ν}∂_{ν}
+ ∂_{μ}ω_{D,σ} + ω_{D,μ}ω_{D,σ} − Γ_{g,μσ}^{ν}ω_{D,ν}} − E_{D}.

We compare (3.4) and (3.6) to see:

A^{ν} =2g^{νμ}ω_{D,μ} − g^{μσ}Γ_{g,μσ}^{ν},  (3.7)
B =g^{μσ}(∂_{μ}ω_{D,σ} + ω_{D,μ}ω_{D,σ} − Γ_{g,μσ}^{ν}ω_{D,ν}) + E_{D}.  (3.8)

We lower indices in (3.7) to determine ω_{D} and solve (3.8) to determine E_{D}.

We express the a_{n}(x, D) in terms of these invariants. Let Ω_{D,ij} be the curvature of the connection ∇_{D} on C^{∞}(M) and let ‘;’ be multiple covariant differentiation with respect to the Levi-Civita connection. We refer to Gilkey [7,8] for the proof of the following Theorem.

Theorem 3.2: Let D = D(∇_{D}, E_{D}) on C^{∞}(M).

(a) a_{0}(x, D) = (4π)^{−m/2}.

(b) a_{2}(x, D) = 6^{−1}(4π)^{−m/2}{τ_{g} + 6E_{D}}.

(c) a_{4}(x, D) = 360^{−1}(4π)^{−m/2}{60(E_{D})_{;kk} + 60τ_{g}E_{D} + 180(E_{D})^{2} + 30Ω_{D,ij}Ω_{D,ij} +12(τ_{g})_{;kk} + 5(τ_{g})^{2} − 2|ρ_{g}|^{2} + 2|R_{g}|^{2}}.

Remark: We refer to Gilkey [7] for the calculation of a_{6}(x, D).

Let D = D(g, ∇) be as defined in §1 and let

θ_{νμ}^{σ} := Γ_{∇,νμ}^{σ} − Γ_{g,νμ}^{σ}.  (3.9)

We define a new connection ∇_{D} with connection 1-form ω_{D} and an endomorphism E_{D} on C^{∞}M so that

D = −(Tr(∇_{D}^{2}) + E_{D}).  (3.10)

Lemma 3.3:

(a) ω_{D,δ} = −\frac{1}{2}g_{νδ}g^{μσ}θ_{μσ}^{ν}.

(b) E_{D} = (m − 1)^{−1}g^{νμ}ρ_{∇,νμ} − g^{νμ}(ω_{D,ν;μ} + ω_{D,ν}ω_{D,μ}).

(c) Ω_{D,νμ} = ω_{D,μ;ν} − ω_{D,ν;μ}.

Proof: We compute:

\begin{matrix}H_{∇} = dx^{ν} ⊗ dx^{μ} ⊗ (∂_{ν}∂_{μ} − Γ_{∇,νμ}^{σ}∂_{σ}), \\ D(g, ∇) = −g^{νμ}(∂_{ν}∂_{μ} − Γ_{∇,νμ}^{σ}∂_{σ} + (m − 1)^{−1}ρ_{∇,νμ}). \end{matrix}  (3.11)

We prove (a) by using Lemma 3.1 to see:

\begin{matrix}ω_{D,δ} = & \frac{1}{2}g_{νδ}(−g^{μσ}Γ_{∇,μσ}^{ν} + g^{μσ}Γ_{g,μσ}^{ν}) \\ = & − \frac{1}{2}g_{νδ}g^{μσ}θ_{μσ}^{ν}. \end{matrix}  (3.12)

The ω_{D,ν} are tensorial since the constant function provides a canonical frame for the trival bundle. We use (a) to compute:

E_{D} = (m − 1)^{−1}g^{νμ}ρ_{∇,νμ} − g^{νμ}(∂_{μ}ω_{D,ν} + ω_{D,ν}ω_{D,μ} − ω_{D,σ}Γ_{g,νμ}^{σ}).  (3.13)

If we choose geodesic polar coordinates centered at the point in question x_{0} ∈ M, Γ_{g} vanishes at x_{0} and ordinary differentiation coincides with partial differentiation; (b) now follows. The proof of (c) is immediate.

Let

E(g, ∇) := E_{D(g,∇)} and Ω(g, ∇) := Ω_{D(g,∇)}  (3.14)

be the associated endomorphisms and curvatures. These are projective invariants of ∇.

Lemma 3.4: If ˜∇ ∈ P^{s}(∇), then E(g, ˜∇) = E(g, ∇) and Ω(g, ˜∇) = Ω(g, ∇).

Proof: Since the association D → E_{D} and D → Ω_{D} is natural within the category of operators of Laplace type, E_{D} and Ω_{D} are invariant under gauge transformation so

E(g, ˜∇) = E_{e^{ϕ}D(g,∇)e^{−ϕ}} = e^{ϕ}E_{D(g,∇)}e^{−ϕ} = E(g, ∇)  (3.15)

and similarly Ω(g, ˜∇) = Ω(g, ∇). We also give a combinatorial proof to illustrate the formulas we have derived. Choose geodesic polar coordinates centered at the point in question of M. Then

\begin{matrix}θ̃_{νμ}^{σ} = & θ_{νμ}^{σ} + ∂_{ν}ϕδ_{μ}^{σ} + ∂_{μ}ϕδ_{ν}^{σ} \\ ω̃_{δ} = & ω_{δ} − \frac{1}{2}g_{νδ}g^{μσ}(θ̃_{μσ}^{ν} − θ_{μσ}^{ν}) = ω_{δ} − ϕ_{;δ} \\ Ω̃_{δσ} = & ∂_{σ}ω̃_{δ} − ∂_{δ}ω̃_{σ} = Ω_{δσ}. \end{matrix}  (3.16)

We complete the proof by computing:

\begin{matrix}Ẽ = & (m − 1)^{−1}Tr_{g}(ρ ˜∇) − g^{νμ}(ω̃_{ν;μ} + ω̃_{μ}ω̃_{ν}) \\ = & E_{D} + Tr_{g}(e^{ϕ}H_{∇}(e^{−ϕ})) + g^{νμ}(ϕ_{;νμ} + 2ω_{ν}ϕ_{;μ} − ϕ_{;ν}ϕ_{;μ}) \\ = & E_{D}. \end{matrix}  (3.17)

The dependence of these tensors upon g is more complicated.

Lemma 3.5: Let g(ϵ) = e^{2ϵψ}g.

(a) ∂_{ϵ}ω_{δ}(g, ∇) = −\frac{1}{2}(m − 2)ψ_{;δ}.

(b) ∂_{ϵ}Ω(g, ∇) = 0.

(c) ∂_{ϵ}E(g, ∇) = −2ψE + \frac{1}{2}(m − 2)ψ_{;ii}.

(d) ∂_{ϵ}dx = mψdx.

(e) ∂_{ϵ}τ = −2ψτ_{g} − 2(m − 1)ψ_{;ii}.

(f) ∂_{ϵ}|ρ_{g}|^{2} = −4ψ|ρ_{g}|^{2} − 2τ_{g}ψ_{;ii} − 2(m − 2)ψ_{;ij}ρ_{g,ij}.

(g) ∂_{ϵ}|R_{g}|^{2} = −4ψ|R_{g}|^{2} − 8ψ_{;ij}ρ_{g,ij}.

Proof: Again, we use geodesic polar coordinates centered at the point in question. We compute

\begin{matrix}\frac{d}{dϵ}|_{ϵ=0}Γ_{g(ϵ),μσ}^{ν}(ϵ) = ψ_{;μ}δ_{σ}^{ν} + ψ_{;σ}δ_{μ}^{ν} − ψ_{;ν}δ_{μσ} + O(x^{2}) \\ \frac{d}{dϵ}|_{ϵ=0}ω_{δ}(g(ϵ), ∇) = ∂_{ϵ} \frac{1}{2}g_{νδ}g^{μσ}(Γ_{g(ϵ)μσ}^{ν}) = −\frac{1}{2}(m − 2)ψ_{;δ}. \end{matrix}  (3.18)

This proves (a); (b) follows. To prove (c), we compute

\begin{aligned}\frac{d}{dϵ}|_{ϵ=0}E = & − \frac{d}{dϵ}|_{ϵ=0}g^{μν}(∂_{ν}ω_{μ} + ω_{ν}ω_{μ} − ω_{σ}Γ_{g,νμ}^{σ}) \\ = & − 2ψE + g^{νμ}(\frac{1}{2}(m − 2)ψ_{;νμ} + (m − 2)ω_{ν}ψ_{;μ} \\ & − ω_{ν}(m − 2)ϕ_{;μ}). \end{aligned}  (3.19)

We refer to Branson and Gilkey [3] for the remaining assertions which are included for the sake of completeness.

Example 3.6: Let m = 2. Let g(ϵ) = e^{2ϵψ}g_{0}. By Theorem 3.2,

a_{0}(x, D) = (4π)^{−1};  (3.20)

this is trivially conformally invariant. We use Lemma 3.5 to study the integrated global invariant a_{2}(D). Since ∂_{ϵ}dx = 2ψdx,

∂_{ϵ}{E_{D}dx} = 0  (3.21)

so ∫_{M}Edx is a conformal invariant of g. We use the Gauss-Bonnet theorem to see

(4π)^{−1}∫_{M}τ_{g}dx = χ(M)  (3.22)

is a topological invariant of M. Thus a_{2}(D) is invariant even though a_{2}(x, D)dx is not pointwise invariant since the derivatives of ψ enter;

∂_{ϵ}τ_{g} = −2ψτ_{g} − ψ_{;ii}.  (3.23)

Example 3.7: Let m = 4. Let g(ϵ) = e^{2ϵψ}g_{0}. We use Lemma 3.5 to see a_{2}(x, ·) transforms properly:

\begin{matrix}∂_{ϵ}(6E_{D}) = −2ψ(6E_{D}) + 6ψ_{;ii}, \\ ∂_{ϵ}τ_{g} = −2ψτ_{g} − 6ψ_{;ii} \\ ∂_{ϵ}a_{2}(x, D(g, ∇)) = −2ψa_{2}(x, D(g, ∇)). \end{matrix}  (3.24)

To study a_{4}(D), we compute:

∂_{ϵ}(60τ_{g}E_{D}dx) = (−360E_{D} + 60τ_{g})ψ_{;ii}dx
∂_{ϵ}(180E_{D}^{2}dx) = 360E_{D}ψ_{;ii}dx
∂_{ϵ}(5τ_{g}^{2}dx) = −60τ_{g}ψ_{;ii}dx  (3.25)
∂_{ϵ}(−2|ρ_{g}|^{2}dx) = 4τ_{g}ψ_{;ii} + 8ρ_{g,ij}ψ_{;ij}dx
∂_{ϵ}(2|R_{g}|^{2}dx) = −16ρ_{g,ij}ψ_{;ij}dx.

This shows

∂_{ϵ}(a_{4}(x, D)dx) = 360^{−1}(4π)^{−2}(4τ_{g}ψ_{;ii} − 8ρ_{ij}ψ_{ij})dx  (3.26)

so a_{4}(x, D)dx is not pointwise conformally invariant. However, when we integrate, this term vanishes:

\begin{matrix}∫_{M}(4τ_{g}ψ_{;ii} − 8ρ_{ij}ψ_{;ij})dx \\ = & ∫_{M}(4τ_{g;ii} − 8ρ_{g,ij;ij})ψdx = 0. \end{matrix}  (3.27)

§4 Heat equation asymptotics for manifolds with boundary

We now consider the case that the boundary of M is non-empty. Let D be an operator of Laplace type on C^{∞}(M). We begin by recalling some results of Branson and Gilkey [3]; in contrast to the case of manifolds without boundary, the odd order invariants do not vanish. Choose a local orthonormal frame on the boundary so the e_{m} is the inward pointing unit normal; indices a, b range from 1 through m − 1 and index the local orthonormal frame for the tangent bundle of the boundary.

The interior invariants are given by Theorem 3.2; they vanish if n is odd. Let ‘:’ denote multiple covariant differentiation tangentially with respect to the Levi-Civita connection of the boundary.

Theorem 4.1 (Branson-Gilkey): With Dirichlet boundary conditions:

(a) a_{0}^{bd}(y, D, B) = 0.

(b) a_{1}^{bd}(y, D, B) = −4^{−1}(4π)^{−(m−1)/2}.

(c) a_{2}^{bd}(y, D, B) = 6^{−1}(4π)^{−m/2}2L_{aa}.

(d) a_{3}^{bd}(y, D, B) = −(384)^{−1}(4π)^{−(m−1)/2}(96E_{D} + 16τ_{g} + 8R_{g,amam} + 7L_{aa}L_{bb} −10L_{ab}L_{ab}).

(e) a_{4}^{bd}(y, D, B) = 360^{−1}(4π)^{−m/2}(−120E_{D;m} − 18τ_{g;m} + 120E_{D}L_{aa} + 20τ_{g}L_{aa} +4R_{g,amam}L_{bb} − 12R_{g,ambm}L_{ab} + 4R_{g,abcb}L_{ac} + 24L_{aa:bb} +40/21L_{aa}L_{bb}L_{cc} − 88/7L_{ab}L_{ab}L_{cc} + 320/21L_{ab}L_{bc}L_{ac}).

Example 4.2: Let m = 2. We use the Gauss-Bonnet theorem to see

a_{2}(D, B) =6^{−1}(4π)^{−1}{∫_{M}(τ_{g} + 6E_{D})dx
+ ∫_{∂M}2L_{aa}dy}
=6^{−1}χ(M) + (4π)^{−1}∫_{M}E_{D}dx.

We showed in §3 that ∫_{M}E_{D}dx was a conformal invariant. Consequently a_{2}(D, B) is a conformal invariant.

Example 4.3: Let m = 3. Let g(ϵ) = e^{2ϵψ}g. We use Lemma 3.5 and the variational formulas of Branson and Gilkey [3] to see

∂_{ϵ}(96E_{D}dy) = 48ψ_{;ii}dy
∂_{ϵ}(16τ_{g}dy) = −64ψ_{;ii}dy
∂_{ϵ}(8R_{g,amam}dy) = (8ψ_{;aa} + 16ψ_{;mm})dy
∂_{ϵ}(7L_{aa}L_{bb}dy) = −28ψ_{;m}L_{aa}dy
∂_{ϵ}(−10L_{ab}L_{ab}dy) = 20ψ_{;m}L_{aa}dy.

We use the identity ψ_{;aa} = ψ_{:aa} − L_{aa}ψ_{;m} to see

∂_{ϵ}a_{3}^{bd}(y, D, B)dy = −8(384)^{−1}(4π)^{−1}ψ_{:aa}dy  (4.1)

so again a_{3}^{bd}(y, D, B)dy is not pointwise conformally invariant. The global integrated invariant is a conformal invariant, however, since

∫_{M}ψ_{:aa}dy = 0.  (4.2)

We change notation slightly to discuss Neumann boundary conditions. Let ∂_{m} be the inward pointing unit normal. Let

\begin{matrix}S(g, ∇) := & S(g, ∇, ∂_{m}) − ω_{D,m} \\ = & aθ_{αm}^{α} + (b + \frac{1}{2})θ_{mm}^{m} + cg^{αβ}L_{αβ} + (d + \frac{1}{2})g_{mm}g^{αβ}θ_{αβ}^{m} \end{matrix}  (4.3)

We express Neumann boundary conditions in terms of covariant differentiation:

Bf = (f_{;m} + Sf)|_{∂M}.  (4.4)

Lemma 4.4: Let S be admissible. Let g(ϵ) = e^{2ϵψ}g_{0} and let ˜∇ ∈ P^{s}(∇).

(a) S(g, ˜∇, ∂_{m}) = S(g, ∇, ∂_{m}).

(b) ∂_{ϵ}S(g(ϵ), ∇) = −ψS(g(ϵ), ∇) + \frac{1}{2}(m − 2)ψ_{;m}.

Proof: We prove (a) by checking

S(g, ˜∇, ∂_{m}) − S(g, ∇, ∂_{m}) = − (ω̃_{D,m} − ω_{D,m}) + S(g, ˜∇, ∂_{m}) − S(g, ∇, ∂_{m})
=∂_{m}ϕ − ∂_{m}ϕ = 0.  (4.5)

The factor of −ψS(g(ϵ), ∇) in (b) comes from differentiating the change in the choice of the unit normal as the metric is varied. Since S is admissible,

S(g, ∇, ∂_{m}) = S(g̃, ∇, ∂_{m}).  (4.6)

We use Lemma 3.5 to study the variation of ω̃_{D,m} − ω_{D,m}.

With Neumann boundary conditions (f_{;m} + Sf)|_{∂M} = 0, the invariants are given by:

Theorem 4.5 (Branson-Gilkey):

(a) a_{0}^{bd}(y, D, B) = 0.

(b) a_{1}^{bd}(y, D, B) = 4^{−1}(4π)^{−(m−1)/2}.

(c) a_{2}^{bd}(y, D, B) = 6^{−1}(4π)^{−m/2}(2L_{aa} + 12S).

(d) a_{3}^{bd}(y, D, B) = (384)^{−1}(4π)^{−(m−1)/2}(96E_{D} + 16τ_{g} + 8R_{g,amam} +13L_{aa}L_{bb} + 2L_{ab}L_{ab} + 96SL_{aa} + 192S^{2}).

(e) a_{4}^{bd}(y, D, B) = 360^{−1}(4π)^{−m/2}(240E_{D;m} + 42τ_{g;m} + 24L_{aa:bb} + 120E_{D}L_{aa} +20τ_{g}L_{aa} + 4R_{g,amam}L_{bb} − 12R_{g,ambm}L_{ab} + 4R_{g,abcb}L_{ac} +40/3L_{aa}L_{bb}L_{cc} + 8L_{ab}L_{ab}L_{cc} + 32/3L_{ab}L_{bc}L_{ac} + 720SE_{D} +120Sτ_{g} + 144SL_{aa}L_{bb} + 48SL_{ab}L_{ab} + 480S^{2}L_{aa} + 480S^{3} + 120S_{:aa}).

Example 4.6: Let m = 3 and let g(ϵ) = e^{2ϵψ}g_{0}. We compute:

∂_{ϵ}(96E_{D}dy) = 48ψ_{;ii}dy
∂_{ϵ}(16τ_{g}dy) = −64ψ_{;ii}dy
∂_{ϵ}(8R_{g,amam}dy) = (8ψ_{;aa} + 16ψ_{;mm})dy
∂_{ϵ}(13L_{aa}L_{bb}dy) = −52ψ_{;m}L_{aa}dy  (4.7)
∂_{ϵ}(2L_{ab}L_{ab}dy) = −4ψ_{;m}L_{aa}dy
∂_{ϵ}(96SL_{aa}dy) = −192ψ_{;m}S + 48ψ_{;m}L_{aa}
∂_{ϵ}(192S^{2}) = 192ψ_{;m}S.

Consequently

∂_{ϵ}{a_{3}^{bd}(y, D, B(S))}dy = −8(384)^{−1}(4π)^{−1}ψ_{:aa}dy  (4.8)

which again is in divergence form.

§5 Isospectral problems for the Levi-Civita connection

In this section, we study the heat equation asymptotics for

D := D(g, ∇^{g});  (5.1)

this operator is not conformally covariant.

Let M be a compact Riemannian manifold without boundary. Let

\begin{matrix}c_{1}(m) = (m − 1)^{−1}(m + 5), \\ c_{2}(m) = (m − 1)^{−2}(m + 5)^{2}, \\ c_{3}(m) = (m − 1)^{−1}(m + 4). \end{matrix}  (5.2)

We recall the notational convention from §3 that Roman letters index orthonormal frames and Greek letters index coordinate frames for the tangent and cotangent bundles; we sum over repeated indices.

Theorem 5.1:

(a) a_{0}(x, D) = (4π)^{−m/2}.

(b) a_{2}(x, D) = 6^{−1}(4π)^{−m/2}c_{1}(m)τ_{g}.

(c) a_{4}(x, D) = 360^{−1}(4π)^{−m/2}{5c_{2}(m)τ_{g}^{2} − 2|ρ_{g}|^{2} + 2|R_{g}|^{2} + 12c_{3}(m)τ_{g;ii}}.

Remark: This shows the invariants we have been studying are non-trivial.

Proof: Since ∇ = ∇^{g}, θ = ω = 0 and consequently Ω = 0. Furthermore, by Lemma 3.3,

E_{D} = (m − 1)^{−1}τ_{g}.  (5.3)

Theorem 5.1 is now a direct consequence of Theorem 3.2.

Berger [1] and Tanno [22] showed that the spaces of constant curvature were characterized by the spectrum of the standard Laplacian if m ≤ 5; we generalize their methods to show the same result holds for the operator we are considering:

Theorem 5.2: Let M_{i} be D isospectral manifolds. Assume M_{1} has constant sectional curvature c.

(a) If m ≤ 5, then M_{2} has constant sectional curvature c.

(b) If m = 6 and if M_{2} is Einstein, then M_{2} has constant sectional curvature c.

Remark: Since the volume of M is a D isospectral invariant, this shows that if M_{1} is isometric to the standard sphere S^{m} or real projective space RP^{m}, then so is M_{2} for m ≤ 5.

Remark: Vigneras [23] has constructed examples of constant curvature isospectral Riemann surfaces of constant sectional curvature −1; these examples are also D isospectral so the spectrum of D does not characterize a Riemann surface.

Proof: Let c and ϵ be real parameters. Define:

\begin{matrix}ρ̃_{g} = & ρ_{g} − (m − 1)cg \\ W_{g,ijkl}(ϵ) = & R_{g,ijkl} − ϵ\{ρ̃_{g,il}δ_{jk} + ρ̃_{g,jk}δ_{il} − ρ̃_{g,ik}δ_{jl} − ρ̃_{g,jl}δ_{ik}\} \\ − c(δ_{il}δ_{jk} − δ_{ik}δ_{jl}). \end{matrix}  (5.4)

Lemma 5.3:

(a) M has constant sectional curvature c ⇔ |W_{g}|^{2} = |ρ̃_{g}|^{2} = 0.

(b) |ρ̃_{g}|^{2} = |ρ_{g}|^{2} + O(c).

(c) |W_{g}|^{2} = |R_{g}|^{2} − (8ϵ + (8 − 4m)ϵ^{2})|ρ̃_{g}|^{2} + 4ϵ^{2}τ_{g}^{2} + O(c).

Proof: If M has constant sectional curvature c, then

R_{g,ijkl} = cδ_{il}δ_{jk} − cδ_{ik}δ_{jl}  (5.5)

so ρ_{g,ij} = (m − 1)cδ_{ij} and ρ̃_{g} = 0. Thus

W_{g,ijkl} = R_{g,ijkl} − cδ_{il}δ_{jk} − cδ_{ik}δ_{jl} = 0.  (5.6)

Conversely, if ρ̃_{g} = 0, then (5.6) holds so R_{g} has constant sectional curvature c. This proves (a); (b) is immediate. We prove (c) by computing:

|W_{g}|^{2} = W_{g,ijkl}W_{g,ijkl} (5.7)

= |R_{g}|^{2} − 2ϵ{ρ̃_{g,il}R_{g,ijjl} + ρ̃_{g,jk}R_{g,ijki} − ρ̃_{g,ik}R_{g,ijkj} − ρ̃_{g,jl}R_{g,ijil}}
+ϵ^{2}{ρ̃_{g,il}ρ̃_{g,il}δ_{jk}δ_{jk} + ρ̃_{g,jk}ρ̃_{g,jk}δ_{il}δ_{il} + ρ̃_{g,ik}ρ̃_{g,ik}δ_{jl}δ_{jl} + ρ̃_{g,jl}ρ̃_{g,jl}δ_{ik}δ_{ik}
+2ρ̃_{g,il}ρ̃_{g,jk}δ_{jk}δ_{il} − 2ρ̃_{g,il}ρ̃_{g,ik}δ_{jk}δ_{jl} − 2ρ̃_{g,il}ρ̃_{g,jl}δ_{jk}δ_{ik}
−2ρ̃_{g,jk}ρ̃_{g,ik}δ_{il}δ_{jl} − 2ρ̃_{g,jk}ρ̃_{g,jl}δ_{il}δ_{ik} + 2ρ̃_{g,ik}ρ̃_{g,jl}δ_{jl}δ_{ik}} + O(c)
= |R_{g}|^{2} − (8ϵ + (8 − 4m)ϵ^{2})|ρ̃_{g}|^{2} + 4ϵ^{2}τ_{g}^{2} + O(c).

If E is a scalar invariant, let

E(M) := ∫_{M}E(x)dx  (5.8)

be the corresponding integrated scalar invariant. By Theorem 5.1,

1(M), τ_{g}(M), (5c_{2}(m)τ_{g}^{2} − 2|ρ_{g}|^{2} + 2|R_{g}|^{2})(M)  (5.9)

are spectral invariants of D. Let

Ψ(ϵ, m) := 8ϵ + (8 − 4m)ϵ^{2} − 1.  (5.10)

We use Lemma 5.3 to decompose:

\begin{matrix}\{5τ_{g}^{2}c_{2}(m) − 2|ρ_{g}|^{2} + 2|R_{g}|^{2}\}(M) \\ = & \{2|W_{g}|^{2} + 2Ψ(ϵ, m)|ρ̃_{g}|^{2} + (5c_{2}(m) − 8ϵ^{2})τ_{g}^{2} + O(c)\}(M). \end{matrix}  (5.11)

The missing terms are multiples of cτ_{g}(M) and c^{2}(M) which are controlled by a_{0} and a_{2}. Consequently, we can find universal constants α_{i} = α_{i}(c, ϵ) so:

\begin{matrix}(α_{0}a_{0} + α_{2}a_{2} + α_{4}a_{4})(D)(M) \\ = & \{2|W_{g}|^{2} + 2Ψ(ϵ, m)|ρ̃_{g}|^{2} \\ + (5c_{2}(m) − 8ϵ^{2})(τ_{g} − m(m − 1)c)^{2}\}(M). \end{matrix}  (5.12)

If we can choose ϵ so Ψ(ϵ, m) and 5c_{2}(m) − 8ϵ^{2} are positive, this spectral invariant will vanish if and only if M has constant sectional curvature c. We note:

Ψ(\frac{1}{2}, 2) = 3 > Ψ(\frac{1}{2}, 3) = 2 > Ψ(\frac{1}{2}, 4) = 1 > 0.  (5.13)

The function Ψ is maximal when ϵ = (m − 2)^{−1}; (m − 2)^{−1} ∈ [0, \frac{1}{2}] for m ≥ 4 and

Ψ((m − 2)^{−1}, m) = 4(m − 2)^{−1} − 1 > 0 for m = 5.  (5.14)

This completes the proof of (a) since

5c_{2}(m) − 8ϵ^{2} ≥ 5c_{2}(m) − 8(\frac{1}{2})^{2} ≥ 5 − 2 > 0.  (5.15)
In the limiting case m = 6, we take ϵ = (m − 2)^{−1} = \frac{1}{4} so
Ψ((m − 2)^{−1}, m) = 0.  (5.16)

Thus the coefficient of |ρ̃_{g}|^{2} vanishes while the coefficients of

|W_{g}|^{2} and (τ_{g} − m(m − 1)c)^{2}  (5.17)

are positive. Thus this spectral invariant vanishes if and only if W_{g} vanishes and τ_{g} = m(m − 1)c is constant. Since M_{2} is assumed Einstein, ρ̃_{g} vanishes on M_{2}.

§6 Isospectral problems in affine differential geometry

Torsion free connections on T M with symmetric Ricci tensor arise naturally in affine differential geometry and motivate the discussion of §1-§4. We review the geometry briefly and refer to [2, 6, 10, 17] for further details; [19] gives a systematic study of relative geometry.

§6.0 Conjugate triples and Codazzi transformations

The notion of a conjugate triple and a Codazzi transformation is fundamental to the study of affine geometry. We refer to [11] and [19, §4.4] for further details; we shall only summarize the needed results here. We clear the notation of §1.1-1.5 as necessary. Let U, V, and W be vector fields.

Definition: (^{1}∇, g,^{2}∇) is a conjugate triple if g is a pseudo Riemannian metric, if ^{1}∇ and ^{2}∇ are torsion free Ricci symmetric connections, and if for all U, V, and W

Ug(V, W) = g(^{1}∇_{U}V, W) + g(V,^{2}∇_{U}W).  (6.1)

In this setting, the Codazzi equations hold, i.e. ^{1}∇g and ^{2}∇g are totally symmetric (0.3) tensors and ∇^{g} = \frac{1}{2}(^{1}∇ +^{2}∇) where ∇^{g} is the Levi-Civita connection of g.

Definition: Two conjugate triples (^{1}∇, g,^{2}∇) and (^{1} ˜∇, g̃,^{2} ˜∇) differ by a Codazzi transformation if there exists a smooth positive function β so that for all U, V

g̃(U, V) = βg(U, V)
1 ˜∇_{U}V −^{1}∇_{U}V = −g(U, V)grad_{g} log(β)
2 ˜∇_{U}V −^{2}∇_{U}V = U(log(β))V + V (log(β))U.

(6.2)

In this setting we write

(^{1}∇, g,^{2}∇) ⟶^{β} (^{1} ˜∇, g̃,^{2} ˜∇).  (6.3)

The metric g̃ is conformally equivalent to g and the connection^{2} ˜∇ is projectively equivalent to ^{2}∇. We note that if (^{1}∇, g,^{2}∇) is a conjugate triple and if (6.2) holds, then (^{1} ˜∇, g̃,^{2} ˜∇) is also a conjugate triple and the Codazzi equations hold.

Remark: The transformation defined by (6.2) was studied in the setting of statistical manifolds by T. Kurose [9] as 1-conformal equivalence. The emphasis of our approach is on that of Codazzi partial differential equations.

The following is an immediate consequence of the results of §1.

Theorem 6.0.1: Let (^{1}∇, g,^{2}∇) be a conjugate triple, let g be a Riemannian metric,

and let (^{1}∇, g,^{2}∇) ⟶^{β} (^{1} ˜∇, g̃,^{2} ˜∇). Let D = D(g,^{2}∇) and D̃ = D(g̃,^{2} ˜∇). Let M be a

compact Riemannian manifold of dimension m. If ∂M ≠ ∅, we impose Dirichlet boundary conditions.

(a) a_{m}(D, B) = a_{m}(D̃, B).

(b) If ∂M = ∅ and if m is odd, ζ′(0, D) = ζ′(0, D̃).

§6.1 Affine differential geometry.

Let A be a real affine space which is modeled on a vector space V of dimension m + 1. Let V^{∗} be the dual space. If a ∈ A, we may identify

T_{a}A = V and T_{a}^{∗}A = V^{∗}.  (6.4)

Let ⟨·, ·⟩ : V^{∗} × V → R be the natural pairing between V^{∗} and V. Let x be a smooth hypersurface immersion of M into A. If P ∈ M, let

C(M)_{P} = {X ∈ V^{∗} : ⟨X, dx(v)⟩ = 0, ∀v ∈ T_{P} M}  (6.5)

be the conormal space at P; we let C(M) be the corresponding conormal line bundle over M. We assume C(M) is trivial and choose a non vanishing conormal field X.

We say the hypersurface x(M) is regular if and only if there exists a conormal field X such that

rank(X, dX) = m + 1  (6.6)

for every point of M; we impose this condition henceforth. It is then clear that (6.6) is satisfied by any conormal field. Furthermore, X is an immersion X : M → V^{∗} which is transversal to X(M). Define y = y(X) : M → V by the conditions:

⟨X, y⟩ = 1 and ⟨dX, y⟩ = 0.  (6.7)

The triple (x, X, y) is called a hypersurface with relative normalization; we remark that y need not be an immersion. The relative structure equations given below contain the fundamental geometric quantities of hypersurface theory: two connections ^{i}∇, the relative shape (Weingarten) operator B, and two symmetric forms g and B̂. Let ^{A}∇ be the flat affine connection on A.

\begin{matrix}^{A}∇_{v}y = dy(v) = −dx(B(v)), \\ ^{A}∇_{w}dx(v) = dx(^{1}∇_{w}v) + g(v, w)y, \\ ^{A}∇_{w}dX(v) = dX(^{2}∇_{w}v) − B̂(v, w)X. \end{matrix}  (6.8)

The first equation is called the Weingarten equation, the second two are the Gauss equations.

Theorem 6.1.1: Let x : M → A be a non-degenerate hypersurface.

(a) If (x, X, y) is a relative normalization, then (^{1}∇, g,^{2}∇) is a conjugate triple.

(b) All possible relative metrics define a conformal class C; all possible relative conormal connections define a projectively flat Ricci symmetric class P.

(c) A transformation of relative normalization

X̃ = βX and ỹ = β^{−1}(y + dX(grad_{g} log(β))

is equivalent to a Codazzi transformation (^{1}∇, g,^{2}∇) ⟶^{β} (^{1} ˜∇, g̃,^{2} ˜∇).

(d) Assume M is compact and the relative metric g is positive definite; if ∂M ≠ ∅ we impose Dirichlet boundary conditions. Then a_{m}(D(g,^{2}∇)) is an affine invariant of the hypersurface which is independent of the normalization chosen.

(e) If m is odd and if ∂M = ∅, then ζ′(0, D) is an affine invariant of the hypersurface which is independent of the normalization chosen.

Proof: We refer to [11] and [19, §4.4 and §5.1] for the proof of (a)-(c); (d) and (e) follow from (c) and from Theorem 6.0.1.

We assume the relative (or Blaschke) metric g is positive definite henceforth; this means that the immersed hypersurface x(M) is locally strongly convex. We will also assume M is compact and without boundary; this implies M is a hyperovaloid or if m = 2 an ovaloid. In §6.6, we relax this restriction slightly and study surfaces with non empty boundary and also surfaces with indefinite metrics. We plan in a later paper to study manifolds with indefinite metrics and with boundary of higher dimensions. We also plan to study the functional determinant as we have not dealt with this invariant here.

We recall the notational convention from §3 that Roman letters index orthonormal frames and Greek letters index coordinate frames for the tangent and cotangent bundles; we sum over repeated indices.

The relative shape (or Weingarten) operator B is self-adjoint with respect to g and is related B̂ by the identity:

B̂(v, w) = g(B(v), w) = g(v, B(w)).  (6.9)

It is useful to define a (1,2) difference tensor A, a totally symmetric relative cubic form Â, and the Tchebychef form T̂ by:

\begin{matrix}A := \frac{1}{2}(^{1}∇ −^{2}∇), \\ Â(v, w, z) = g(A(v, w), z), \\ T̂(z) = m^{−1}Tr_{g}(A(z, ·)). \end{matrix}  (6.10)

Let ‘;’ denote multiple covariant differentiation with respect to the Levi-Civita connection ∇^{g}. T̂ has the following useful symmetry property [20]:

T̂_{i;j} = T̂_{j;i}.  (6.11)

Let ρ_{·,ij} be the Ricci tensors of ^{·}∇ relative to a local orthonormal frame. We use the metric to raise and lower indices and identify B̂ = B. Then:

ρ_{1,ij} = δ_{ij}B_{kk} − B_{ij} and ρ_{2,ij} = (m − 1)B_{ij}.  (6.12)

We denote the normalized mean curvature by

H := m^{−1}B_{ii};  (6.13)

the normalized traces are then equal:

(m − 1)^{−1}Tr_{g}(ρ_{1}) = (m − 1)^{−1}Tr_{g}(ρ_{2}) = mH.  (6.14)

Since the ^{i}∇ are torsion free, Ricci symmetric connections, we can apply the results of §1-§4 to this setting. Let

1D := D_{g,^{1}∇}, ^{2}D := D_{g,^{2}∇}, and ^{g}D := D_{g,^{g}∇}  (6.15)

be the associated operators of Laplace type on C^{∞}(M); these operators and their spectra are affine invariants of (x, X, y).

We now compute the expressions of Lemma 3.3. Modulo a few changes of sign, the quantities for ^{1}∇ and ^{2}∇ are remarkably similar:

Lemma 6.1.2: Let D = ^{1}D and let ϵ = 1 or let D = ^{2}D and let ϵ = −1. Then:

\begin{matrix}^{D}θ = ϵA, ω_{D} = −\frac{1}{2}ϵmT̂, Ω_{D} = 0, \\ E_{D} = mH − \frac{1}{4}m^{2}|T̂|^{2} + \frac{1}{2}ϵmT̂_{i;i}. \end{matrix}

Remark: This means the connections ∇_{^{1}D} and ∇_{^{2}D} are conjugate with respect to the flat metric and flat connection on the trivial bundle;

(∇_{^{1}D}f_{1})f_{2} + f_{1}(∇_{^{2}D}f_{2}) = d(f_{1}f_{2}) for f_{i} ∈ C^{∞}(M).  (6.16)

Proof: This follows from Lemma 3.3 and (6.10), (6.11), and (6.14).

We combine Lemma 6.1.2 and Theorem 3.2 to see:

Theorem 6.1.3: Let x : M → A be a hyperovaloid. Let D = ^{1}D and let ϵ = 1 or let D = ^{2}D and let ϵ = −1. Then:

(a) a_{0}(x, D) = (4π)^{−m/2}.

a_{0}(D) = (4π)^{−m/2}1(M).

(b) a_{2}(x, D) = (4π)^{−m/2}\{6^{−1}τ_{g} + mH − \frac{1}{4}m^{2}|T̂|^{2} + \frac{1}{2}ϵmT̂_{i;i}\}. a_{2}(D) = (4π)^{−m/2}\{6^{−1}τ_{g} + mH − \frac{1}{4}m^{2}|T̂|^{2}\}(M).

(c) a_{4}(x, D) = (4π)^{−m/2}360^{−1}\{60τ_{g}(mH − \frac{1}{4}m^{2}|T̂|^{2} + \frac{1}{2}ϵmT̂_{i;i} )

+180(mH − \frac{1}{4}m^{2}|T̂|^{2} + \frac{1}{2}ϵmT̂_{i;i})^{2} + 60(mH − \frac{1}{4}m^{2}|T̂|^{2} + \frac{1}{2}ϵmT̂_{i;i})_{;jj} +12τ_{g;kk} + 5τ_{g}^{2} − 2|ρ_{g}|^{2} + 2|R_{g}|^{2}}.

Remark: a_{0}(^{1}D) = a_{0}(^{2}D) and a_{2}(^{1}D) = a_{2}(^{2}D); a_{4}(^{1}D) ≠ a_{4}(^{2}D) in general.

§6.2 Equiaffine geometry

We fix a volume form or determinant on A. Then there is, up to orientation, a unique equiaffine unimodular normalization which is invariant under the unimodular group. Admitting arbitrary volume forms, all such normalizations differ by non-zero constant factors. The class of equiaffine normalizations can be characterized within the class of relative normalizations by the vanishing of the Tchebychev form. We call such a hypersurface with equiaffine normalization a Blaschke hypersurface.

The vanishing of T simplifies the local invariants greatly. Recall that if P is a local invariant, then

P (M) = ∫_{M}P (x)dν_{x}.  (6.17)

Lemma 6.2.1: Let x : M → A be a hyperovaloid. With equiaffine normalization:

(a) ^{1}D = ^{2}D.

(b) a_{2}(^{2}D) = (4π)^{−m/2}{6^{−1}τ_{g} + mH}(M).

(c) a_{4}(^{2}D) = (4π)^{−m/2}360^{−1}{60mHτ_{g} + 180m^{2}H^{2} + 5τ_{g}^{2} − 2|ρ_{g}|^{2} + 2|R_{g}|^{2}}(M).

Proof: (a) follows from Lemma 6.1.2; (b) and (c) follow from Theorem 6.1.3.

Remark: In fact (a) is a characterization of the equiaffine normalization within the relative class since ^{1}D = ^{2}D if and only if T̂ = 0.

We draw the following consequence:

Corollary 6.2.2: Let D = ^{1}D or D = ^{2}D.

(a) Let (x,X,y) be a relative normalization. Then

(4π)^{m/2}\{a_{2}(D) − \frac{m−1}{m+5}a_{2}(^{g}D)\} ≤ mH(M).

Equality holds if and only if the normalization is equiaffine.

(b) If the normalization is equiaffine, then

m(m + 5)H(M) ≤ 6(4π)^{m/2}a_{2}(D) ≤ m(m + 5)(H + J)(M).

Equality holds on the left or on the right hand side if and only if the hyperovaloid is an ellipsoid.

Proof: We use Theorem 6.1.3 and Theorem 5.1 to prove (a) by computing:

(4π)^{m/2}\{a_{2}(^{·}D) − \frac{m−1}{m+5}a_{2}(^{g}D)\} = \{mH − \frac{1}{4}m^{2}|T̂|^{2}\}(M).  (6.18)

We use the relative Pick invariant J to prove (b). Let

J = {m(m − 1)}^{−1}|A|^{2};  (6.19)

J ≥ 0. Then the equiaffine theorema egregium yields

τ_{g} = m(m − 1)κ = m(m − 1)(H + J).  (6.20)

Consequently

\begin{matrix}m(m + 5)H ≤ m(m + 5)H + m(m − 1)J = τ_{g} + 6mH, \\ τ_{g} + 6mH = (1 + 6(m − 1)^{−1})τ_{g} − 6mJ ≤ (m + 5)(m − 1)^{−1}τ_{g}. \end{matrix}  (6.21)

The two inequalities of (b) now follow. Furthermore, equality holds if and only if J ≡ 0 or equivalently A ≡ 0. This means the hyperovaloid is an ellipsoid by a theorem of Maschke [19, §7.1.1].

§6.3 Centroaffine geometry

We take y = −x to define the centroaffine normalization. This geometry is invariant with respect to the subgroup of regular affine mappings which fix the origin of A. Let X_{e} be the equiaffine conormal field and let

ζ = −⟨x, X_{e}⟩  (6.22)

be the equiaffine support function. We choose the orientation so that ζ > 0. Then

B = g, mH = m, and T̂ = \frac{2+m}{2m}d ln(ζ).  (6.23)

Lemma 6.3.1: Let D = ^{1}D or D = ^{2}D. With the centroaffine normalization:

(a) a_{0}(D) = (4π)^{−m/2}1(M). a_{2}(D) = (4π)^{−m/2}\{6^{−1}τ_{g} + m − \frac{1}{4}m^{2}|T̂|^{2}\}(M).

(b) 6(4π)^{m/2}{a_{2}(D) − ma_{0}(D)} ≤ τ_{g}(M) with equality if and only if the hyperovaloid is an ellipsoid.

(c) Let x(M^{2}) and x̃(M^{2}) be ovaloids in 3 space with centroaffine normalization which are D isospectral. Then x is an ellipsoid if and only if x̃ is an ellipsoid.

Proof: (a) follows from Theorem 6.1.3 and equation (6.23). We compute:

6(4π)^{m/2}\{a_{2}(D) − ma_{0}(D\} = (τ_{g} − \frac{3m^{2}}{2} |T̂|^{2})(M).  (6.24)

This yields the desired inequality; equality holds when T̂ = 0. This is equivalent to assuming the support function is constant which means the hyperovaloid is an ellipsoid, see [5, §III 5.5] and [19, §7.2].

If m = 2, the total scalar curvatures are equal by the Gauss-Bonnet theorem. For the centroaffine normalization, we have

Tr_{g}(ρ_{^{1}∇}) = m.  (6.25)

If x(M^{2}) is an ellipsoid, then T̂ = 0 so |T̂|^{2}[x(M^{2})] = 0 and hence

|T̂|^{2}[x̃(M^{2})] = 0.  (6.26)

Since m = 2,

5τ_{g}^{2} − 2|ρ_{g}|^{2} + 2|R_{g}|^{2} = 6τ_{g}^{2}.  (6.27)

It now follows that x̃(M^{2}) is an ellipsoid.

Remark: If m = 2, τ_{g}(M) = 8π by the Gauss-Bonnet theorem so (b) becomes the following inequality which is determined by spectral data:

3(a_{2}(D) − 2a_{0}(D)) ≤ 1.  (6.28)

§6.4 Polar hypersurfaces

Consider a pair of non-degenerate hypersurfaces x : M → V and x^{∗} : M → V^{∗} such that

⟨x, x^{∗}⟩ = −1, ⟨x^{∗}, dx⟩ = 0, and ⟨dx^{∗}, x⟩ = 0.  (6.29)

Such a pair is called a polar pair. These relations are satisfied for a non degenerate hypersurface x : M → V and its centroaffine conormal map x^{∗} := X : M → V^{∗}. This indicates the important role which centroaffine differential geometry has for the investigation of polar pairs. They also provide interesting examples to illustrate the spectral invariants we have studied in the preceeding sections. We recall some facts about the controlling geometry of polar pairs and refer to [12, §7.2] for further details.

\begin{matrix}B̂ = g = g^{∗} = B̂^{∗}, ^{1}∇ = ^{2}∇^{∗}, ^{2}∇ = ^{1}∇^{∗}, A = −A^{∗}, \\ T̂ = −T̂^{∗}, J = J^{∗}, ρ_{^{1}∇} = ρ_{^{2}∇} = ρ_{^{1}∇^{∗}} = ρ_{^{2}∇^{∗}}, \\ ^{1}R = ^{2}R = ^{1}R^{∗} = ^{2}R^{∗}. \end{matrix}  (6.30)

Thus as an immediate consequence we see that

\begin{matrix}D(^{1}∇, g) = D(^{2}∇^{∗}, g^{∗}) & , D(^{2}∇, g) = D(^{1}∇^{∗}, g^{∗}), \\ a_{2}(D(^{1}∇, g)) & = a_{2}(D(^{1}∇^{∗}, g^{∗})). \end{matrix}  (6.31)

In centroaffine geometry, the spectral coefficients a_{n}(x, ·) for any of the operators we have been considering and for any n are determined if the metric and the Tchebychev field are given. This is of interest because of the following two global uniqueness results:

Lemma 6.4.1 (see [18, §4.5]): Let x_{i} : M → V be hyperovaloids with centroaffine normalization.

(a) If g_{1} = g_{2} and T̂_{1} = T̂_{2}, then x_{1} and x_{2} are centroaffinely equivalent.

(b) If g_{1} = g_{2} and T̂_{1} = −T̂_{2}, then x_{2} is centroaffinely equivalent to the polar hyperovaloid x_{1}^{∗} of x_{1}.

§6.5 Two-dimensional manifolds

Let M be a simply connected compact surface and let x be an ovaloid. Let P (M) be the Pick functional (see [2, p. 174]);

P (M) := (J − \frac{3}{2}|T̂|_{g}^{2})(M).  (6.32)

Theorem 6.5.1:

(a) P (M) = \frac{2}{3}π(7 − 3a_{2}(^{2}D)).

(b) (2H − |T̂|_{g}^{2})(M) = 4π(a_{2}(^{2}D) − \frac{1}{3}).

(c) (2H − |T̂|^{2})(M) and P (M) are independent of the relative normalization chosen for the ovaloid.

(d) P (M) is non-negative; P (M) vanishes if and only if the ovaloid is an ellipsoid; thus the Pick functional measures the deviation from an ellipsoid.

(e) a_{2}(^{2}D) ≤ 7/3. If a_{2}(^{2}D) = 7/3, then x is an ellipsoid.

Remark: We recall that a_{2}(^{1}D) = a_{2}(^{2}D) so the conclusions of Theorem 6.5.1 also hold for the operator ^{1}D.

Proof: We use Theorem 6.1.3 and set m = 2 to see:

a_{2}(^{2}D) = (4π)^{−1}{6^{−1}τ_{g} + 2H − |T̂|^{2}}(M).  (6.33)

The relative theorema egregium [19, (4.12.2.2)] shows:

τ_{g} = 2(J + H − 2|T̂|_{g}^{2}).  (6.34)

By the Gauss-Bonnet theorem,

τ_{g}(M) = 8π;  (6.35)

(a) and (b) follow from (6.33), (6.34), and (6.35).By Theorem 6.1.1, a_{2}(g,^{2}∇) is an affine invariant of the embedding x; (c) now follows from (a) and (b). We note that (c) can also be derived directly using [19, §5.1].

To prove the remaining assertions, we recall [20, §IV.6] that the symmetric (1.2) tensor field

Ã(v, w) := A(v, w) − \frac{m}{m+2}(T̂(v)w + T̂(w)v + g(v, w)T)  (6.36)

is independent of the relative normalization and that Â ≡ 0 characterizes quadrics. We estimate:

0 ≤ |Ã|_{g}^{2} = m(m − 1)J − \frac{3m^{2}}{m+2}|T̂|_{g}^{2}.  (6.37)

Consequently, if m = 2,

2P (M) = |Ã|_{g}^{2}(M);  (6.38)

(d) and (e) now follow.

Theorem 6.5.1 has important consequences as it permits a comparison of different geometric structures (Euclidean, equiaffine, centroaffine) on a given ovaloid. Recall that the Tchebychev forms satisfy the following relations in dimension 2; we refer to [12] and [19, §6.1-2] for details. We denote the Euclidean, centroaffine, and equiaffine structures by ‘E’, ‘c’, and ‘e’. Let K = \frac{1}{2}τ_{g(E)} be the Euclidean Gauss curvature.

(a) Euclidean: T̂_{E} = −\frac{1}{4}d ln(K) for K the Euclidean Gauss curvature.

(b) Centroaffine: T̂_{c} = d ln(ζ_{e}) for ζ_{e} the equiaffine support function.

(c) Equiaffine: T̂_{e} = 0. Let x be an ovaloid with H_{e} as equiaffine and H_{E} as Euclidean mean curvature. Let dν_{e} be the equiaffine volume element, dν_{I} the volume element with respect to the first fundamental form, and dν_{II} the volume element with respect to the second fundamental form II; this is the relative metric with respect to the Euclidean normalization. Let |_{·} be the norm of a tensor with respect to the form under consideration. We use Theorem 6.5.1 to see

∫_{M}2H_{e}dν_{e} = ∫_{M}\{2H_{E} − \frac{1}{16}|d ln(K)|_{II}^{2}\}dν_{II}  (6.39)

is the conserved quantity. K is a Euclidean invariant; on the other hand, the Tchebychev form T (E) is an affine invariant. From [19, §5.1] we can compare different relative geometries and get

∫_{M}|d ln(K)|_{e}^{2}dν_{e} = ∫_{M}|d ln(K)|_{II}^{2}dν_{II}.  (6.40)

This gives a lower bound for the Willmore functional in terms of equiaffine data.

Corollary 6.5.2: Let x be an ovaloid. Then

∫_{M}H_{e}dν_{e} + \frac{1}{8}∫_{M}|d ln(K)|_{e}^{2}dν_{e} ≤ ∫_{M}H_{E}^{2}dν_{I}.

Moreover ∫_{M}H_{e}dν_{e} = ∫_{M}H_{E}^{2}dν_{I} if and only if x is a sphere.

Proof: By definition, dν_{II} = K^{1/2}dν_{I} and K^{1/2} ≤ H_{E}. Equality holds if and only if d ln(K) = 0.

In our next step, we compare the equiaffine and the centroaffine geometry of an ovaloid with origin inside the ovaloid; thus the support function ζ = ⟨x, X⟩ (in any relative geometry) is positive by an appropriate choice of the orientation of the conormal. We denote the equiaffine structures by “e” and the centroaffine structures by “c”. Then, see [19, §5-6] and [12],

ζ_{e}g_{c} = g_{e}, ^{e}ζdν_{c} = dν_{e}, |^{c}T |_{c}^{2}dν_{c} = |(d ln(^{e}ρ)|_{e}^{2}dν_{e}.  (6.41)

Thus Theorem 6.5.1 gives another upper bound for the total equiaffine mean curvature:

Corollary 6.5.3: ∫_{M}H_{e}dν_{e} ≤ vol(M, g_{c}) with equality if and only if the ovaloid is an ellipsoid.

Proof: We see that

∫_{M}\{H_{e} + \frac{1}{2}|d ln(ρ_{e})|_{e}^{2}\}dν_{e} = vol(M, g_{c}).  (6.42)

Equality holds if and only if d ln(ρ_{e}) = 0.

Remark: There is a similar relation between the Euclidean and the centroaffine data which we do not give explicitly.

§6.6 Two-dimensional manifolds with boundary

Throughout this section, we have assumed M compact without boundary. We conclude by generalizing Theorem 6.5.1 to surfaces with boundary. Let T_{N} be the component of T with respect to the inward pointing unit normal on M.

Theorem 6.6.1: (2H −|T̂|^{2})(M)+(T_{N})(∂M) is independent of the normalization chosen.

Proof: We impose Dirichlet boundary conditions; by Theorems 1.1 and 1.2, a_{2}(^{2}D) is an affine invariant. We use Theorem 4.1 to determine the additional boundary integrand and Theorem 6.1.3 to determine the interior integrand to compute:

\begin{matrix}a_{2}(^{2}D) = & (4π)^{−m/2}\{3^{−1}κ_{g}(∂M) \\ + (6^{−1}τ_{g} + mH − \frac{1}{4}m^{2}|T̂|^{2} − \frac{1}{2}mT_{i;i})(M)\} \end{matrix}  (6.43)

where κ_{g} is the geodesic curvature of the boundary. The Gauss-Bonnet theorem for manifolds with boundary yields

(2κ)(∂M) + τ_{g}(M) = (4π)χ(M).  (6.44)

We integrate by parts to see

−\frac{1}{2}mT_{i;i}(M) = \frac{1}{2}mT_{N}(M).  (6.45)

Remark: The direct calculation done in §3 shows that E_{D} is invariant under projective transformations and E_{D}dvol is invariant under conformal transformations. The calculation did not use the assumption that the metric was positive definite; consequently Theorems 6.5.1 and 6.6.1 hold for indefinite metrics as well.

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Received August 1, 1994.