Euclidean Embeddings and Riemannian Bergman Metrics
Consider the sum of the first eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree to be those metrics induced by such embeddings. Our main result is to identify a natural sequence of Bergman metrics approximating any given Riemannian metric. In particular we have constructed finite dimensional symmetric space approximations to the space of all Riemannian metrics. Moreover the construction induces a Riemannian metric on that infinite dimensional manifold which we compute explicitly.
Let be a closed smooth manifold and the space of all smooth Riemannian metrics on . The main result of this thesis is to use a reference Riemannian metric to define and analyze a sequence of finite dimensional approximations of which we call (Riemannian) Bergman spaces.
Let be the sum of the first real eigenspaces for the Laplace-Beltrami operator, and let be the dimension of this vector space. Then an ordered basis of eigenfunctions for determines a map to Euclidean space:
It follows from the density of eigenfunctions that, for sufficiently large, such a map is an embedding of the manifold into Euclidean space.
We define, for the Bergman spaces to be the collection of pullbacks of the Euclidean metric by bases of eigenfunctions:
If two bases differ by an orthogonal transformation then the pullback metrics are the same so we have a natural map from inner products to Bergman metrics
defined by taking an inner product to the pullback of the Euclidean metric by an orthonormal basis. Each Bergman space is thus the image of the symmetric space in .
The main consequence of our results is Corollary LABEL:DensityCor that the Bergman spaces are dense in in the topology
and there is a natural approximation map defined by (LABEL:HilbDef). Consequently the symmetric space metrics on induce a Riemannian metric on , which we shall compute explicitly in Theorem 3.
Recall that is a contractible cone in the space of all symmetric covariant two tensor fields. Hence as it is an open subset of a Fréchet space so it is a Fréchet manifold of infinite dimension [Clarke]. This space, as well as its quotient by the group of diffeomorphisms of , has been studied extensively. has the structure of a fiber bundle over the space of smooth normalized volume forms on with fiber [Ebin]. Model theories of quantum gravity [DeWitt] are concerned with the evolution of measures on . There is a natural Riemannian metric on the space of Riemannian metrics called the metric and for which the curvature and geodesics are known [Freed], [Michor]. Geometric flows such as Ricci flow are the integral curves of a vector fields on , at least for short time.
By an approximation of a space we mean simply a sequence of finite dimensional subspaces whose union is dense, together with a compatible sequence of approximation maps. In general approximating subspaces need not be increasing, though our Bergman spaces will have the property . To put our construction in context we review some important examples of approximations in analysis and geometry.
The inspiration and most relevant approximation for us is that of Bergman spaces of Kähler metrics. Let be a Kähler manifold and the space of Kähler metrics cohomologous to . By the lemma we can identify Kähler metrics with potentials:
Moreover we can identify with the space of positive Hermitian metrics on via the map taking such a metric to its associated curvature form.
Then the theory uses an ample line bundle to construct a sequence of spaces approximating as follows. There is a canonical map from to the projectivization of the dual of the space of holomorphic sections of
By the Kodaira embedding theorem, is an embedding and an inner product on induces a Kähler metric on which we may pull back.
Thus there is a map from , the space of Hermitian inner products on , to the space of Kähler metrics on . Namely in the notation of Donaldson we define
by taking an inner product to where denotes the induced metric on the projective space.
The space of Kähler Bergman metrics of height is defined as the image of the map:
There is also a map
which takes a positive Hermitian metric to its associated inner product
Then Tian’s asymptotic isometry theorem [Tian] states that tends back to in the topology. This was later improved to a asymptotic expansion [ZelditchTian]. More precisely we have for any
In particular the union is dense in .
Also the reference metric induces a reference Hermitian metric which we use to identify with the symmetric space . Thus the construction induces a Riemannian metric on given as a limit of pullbacks of the symmetric space metrics:
In fact it follows from [ZelditchTian] (cf. [ChenSun]) that the limiting metric is the well-known Mabucchi-Semmes-Donaldson metric
Below we will briefly mention some of the many applications of these Bergman metrics to important problems in Kähler geometry.
A simpler example of an approximation is given by Fourier series which approximate where is a closed Riemannian manifold. Here the approximating subspaces are the finite dimensional spaces of (smooth) functions spanned by the first eigenfunctions. The approximating maps are given by spectral truncation.
Using Fourier series we can approximate a conformal class of Riemannian metrics on a manifold. Namely fix a reference metric and let be the conformal class of . Then we have the parameterization
and so an approximation of , such as Fourier series, induces an approximation of . A similar setup where the manifold is replaced by a planar domain was used by Sheffield to define Gaussian free fields [Sheffield].
Let be a closed Riemannian manifold of dimension and the associated volume form and the associated non-negative Laplace-Beltrami operator. Unless noted otherwise, metric-dependent objects such as the cosphere bundle are defined with respect to the reference metric . Let be orthonormal Laplace eigenfunctions with eigenvalues ,
We will also use to denote the increasing sequence of distinct eigenvalues of . Let be the eigenfunctions of degree N, that is the eigenspace
and let be the multiplicty of . For define to be the eigenfunctions up to degree N
and . We endow these vector spaces with the restricted inner product which we denote by . Define and to be orthogonal projection from onto and , respectively.
be (ordered) orthonormal bases for , respectively. These may be viewed as maps from to Euclidean space with Euclidean metric .
In Proposition 5 we show that is both injective and an immersion, that is an embedding, for sufficiently large. Thus for such we may consider the pullback of the Euclidean metric on a general Riemannian manifold.
1.4. Previous Results
Assume that is an isotropy irreducible Riemannian manifold, whose definition we shall recall presently. Under this assumption Takahashi [Takahashi] proved that for every we have
for . That is the pullback of the Euclidean metric by the orthonormal basis is, up to rescaling, an isometry of . And thus so is :
Let be the Riemannian isometry group. Recall that the isotropy group at a point in is the subgroup of isometries which fix . This group acts (by the derivative) on the tangent space . A connected Riemannian manifold is said to be isotropy irreducible if the isometry group acts transitively on and the isotropy action at each fixed point is irreducible. Hence such spaces are homogeneous and can be written as . An example is the dimensional sphere .
In the case of the round sphere, the parity of spherical harmonics implies that the metric induced by any basis for is always even. Hence such metrics will not approximate all of . Because of this we focus on pullback metrics by bases for the larger spaces of eigenfunctions .
Zelditch [ZelditchWaves] gave a generalization of (1.1) by showing that the sequence of maps is asymptotically an isometry:
as . We will recall the proof of this result in Proposition 9 below.
A related result about embedding a manifold into using the heat kernel was obtained in [BBG]. Also of interest are so-called eigenmaps, which are employed in a variety of applications to locally parameterize data which lie on a low dimensional manifold embedded in a high dimensional ambient Euclidean space. See [JMS] for mathematical results in that direction.
1.5. Riemannian Bergman Metrics
Equation (1.2) shows that an arbitrary Riemannian metric may be approximated using its own eigenfunctions . However this approximation scheme is not finite-dimensional because the set of all as varies is not.
We remedy this by fixing which we call the reference metric and studying pullbacks by arbitrary bases. That is we define the Bergman metrics of degree to be
Of course if is a basis for then for some . Hence if is an embedding, which it is for by Proposition 5, then so is and so for such the Bergman metrics are Riemannian metrics.
Also note that since if is a basis for it may be extended to a basis for as and then
An equivalent and somewhat more natural way of understanding Bergman metrics is to start with the canoncial map from to the dual of eigenfunctions up to degree
assigning to each point the evaluation functional, and to pull back an inner product on to a Riemannian metric on .
The inner product on induces an inner product on the dual , which we also denote , and we have
To see this let . Then we have
by definition of the dual inner product. In fact that definition is all we used and so the above is true with replaced by any inner product and with a corresponding orthonormal basis.
Concretely, if is positive-definite and symmetric then is another inner product on , corresponding to on the dual where acts by precomposition. Then if is a self-adjoint square root we have
is an orthonormal basis with respect to and so we have
In analogy with the Kähler case we define a map taking inner products on to Riemannian Bergman metrics of degree
given by pulling back under :
Equivalently by (1.7), takes an inner product on to the metric given by pulling back the Euclidean metric on by an orthonormal basis for the corresponding inner product on .
1.6. Summary of Results
Using the tools of semiclassical analysis we will analyze applied to inner products of the form as . Here is a pseudodifferential operator of order zero (hence bounded on ) so is a linear operator on which plays the role of in (1.6), (1.7) above. The inner product is implicitly restricted to . For convenience we will simply write and the inner product will be implicitly restricted to .
Of course for to be an inner product on in the limit, must be symmetric and for . If satisfies the last condition we say that it is positive-definite. Let to denote the principal symbol of .
Our main calculation is the following limit formula for the map defined in (1.8).
A If is a positive-definite symmetric order zero pseudodifferential operator then
Here is the hypersurface volume form on the -cosphere with respect to the volume form on . We view the right side of the theorem as a weighted average of rank one symmetric tensors . Each direction of the cosphere is weighted by the principal symbol of the pseudodifferential operator in that direction. When is the identity the average is unweighted and the theorem reduces to the asymptotic isometry (1.2) above.
The theorem may be equivalently reformulated in terms of orthonormal bases as follows. For as in the theorem, is a sequence of inner products on . Then as in (1.6)
is a corresponding sequence of orthonormal bases, where the square root is taken as an operator on . The theorem says that as the associated pullbacks of the Euclidean metrics are given by
Our next result shows that the integral transform appearing on the right side of Theorem 1 can be inverted.
Let be a Riemannian metric on . Then if is a positive-definite symmetric order zero pseudodifferential operator with principal symbol
for , we have
In particular the Bergman metrics are dense in the space of Riemannian metrics. Continuing the analogy to the Kähler case, we will define (LABEL:HilbDef) a sequence of maps taking a Riemannian metric to an inner product
and satisfying (cf. Corollary LABEL:HilbCor)
Given the maps we can induce a Riemannian metric on by pulling back the symmetric space metric on . Namely we define the length of in the tangent space to on by
It is not a priori clear that the limit exists or that the limiting metric is positive-definite. We have the following result.
The induced metric on is positive-definite and given by
Moreover the metric is independent of the choice of quantization in the definition of the map.
In our context it is somewhat more natural to use the induced metric on to measure lengths of tangent vectors which come from identifying Riemannian metrics with the metrics they induce on the cotangent bundle. Namely let be a tangent vector to in the latter, that is is a symmetric contravariant 2-tensor. Then the identification takes to the tangent vector to given by
where the right hand side is interpreted in local coordinates as multiplication of matrices and then easily seen to be invariant under change of coordinates. In this notation Theorem 3 reads
Lastly we return to bases for the individual eigenspaces . It is only feasible for us to analyze such maps in the case that is the round sphere because in that case the spectral projector is a Fourier integral operator.
Analogously to we define for individual eigenspaces
and prove the following theorem.
Suppose is the sphere and is the round metric. Let be an order zero pseudodifferential operator which is positive-definite and symmetric on for sufficiently large. Then we have
as where is the periodic geodesic flow of .
This generalizes Takahashi’s result for orthonormal bases on the round sphere. Moreover, the theorem leads to a different proof of Theorem 1 with improvement in remainder to in the case of the round sphere.
1.7. Analogy and Motivation
Here we discuss the analogy between Kähler and Riemannian Bergman metrics and review some of the applications of Bergman metrics in Kähler geometry which we hope will help to motivate our Riemannian construction.
The basic connection between Kähler and Riemannian Bergman metrics is that the space of sections of high power of a line bundle has been replaced by the space of eigenfunctions up to some degree . Note that the definition of Kähler Bergman metrics depend only on the polarisation . A reference metric is used only to identify the Bergman space with . In contrast, eigenfunctions do depend on the choice of reference metric hence so do Riemannian Bergman metrics.
According to Tian, the philosophy of Bergman metrics in Kähler geometry is to use the finite dimensional geometries of projective spaces and to approximate the infinite dimensional space of Kähler metrics . This was done most famously by Donaldson [Donaldson1], [Donaldson2] to relate the existence of Kähler metrics of constant scalar curvature to the existence of balanced projective embeddings.
And so naturally our hope in developing Riemannian Bergman metrics is that they will serve an analogous role. Namely that we might use the finite dimensional geometry of Euclidean space and to better understand the infinite dimensional space of Riemannian metrics . The first step in that direction is Theorem 3 where we compute the Riemannian metric on induced by the Bergman spaces.
Zelditch and collaborators [ZelditchRandom] proposed a way of rigorously defining and calculating path integrals over the space of Kähler metrics via limits of integrals over the finite dimensional Kähler Bergman spaces. We speculate that our construction might be similarly used to give an approximation to Sheffield’s Liouville quantum gravity [Sheffield].
In §2 we lay the groundwork for analyzing Bergman metrics via spectral projection kernels. In §3 we recall Zelditch’s asymptotic isometry. The proof of our asymptotic expansion, Theorem 1, is found in §4. Its inversion, Theorem 2, is proven in §5. Then in §LABEL:MetricSection we prove Theorem 3 calculating the induced metric on the space of metrics. Lastly in §LABEL:SphereSection we look at individual eigenbasis maps on the round sphere and prove Theorem 4.
2. The operator
First we prove the aforementioned fact that the eigenbasis maps are eventually embeddings.
For sufficiently large, if is a basis for then is (i) an immersion and (ii) injective.
Because any two bases are related by a non-degenerate linear transformation, it suffices to consider , the orthonormal basis.
(i) Note that if is not an immersion for some then neither is for all . So supposing is not an immersion for arbitrarily large , then for each there is a unit tangent directions such that
Then since is compact there is a subsequence of converging to . So we have
But this is a contradiction because is a basis for in the topology ([Shubin], Proposition 20.1). Hence is an immersion for .
(ii) Similarly, if is not injective for some then neither is for all . So supposing is not injective for arbitrarily large , we have for each points such that
Then since is compact there are subsequences of and converging to and , respectively. So
If this contradicts the fact that is a basis.
If let us work in a coordinate patch containing . Then for sufficiently large . For such view as a tangent vector at and define
Then by compactness of there is a subsequence such that .
We invoke the mean value theorem: for any and every there is a point on the line from to such that has derivative vanishing at that point along that line:
For any , as we have . So
for all , again contradicting the fact that is a basis.
Note the formula for the pullback of the Euclidean metric by a map is given by
It will be useful for us to have an alternate expression for this. To that end we define a differential operator which takes a function on the product and produces a covariant 2-tensor on . That is let
Given we have
or in local coordinates
Note that if is a symmetric function then is a symmetric tensor. Moreover if is the kernel of an operator on we have
are the matrix entries of with respect to the basis . When the operator is smoothing (that is is smooth) we can apply . In particular we have
Our interest in stems from the following observation.
Let . Then we have
where we view as an operator kernel.
This is a straightforward calculation. Note that both sides of the claim are invariant under orthogonal transformation of so let be a positive definite symmetric representative for . Then
as desired. ∎
Note that if is a positive definite and symmetric (pseudodifferential) operator on all of we can apply the lemma to to obtain
3. Asymptotic Isometry
We begin by recalling the proof of ([ZelditchWaves], Proposition 2.3) that the orthonormal basis maps are asymptotically isometric. Note that the original result assumed aperiodicity of the geodesic flow in order to improve the remainder term. Our proof starts with a slightly more general calculation that will be central to the Bergman asymptotics of the next section.
Fix to be a positive Schwartz function with Fourier transform supported where where is the injectivity radius of and . Such a function exists ([Hormander3], Section 17.5) and is the staple of the Fourier Tauberian arguments which are standard in spectral asymptotics.
This is a standard calculation with a parametrix for the wave group and the method of stationary phase. Let and notice that
Let be the Fourier integral operator parametrix given by Hörmander [Hormander4]. That is for within the injectivity radius of we have
where is the Euclidean volume form with respect to the inner product on . For let
denote the Riemannian exponential map and
denote geodesic flow on the cotangent bundle. With this notation, the phase and amplitude in 3.3 may be written as
for some smooth functions . We will use the following facts:
By parametrix we mean that
So since the Fourier transform maps Schwartz functions to Schwartz functions, we can substitute for and substitute to get
modulo . The leading term in of the symbol comes from differentiating the phase and so we have
We apply stationary phase and note that the phase is the same as in the proof of the Weyl law. That is for fixed the unique stationary point is and and it is non-degenerate with Hessian signature zero. Hence we have
as desired. ∎
We will need the following simple lemma concerning the average of rank one tensors over a sphere.
Let be an -dimensional (real) inner product space and the corresponding cosphere. Then
In normal coordinates where is the identity matrix we have
and note that for
while when we have
Now we can deduce the limit of Bergman metrics corresponding to orthonormal bases.
Proposition 9 ([ZelditchWaves], Proposition 2.3).
Since is increasing in we can subtract to get the useful estimate
for some positive constant .
4. Bergman Asymptotics
Next we derive the Bergman metric asymptotics:
If is a positive-definite and symmetric order zero pseudodifferential operator we have
The proof of the theorem combines two standard arguments. The first, which is formalized in Lemma 10 below, is that
is “lower order” and so for the sake of asymptotics it suffices to study whose smoothed out asymptotics we have already worked out. Then we need to adapt the Tauberian argument to this setting since
is not in general non-decreasing.