# Mean Curvature and the Wave Invariants of the Basic Spectrum for a
Riemannian Foliation

M. R. Sandoval  Department of Mathematics, Trinity College
Hartford, CT 06106 United States

###### Abstract

Given a (possibly singular) Riemannian foliation $\mathcal{F}$ with closed
leaves on a compact manifold $M$ with an adapted metric, we investigate the
wave trace invariants for the basic Laplacian about a non-zero period. We
compare them to the wave invariants of the underlying Riemannian orbifold that
exists when the leaves in the regular region are identified to points,
equipped with the metric that is transverse to the leaves of the foliation.
Recalling that the basic Laplacian differs from the underlying orbifold
Laplacian by a term that is the mean curvature vector field associated to the
foliation, we show that the first wave invariant about any non-zero period $T$
corresponding to geodesics perpendicular to the leaves that all lie entirely
in the regular region of $M$ is independent of the mean curvature vector field
and depends only on the underlying orbifold structure of the leaf space
quotient. Similar results hold on the singular strata whenever the transverse
geodesic flow remains confined to the strata defined by the leaf dimension of
the foliation. Conversely, closed geodesics that pass through the exceptional
leaves depend on the full laplacian, including the leaf-wise metric on the
ambient space.We also discuss families of representations that yield the same
leaf space and basic spectrum. We use this to give conditions under which non-
trivial isotropy for orbifold quotients can be detected.

###### keywords:

Spectral geometry, Laplace operator, $G$-invariant spectrum, orbifolds, orbit
spaces, group actions, singular Riemannian foliations

###### MSC:

[2010] 58J50 , 58J53 , 22D99 , 53C12

##  1 Introduction

###  1.1 Setting

Let $(M,\mathcal{F})$ be a (possibly) singular Riemannian foliation
$\mathcal{F}$ with closed, but not necessarily connected, leaves on a compact
manifold, $M$, equipped with an adapted Riemannian metric $g.$ (See [20],
Chapter 6, for basic definitions.) Recall that a Riemannian foliation is
singular if and only if the leaves do not have constant dimension; otherwise
the foliation is said to be regular. We note that if the leaves are not
closed, we may always take the closures of the leaves; by [7], the foliation
by leaf closures defines a singular Riemannian foliation on $M$. Hereafter, we
will refer to $M$ as the ambient manifold, and denote its dimension by $n.$
The leaf space $Q$ of $(M,\mathcal{F})$ is the space obtained by identifying
the leaves of $\mathcal{F}$ to points and with the closed leaf hypothesis, it
is a Hausdorff space, whose dimension we denote by $q$; furthermore, it is
naturally equipped with a metric, $g_{Q}$ inherited from the adapted metric on
$M$ via the restriction of the adapted metric to the vectors that are
orthogonal to the leaves. This metric is independent of the leaf-wise
coordinates, by definition of the adapted metric. We denote the leaf space
quotient map by $\pi_{Q}:M\rightarrow Q.$ The quotient $Q$ can be thought of
as a potentially singular generalization of a Riemannian manifold.

Examples of types of singular spaces that arise as leaf spaces in this way
include some Alexandrov spaces, space of objects of proper Lie groupoids
equipped with a transversally invariant Riemannian metric, Riemannian
orbifolds, and (of course) manifolds. An important feature of this setting is
that the same leaf space $(Q,g_{Q})$ may be obtained as a leaf space quotient
in more than one way–that is, very different foliated spaces
$(M_{1},\mathcal{F}_{1})$ and $(M_{2},\mathcal{F}_{2})$ may produce the same
quotient $Q=M_{1}/\mathcal{F}_{1}=M_{2}/\mathcal{F}_{2}$ with the same induced
metric $g_{Q}$. We refer to a particular Riemannian foliation
$(M,\mathcal{F},g)$ that gives rise to $Q$ as a representation of $(Q,g_{Q}).$
In fact, one can generate infinite families of representations that produce
the same $(Q,g_{Q})$. (See Section 2 for examples.)

Associated to each representation of $(Q,g_{Q})$ given by $(M,\mathcal{F},g)$,
we have the following spectral problem. Consider the functions on $M$ that are
constant along the leaves of the foliation; these are known as basic
functions, and denoted by $C^{\infty}_{B}(M,\mathcal{F}).$ Such functions
naturally descend to functions on the leaf space $Q=M/\mathcal{F}$. The
restriction of the Laplacian on $M$ to $C^{\infty}_{B}(M,\mathcal{F})$ is
known as the basic Laplacian, $\Delta_{B}$, and its spectrum is known as the
basic spectrum, denoted by $spec_{B}(M,\mathcal{F}).$ It is well-known that
the basic spectrum depends to some extent on the choice of the representation
$(M,\mathcal{F},g),$ including the leaf-wise metric, and generalizes the
Laplace spectrum on functions on $M/\mathcal{F}$, when the leaf space has the
structure of a manifold or an orbifold.

Before proceeding further, we note that in the literature on singular
Riemannian foliations that the leaf space has an orbifold structure if and
only if it has the property of being infinitesimally polar, see [19]. This is
a property of the quotient, which is independent of the choice of
representation; thus, having a Riemannian orbifold structure is a property of
the leaf space quotient, in the sense that this property holds for all
representations as singular or regular)Riemannian foliations that yield the
same quotient and transverse metric. This yields a much broader definition of
a Riemannian orbifold than the traditional definition found in the orbifold
literature. (See, for example, [3]) We use this more flexible definition of
orbifold here, as it allows for a larger, more interesting version of spectral
theory.

###  1.2 Families of Spectral Problems

For a given representation $(M,\mathcal{F},g)$ of $(Q,g_{Q}),$ we can consider
the region where the leaf dimension is maximal, denoted by $M_{reg}$. On
$M_{reg}$, which is open and dense in $M$, we have distinguished coordinates
of the form $(x,y)$ where $y$ denotes the leafwise coordinates and $x$ denotes
the transverse coordinates. With respect to these coordinates, the metric can
be written as $g(x,y)=g_{Q}(x)+g_{L}(x,y)$ where $g_{L}(x,y)$ denotes the
leaf-wise metric and $g_{Q}(x)$ is the transverse metric; this is due to the
nature of the adapted metric on $M$. For every choice of a representation of
$(Q,g_{Q})$ we obtain a basic spectrum. If we consider all such
representations of $(Q,g_{Q})$, we obtain a family of spectral problems (and
basic spectra), which are all associated to the metric structure of
$(Q,g_{Q})$. Denote the set of all such representations by
$\mathcal{R}(Q,g_{Q})$.

It is natural to ask the following questions:

  1. (1)

To what extent does the basic spectrum of a given representation detect the
properties of the ambient space $M$ versus the properties of the leaf space
$Q$?

  2. (2)

Can one assign a spectral invariant to $(Q,g_{Q})$ by considering the family
of basic spectra or a suitably chosen subfamily? A spectral invariant of the
quotient $Q$ would have to be associated to the family of all representations
in some natural way. For example, it could either depend on a particular
canonically chosen representation (such as the orthonormal frame bundle
$M=\mathcal{F}r(\mathcal{O})$ and the $O(q)$ for an orbifold $\mathcal{O}$),
or some subfamily, or be independent of all representations.

  3. (3)

For an orbifold quotient $Q=\mathcal{O}$, to what extent can the family of
basic spectra be used to distinguish properties of $(Q,g_{Q})$ or its singular
sets111There are different notions of singular sets arising from the
literature of singular Riemannian foliations versus the corresponding notion
from the literature on orbifolds. These distinctions are clarified in Section
2. Both sets of ideas will be explored in this paper.?

We will focus primarily on two types of leaf space quotients: when $Q$ is
either a manifold or an orbifold (with non-empty orbifold singular set). These
situations are distinguished by the fact that the family of representations in
these cases will always contain a representation that is a regular Riemannian
foliation–namely the orthonormal frame bundle representation which is an
example of a geometric resolution of the quotient. (See [18] for definitions.)
In this case, it is always possible to assign spectral invariants to
$(Q,g_{Q})$ via the basic wave trace invariants in a well-defined way that is
independent of the representation. In fact, manifold and orbifold quotients
are characterized by the existence of a representation that is a geometric
resolution.) Basic wave invariants can be assigned via the geometric
resolution for each value of $T$ in the length spectrum of $Q$ which we denote
by $Lspec(Q,g_{Q})$. For leaf spaces that are more singular than than orbifold
quotients, it is not clear that there is a way to assign spectral invariants
to $(Q,g_{Q})$ due to the lack of a representation that is a geometric
resolution.

To shed light on the questions above, we briefly review the definition of
singular sets in the sense of singular Riemannian foliations. We refer the
reader to [20] for standard definitions.

######  Definition 1.1.

Given a representation of $(Q,g_{Q})$ denoted by $(M,\mathcal{F}),$ we
consider first the stratification of the ambient space $M$.

  * 1.

A foliation for which the leaf dimension is constant is a regular Riemannian
foliation and the leaf space is, at worst, an orbifold. Thus, if a Riemannian
foliation is regular, then its leaf space quotient is an orbifold (possibly
with singularities), however the converse is false: an orbifold can also be
represented by a singular Riemannian foliation. (See, for example, [16].)

  * 2.

If a Riemannian foliation is singular rather than regular, then let $d(x)$
denote the function that assigns to every $x\in M$ the dimension of the leaf
containing $x,$ with values $k,$ ranging over $0\leq k_{1}\leq k\leq k_{N}<n,$
with $k_{1}$ and $k_{N}$ denoting the minimal and maximal values for $k$,
respectively. (Here we assume there are no dense leaves so that $k_{N}<n$.)
The function $d(x)$ is lower semi-continuous on $M,$ (see [20], Chapter 5),
and it defines a stratification of $M$ with each stratum defined as
$\Sigma_{k}=d^{-1}(k)$. Thus, each stratum $\Sigma_{k}$ is the (possibly
disconnected) union of closed leaves of dimension $k$, and is an embedded
manifold that is regularly foliated by the $k$-dimensional leaves, denoted by
$(\Sigma_{k},\mathcal{F}_{k})$. The images of these sets in the quotient are
orbifolds $\mathcal{O}_{k}=\Sigma_{k}/\mathcal{F}_{k}.$

  * 3.

The lower semi-continuity of $d(x)$ implies that the stratum $M_{reg}$ or
$\Sigma_{k_{N}},$ is an open, dense set and that that for each stratum,
$\Sigma_{k}$, its closure $\overline{\Sigma_{k}}\subset\cup_{\ell\leq
k}\Sigma_{\ell}.$ The strata which are in the complement of the regular
stratum are referred to collectively as the singular strata, and their leaves
are known as exceptional leaves.

######  Remark 1.2.

We emphasize that the singular sets depend on the choice of the
representation. For a given $Q,$ the stratification is not necessarily
constant across the family of representations, unless $Q$ is a manifold or an
orbifold that does not admit singular representations. (Such an orbifold would
probably have to be a very good orbifold, given that the representations are
global quotients.) In fact, some representations are better than others
depending on whether or not one wishes capture the orbifold singularities in
the regular stratum or the singular strata. In Section 2.1, we illustrate this
phenomenon for various examples. The main idea here is to use the fact that
some representations are preferred over others for answering particular
questions. For the purpose of using the wave invariants to address the first
two questions, we will be interested in choosing representations whose strata
are well-situated with respect to the dynamics of the geodesic flow across
$M$, as we shall see below. To address the third question, we will use a
representation of an orbifold that allows us to extract the length spectrum of
closed geodesics in $Q$ whose length is affected by passing through an
orbifold singularity. We will describe this phenomenon below and prove the
corresponding results in the Section 4.

First, we describe how two representations in a family can yield the same
basic spectrum.

######  Remark 1.3.

To that end, we make the following remarks.

  1. 1.

The basic spectrum is affected by leaf-wise data–both the leaf dimension and
the leaf volume. The role of the leaf volumes in spectral questions involving
the basic spectrum is evident from the well-known formula which is valid on
$M_{reg}$ relating the basic Laplacian on the ambient space $M$, $\Delta_{B}$,
with the Laplacian on $\mathcal{O}$, denoted by $\Delta_{\mathcal{O}}$:

$\Delta_{B}f=\Delta_{\mathcal{O}}f-H_{*}(f),$ (1)

where $f$ is a basic function on $M$, which descends to a function on the leaf
space, also denoted by $f,$ and similarly $H$ also descends to a vector field
on $\mathcal{O},$ denoted by $H_{*}.$

  2. 2.

It has been shown in [8] that the mean curvature vector field, $H$, can be
expressed as in terms of the leaf volume as follows

$H(x)=\nabla\log\Bigl{(}\frac{1}{Lvol(x)}\Bigr{)},$ (2)

where $Lvol(x)$ denotes the volume of the leaf through the point $x\in
M_{reg}$. Thus, $H$ is a conservative vector field of a smooth function on
$M_{reg}$, and its dual 1-form (the mean curvature form) is exact.
Furthermore, it is also known that when $H$ is basic, the basic spectrum is a
subset of the Laplace spectrum on $M$, equipped with the adapted metric.

  3. 3.

Next, we quote the following result from [2] which gives criteria for when two
representations of the same leaf space have the same basic spectra, given a
suitable map, called an SRF isometry, (see [2] for definitions) between the
leaf spaces:

Theorem: [Adelstein-Sandoval, [2]] Let $(M_{1},\mathcal{F}_{1})$ and
$(M_{2},\mathcal{F}_{2})$ be two singular Riemannian foliations with mean
curvature vector fields $H_{1}$ and $H_{2}$, respectively, and let
$\varphi\colon M_{1}/\mathcal{F}_{1}\to M_{2}/\mathcal{F}_{2}$ be a smooth SRF
isometry satisfying the following two conditions: (1) $H_{1}$ and $H_{2}$ are
basic vector fields, and (2) $d\varphi(H_{1*})=H_{2*}$ on an open, dense set.
Then the leaf spaces are basic isospectral, i.e.
$spec_{B}(M_{1},\mathcal{F}_{1})=spec_{B}(M_{2},\mathcal{F}_{2}).$

(See [2] for the definition of SRF isometry.) A singular Riemannian foliation
that satisfies condition (1) is said to be a generalized isoparametric
singular Riemannian foliation. We will now assume henceforward that any
singular Riemannian foliation that represents $Q$ is a generalized
isoparametric singular Riemannian foliation.

  4. 4.

The formula (1) and the above result imply that the basic spectrum is
precisely the orbifold spectrum when $H$ is identically zero, but otherwise is
likely to yield a different spectrum altogether. (See Example 1, [2]).

Our first result demonstrates that the set $\mathcal{R}(Q,g_{Q})$ is infinite
and contains an infinite basic isospectral family. It is proved in the Section
4 and will have applications to both Questions (2) and (3) above:

######  Theorem 1.4.

For any leaf space $(Q,g_{Q})$, where $Q=M/\mathcal{F}$ with mean curvature
vector field $H$, there exists an infinite subfamily of representations
$\mathcal{R}_{\mu}(Q,g_{Q},H)\subset\mathcal{R}(Q,g_{Q})$ indexed by
$\mu\in\mathbb{R}^{+}=(0,\infty)$ that are all basic isospectral and whose
ambient space is $M$ but with the leaf-wise metric rescaled by constant $\mu$
on $M_{reg}$.

We will show that the basic spectra contain geometric information about the
orbifold structure of the underlying leaf space for relatively closed curves
that avoid exceptional leaves, even when $H$ is not identically zero.
Furthermore, we describe under what conditions the leading wave invariant of
the basic spectrum detects properties of the underlying orbifold
$\mathcal{O}\subseteq Q$ versus those of the ambient space $M$.

The approach taken in this paper is to use the wave trace invariants about a
non-zero period $T$ associated to certain relatively closed geodesics that are
orthogonal to the the leaves of the foliation; such curves are relatively
closed with respect to the foliation (for definitions and more detail, see
Section 2). We will denote the set of such relatively closed geodesics by
$\mathcal{RT}(M,\mathcal{F}),$ which is equal to $Lspec(Q,g_{Q})$.

Note also: the set of relatively closed geodesics in $M$ of length $T$ is
stratified by leaf dimension as well.

###  1.3 Detecting Quotient Space Versus Ambient Space Geometry

Our next set of results address Question (1). In particular, we show that the
first basic wave invariant is independent of the mean curvature vector field,
and, thus, does not detect the variability of leaf volumes in the regular
region.

######  Theorem 1.5.

Let $T\in\mathcal{RT}(M,\mathcal{F})$ be a period of a relatively closed
geodesic that satisfies the following two conditions: (1) the geodesic
projects to a geodesic in $M/\mathcal{F}$ that is contained in the image of
the regular region, and (2) the set of closed geodesics in $Q$ of length $T$
has maximal dimension (in $Q$). Then, under standard clean intersection
hypothesis, the basic wave trace expansion localized about $T$ has degree and
leading coefficient $\sigma_{0}(T)$ given by the corresponding wave invariant
associated to the orbifold $\mathcal{O}=M_{reg}/\mathcal{F}.$

We also have the following, which says that the corresponding wave invariant
is independent of the representation:

######  Corollary 1.6.

Suppose $(M^{\prime},\mathcal{F}^{\prime})$ is another singular Riemannian
foliation representing the leaf space of $(M,\mathcal{F})$ that has the same
transverse metric $g_{Q}$, and for which the set of relatively closed geodesic
curves of length $T$ are contained in the regular regions for both
representations. Then under the same hypotheses as the previous theorem on the
curves of length $T$, the two representations will yield the same leading term
for the basic wave trace. This will be true even if
$M^{\prime}_{reg}\not=M_{reg}.$ Here we denote
$M_{reg}^{\prime}/\mathcal{F}^{\prime}=\mathcal{O}^{\prime}.$ Thus, the
leading term in the basic wave trace for this $T$ is independent of the
representation of the orbifold $\mathcal{O}\cap\mathcal{O}^{\prime}$.

######  Remark 1.7.

Having answered the question of whether or not the first wave invariant of the
basic spectrum detects the mean curvature vector field, it is natural to ask
if it detects changes in leaf dimension. Perhaps surprisingly in light of
Remark 1.3, item 3, the answer is possibly yes, in the sense that the first
wave invariant of singularities corresponding to closed geodesics that pass
through exceptional leaves (and thus experience a change in leaf dimension)
may depend on the leaf-wise metric depending on the whether or not the
projection of the component of fixed point set of the time $T$-geodesic flow
that is largest in $Q$ contains a geodesic that passes through an exceptional
leaf, as we shall see in the theorem below.

More generally, for a given representation $(M,\mathcal{F})$ with
$Q=M/\mathcal{F}$, we can decompose $Lspec(Q,g_{Q})$ into the set of
relatively closed periods that correspond to geodesics that avoid exceptional
leaves, $\mathcal{RT}_{orb}(M,\mathcal{F})$, and its complement–those for
which there is a geodesic curve of length $T$ that passes through exceptional
leaves, $\mathcal{RT}_{exc}(M,\mathcal{F}).$ Then we have the following
disjoint decomposition of the length spectrum of $(Q,g_{Q})$:

$Lspec(Q,g_{Q})=\mathcal{RT}_{orb}(M,\mathcal{F})\cup\mathcal{RT}_{exc}(M,\mathcal{F})$
(3)

######  Theorem 1.8.

The basic wave trace $Trace(U_{B}(t))$ then admits an expansion near $T\in
Lspec(Q,g_{Q})$ as a sum of lagrangian distributions on $\mathbb{R}$ that
whose degree is equal to $-\frac{1}{4}-\frac{e^{T}}{2}$ where $e^{T}$ the
maximum dimension of the projection of the fixed point set of the time $T$
geodesic flow in $M$ to $Q$. Furthermore, the basic wave trace decomposes as
follows:

$TrU_{B}(t)=I(t)+II(t)$ (4)

where the leading term of $I(t)$ depends on the underlying quotient orbifolds
$\mathcal{O}_{k}$, and the leading term of $II(t)$ may depend on the
representation and the ambient space. $I(t)$ is non-zero only when there exist
contributions of maximum dimension in $Q$ that arise from geodesics that avoid
exceptional leaves.

######  Remark 1.9.

We note the following:

  1. 1.

We will defer the precise statement of this trace formula, which is rather
lengthy and technical, and its proof until Section 3.

  2. 2.

For manifold quotients, all the representations are regular (See Section 2.1),
and thus,
$\mathcal{RT}_{exc}(M,\mathcal{F})\cup\mathcal{RT}_{b}(M,\mathcal{F})=\emptyset,$
and the first wave invariant is independent of the mean curvature form and the
leaf volume for each $T\in Lspec(Q,g_{Q})$.

  3. 3.

Similarly for orbifold quotients for which all the representation in
$\mathcal{R}(Q,g_{Q})$ are regular Riemannian foliations, the leading wave
invariant is independent of the mean curvature term for every $T$. The first
wave invariant is constant for all representations in $\mathcal{R}(Q,g_{Q})$.
Hence such quotients are are spectrally distinct from quotients for which
$\mathcal{RT}_{exc}(M,\mathcal{F})\not=\emptyset.$

  4. 4.

If all the geodesics of lengths $T\in\mathcal{RT}_{exc}(M,\mathcal{F})$ pass
through exceptional leaves then the basic wave trace can be localized into
parts that depend on either the underlying quotient orbifolds or the ambient
space. Such representations are preferable for answering Question (1) above.

###  1.4 Detecting Orbifold Singularities

With regard to Question (3) above, we now restrict our attention to leaf
spaces that are either orbifolds or manifolds. The goal here is to find
orbifold singularities. Here, we keep in mind that there are two aspects to
the basic wave trace invariants which arise from the expansion of the wave
trace as a Lagrangian distribution about its singularities: (1) the location
of those singularities which are contained in the length spectrum of $Q,$ and
(2) the particular form of the expansion itself. Detecting the presence of
orbifold singularities in this case is less about the form of the expansion
than about the length spectrum. Notionally, one way to detect orbifold
singularities is to find a closed curve geodesic in $Q$ that arises from a
curve in the ambient space $M$ that is relatively closed, but not smoothly
closed. Here we will take $M$ to be $\mathcal{F}r(Q)$ and the foliation to be
that arising from the orbits of the usual $O(q)$ action. A closed geodesic
curve that passes through an orbifold singularity, say $\overline{p}\in Q$
will be one whose image in $\mathcal{F}r(Q)$ is relatively closed with respect
to the foliation but is not smoothly closed. If $\pi(p)=\overline{p}$, then
this curve will be closed in $Q$ because of the action of some element of the
isotropy group at $p,$ denoted by $G_{p}.$ If such geodesics exist for $Q$
then we can detect an orbifold singularity at $\overline{p}$. For quotients
$Q$ with no orbifold singularities, such curves do not exist. The problem here
is how to distinguish the smoothly closed curves in the frame bundle from the
relatively closed curves in $\mathcal{RT}(\mathcal{F}r(Q),\mathcal{F}).$ Let
$Lspec^{\perp}(\mathcal{F}r(Q),g)$ denote the set of lengths of smoothly
closed geodesic curves in $\mathcal{F}r(Q)$ that are orthogonal to the orbits
of the $O(q)$ action with respect to an $O(q)$-invariant metric $g$.

Then we have the following as an application of the above result and the
general approach of looking at families of representations, we will show the
following as a consequence of Theorem 1.4.

######  Theorem 1.10.

Given a Riemannian orbifold $(Q,g_{Q})$ with $O(q)$ frame bundle
$\mathcal{F}r(\mathcal{O}),$ then in the notation above,

$Lspec^{\perp}(\mathcal{F}r(Q),O(q))=\bigcap_{\mu\in(0,\infty)}Lspec(\mathcal{F}r(Q),g_{\mu})$
(5)

and thus is independent of the choice of representation in the subfamily
$\mathcal{R}_{\mu}(\mathcal{O},g_{\mathcal{O}},H=0)$ and depends only on $Q$.
Furthermore, if there exists a length $T\in\mathcal{RT}(\mathcal{F}r(Q),O(q))$
that is not in $Lspec^{\perp}(\mathcal{F}r(Q),O(q)),$ then $Q$ is an orbifold
with non-empty singular set, and there exists a closed geodesic curve of
length $T$ that contains an orbifold singularity, $\overline{p}$.

We can also detect the order of an element in $G_{p}$ as follows:

######  Corollary 1.11.

Under the hypotheses of the previous theorem, there exists a smallest integer
$k>1$ such that for some $T\in Lspec^{\perp}(\mathcal{F}r(\mathcal{O}),O(q))$,
but $jT\not\in Lspec^{\perp}(\mathcal{F}r(\mathcal{O}),O(q)),$ for any
$1<j<k,$ and thus $G_{p}$ contains an element of order $k$.

###  1.5 General Remarks and Organization

We use the wave invariants for the basic spectrum as opposed to the basic heat
invariants (or, equivalently, the wave invariants at the zero singularity),
because they have some advantages in this context. One advantage is that the
wave invariants about a non-zero $T$ are generally neither too local nor too
global in nature. One problem with the heat invariants (and the wave trace
invariants about $T=0$ is that they involve functions (the local heat
invariants) that must be integrated over the entire manifold, and thus over
the singular strata, which can be problematic. As K. Richardson has shown in
[23], the local heat invariants, for example, are not always integrable. On
the other hand, the wave invariants over non-zero $T$ involve integrals over
the sets in $S^{*}M$ fixed by the time $T$ hamiltonian flow of the transverse
metric, and objects defined thereon. In this paper, we will focus on certain
curves whose fixed point sets are well-behaved with respect to the singular
sets. This approach takes advantage of the fact that the dynamics of the
hamiltonian flow of the transverse metric behave rather nicely with respect to
the singular strata; see Section 2 for the particulars.

One disadvantage to the wave invariants at non-zero $T$ is that one cannot be
guaranteed that they necessarily appear in the asymptotic expansion; it is
always possible that the singularities associated to different curves of the
same length could cancel each other out, although it is known in many specific
instances that some they do not cancel, [32], and generically, they do not
cancel. Thus, the results described here that involve using the wave trace
expansion localized to a particular non-zero period $T$ can only be used to
distinguish spectra of different leaf spaces, in principle, assuming the
coefficients in the asymptotics do not cancel.

The results here are concerned almost entirely with the first wave invariant.
The lower order terms in the wave trace expansion are quite complex even in
the case of a manifold (see for example, [36]). It is entirely possible that
the variation of leaf volumes via the mean curvature form can be detected in
the lower order terms.

This paper represents a continuation of the author’s previous work on the
basic spectrum involving some special cases of the basic wave trace for a
singular Riemannian foliation in [26], [27], [28], and is related to [1], [2].
Similar work with the heat invariants has been done by [21], [23], and [24].
The asymptotics of the heat kernel for orbifolds has been studied in [11],
where it was shown that the heat invariants of the singular sets of orbifolds
can detect information about the singular sets. The orbifold results here are
also related to the results of [31].

The paper proceeds as follows. In Section 2, we illustrate the existence of
many representations of a leaf space quotient $(Q,g_{Q})$ with examples, and
the proof of Theorem 1.4. Section 3 contains a review the necessary background
for the main tool for proving Theorem 1.5, Corollary 1.6, and 1.8, and the
statement and proof of the the generalization of the wave trace for singular
Riemannian foliations with disconnected leaves. In Section 4, we prove Theorem
1.10, and the related corollary.

##  2 Background

###  2.1 Examples of Representations of the Same Leaf Space

Here we provide a series of examples to illustrate how leaf space quotients
can be represented in different ways as leaf spaces of Riemannian folliations.
The examples that follow include foliations with both connected and
disconnected leaves, as well as examples with minimal and non-minimal leaves.
We begin with the following definitions, which can be found in the literature.

######  Definition 2.12.

A (singular) Riemannian foliation with disconnected leaves is the triple
$(M,\mathcal{F},\Gamma)$ where $(M,\mathcal{F})$ is a singular Riemannian
foliation with connected leaves and $\Gamma$ is a discrete group of isometries
of $Q=M/\mathcal{F}$ that is used to “glue” together the leaves of
$\mathcal{F}$ to form new disconnected leaves as follows. The group $\Gamma$
extends naturally to an action on the leaves of $\mathcal{F}$ so that the
orbits of $\Gamma$ form the disconnected leaves of the foliation. In other
words, if $L$ is a leaf for $\mathcal{F}$ then we define the corresponding
leaf of the disconnected foliation to be $L^{\prime}:=\Gamma\cdot L.$

In the case of disconnected leaves, the mean curvature vector field and leaf
space isometries can be defined as before. But, as we shall illustrate with
examples, we will need to distinguish between principal leaves in a foliation
with connected leaves versus the principal leaves in the disconnected case.

######  Definition 2.13.

disconnected2 The principal leaves of a disconnected singular Riemannian
foliation are the sets $L^{\prime}$ formed by taking the disjoint union of the
components (which we denote we by $L$) that have maximal dimension and trivial
leaf holonomy, and have the additional property that if there exists an
$f\in\Gamma$ such that $f(L)=L$ for some component $L$ of $L^{\prime}$, then
$f$ must be the identity in $\Gamma$.

######  Example 2.14.

If $Q$ of dimension $q$ is a closed compact manifold $X$ (without boundary)
then we always have the following representations.

  1. 1.

We can take the ambient space $M$ to be the $O(q)$ frame bundle with the
fibres of the bundle to be the leaves, and the usual metric will be bundle-
like for the regular Riemannian foliation. Note that the leaves are
disconnected with $\Gamma=\mathcal{Z}_{2}\cong O(q)/SO(q),$ but if $X$ is
orientable, then the $SO(q)$ frame bundle with fibres as the connected leaves
forms another regular Riemannian foliation with adapted metric.

  2. 2.

In a similar way, any bundle over $X$ with compact fibres can serve as the
ambient space and forms a regular Riemannian foliation with appropriate
metric.

  3. 3.

At the other extreme, we can consider any finite covering map over $X,$ of
degree $k$. The cover $\tilde{X}$ is the ambient space of a regular Riemannian
foliation with totally disconnected leaves and $\Gamma=\mathbb{Z}_{k}$ acting
trivially.

  4. 4.

The trivial foliation of a manifold by points.

  5. 5.

Any simple foliation on a connected (or disconnected) space with
$p:M\rightarrow X$.

  6. 6.

Any Riemannian submersion $p:M\rightarrow X$ with connected fibres, with
manifold base $X$ (so with trivial holonomy).

  7. 7.

Homogeneous spaces, where $X=G/N$ is a Lie group where $N\lhd G$ are Lie
groups and $G$ compact.

Many of the above examples are minimal representations by which we mean that
the corresponding mean vector field is identically zero. Given any minimal
representation one can also create a non-minimal representation by replacing
the leaf-wise metric with one that has variable leaf volumes.

In the above cases, all the examples have only principal leaves, and thus
yield only regular Riemannian foliations.

For representations with disconnected leaves, we note that on the connected
portions of the leaves, the leaf holonomy is trivial and the action by
$\Gamma$ satisfied the condition above. We note that in these disconnected
examples, the action of $\Gamma$ induces an action on the subspace of the
tangent space that is normal to the leaves, and that this action is trivial.
If it were otherwise, the leaf space would have an orbifold singularity. In
all cases above, the entire ambient space forms the regular stratum, and all
the leaves are principal.

Next we consider orbifold quotients, and we note that a regular Riemannian
foliation can result in an orbifold quotient.

######  Example 2.15.

If $Q$ is an orbifold $\mathcal{O}$ then we have the following examples of
representations.

  1. 1.

We can take the ambient space $M$ to again be the $O(q)$ frame bundle, in the
nonorientable case and the $SO(q)$ frame bundle otherwise. These
representations are regular, but they may have non-principal leaves with
$\Gamma=\mathbb{Z}_{2}$ acting with fixed components in the non-orientable
case for these leaves. Here, the regular region is the entire ambient space,
and the mean curvature vector field is identically zero. We note that in the
non-orientable case the leaf holonomy induced by the action of $\Gamma$ will
not be trivial.

  2. 2.

If $\mathcal{O}$ has as its underlying space $|\mathcal{O}|,$ a manifold with
totally geodesic boundary, and the usual $Z_{2}$ action obtained after
doubling $|\mathcal{O}|$ over its boundary, then this yields another example
of a regular foliation. Here, the non-principle leaves are those over the
boundary points, and principal leaves are on on the interior.

  3. 3.

As an example of the above, the interval $\mathcal{O}=[-\pi,\pi]$ is an
orbifold with reflectors at the boundary points corresponds to an orbifold as
traditionally defined in the orbifold literature. If we represent this same
quotient as the leaf space that arises by identifying the lines of constant
latitude on a standard 2-sphere (here have the action of $S^{1}$ acting by
rotations through the poles) then we have a singular Riemannian foliation with
two strata: the regular region is the sphere with the poles deleted, and the
singular stratum consists of the poles. Here, the mean curvature vector field
arises from the taking the gradient of the function that is equal to the
inverse of the circumferences of the latitude circles.

  4. 4.

Homogeneous spaces with orbifold quotients, $\mathcal{O}=G/H$ where are $G$
and $H$ Lie groups. If the orbits are either principal or exceptional, then
the representation will be regular. This example will have one stratum, but
possibly variable orbit volumes. If there are singular orbits, then the
quotient will be a singular quotient with the singular orbits comprising the
singular strata.

  5. 5.

Any singular Riemannian foliation that is infinitesimally polar ([18]).

  6. 6.

We always have the foliation by points, once more.

  7. 7.

For any quotient given by the leaf space of a homogenous singular Riemannian
foliation, one can generate new representations by reducing the action in some
way or by fattening up the leaves due to the method of K. Richardson in [24].

Orbifolds are characterized by the property that one can find a geometric
resolution. In the above, the frame bundle representation is always a
geometric resolution. An orbifold may also admit representations with non-zero
mean curvature vector fields.

######  Remark 2.16.

We note that orbifolds come with a natural stratification arising from
isotropy type, however, this stratification does not necessarily coincide with
the stratification of a (singular) Riemannian foliation. For the purposes of
studying the orbifold singularities via the basic wave trace invariants, we
must seek representations that best capture the dynamics of the geodesic flow,
if possible. Often this representation is one for which the orbifold singular
set of interest coincides with the image under $\pi_{Q}$ of one of the
orbifolds $O_{k}$ and the dynamics of the geodesic flow are such that the
relatively closed curves do not pass through exceptional leaves.

######  Remark 2.17.

While manifolds only admit regular representations, it is not known if there
exist orbifolds that only admit representations as a regular quotient. We
conjecture that if such an orbifold exists it must be a very good, orientable
orbifold with isolated singularities because otherwise the principle isotropy
reduction of the usual $O(q)$ action on the frame bundle would yield singular
strata, (See [17] for the principle isotropy reduction.) If it were the case
that there are no such orbifolds that are not manifolds, then the property of
$Q$ admitting only regular Riemannian foliation representations would
distinguish manifolds from orbifolds in a way that might prove useful.

If $Q$ with a fixed metric $g_{Q}$ is not a Riemannian orbifold or manifold,
then it cannot be represented as the leaf space of a regular Riemannan
foliation. This is implied by the work A. Lytchak of [18]. The quotient
$(Q,g_{Q})$ must therefore have a stratification with singular strata, and
therefore it must also have a non-zero mean curvature vector field, since due
to the main result of [35], there are no singular Riemannian foliations with
minimal leaves–they must be regular.

######  Remark 2.18.

We note that in [1], it was shown that there exist representations of orbifold
and non-orbifold quotients that are basic isospectral using a generalization
of Sunada’s method.

We refer to Theorems 6.2 and 6.7 of [4], which together characterize the
property of a point in $Q$ being belonging to an orbifold for a leaf space
quotient $Q=M/\mathcal{F}.$ We note that one can define manifold, orbifold,
and non-orbifold points as follows.

######  Definition 2.19.

Let $Q=M/\mathcal{F}$ be a leaf space quotient with $\pi:M\rightarrow Q$ and
$\pi(x)=\bar{x}$.

  1. 1.

We define $Q_{orb}$ to be the set

$\\{\bar{x}\in Q\,|\,\forall\,(M,\mathcal{F})\text{ with
}Q=M/\mathcal{F}\text{ where the infinitesimal foliation at }x\text{ is
polar}.\\}$

See Definition 2.5 of [4], for definition of the infinitesimal foliation.

  2. 2.

Similarly we define $Q_{man}$ to be the set

$\begin{split}\\{\bar{x}\in Q\,|\,\forall\,(M,\mathcal{F})\text{ with
}Q=M/\mathcal{F}\text{ where the infinitesimal foliation at }&x\text{ is polar
}\\\ \quad\text{ and the leaf is principal}\\}\end{split}$

  3. 3.

Finally, $Q_{non-orb}=(Q_{orb})^{c}$.

We conclude this section with the proof of Theorem 1.4.

###### Proof of Theorem 1.4..

First suppose $Q$ is given a regular Riemannian foliation with leaf dimension
equal to $k=k_{N}.$ Then $Q=M/\mathcal{F}$ with $H=-\nabla\log(Lvol(x)),$
which may be zero. Let $g(x,y)=g_{Q}(x)+g_{L}(x,y)$ be the (bundle-like)
metric on $M$, where $x$ and $y$ denote the transverse and leaf-wise
coordinates, respectively. Then for each $\mu\in\mathbb{R}^{+}$ one can alter
the metric so that over the regular region we replace the leaf-wise metric
$g_{L}(x,y)$ by $\rho g_{L}(x,y)$ been rescaled conformally by some positive
conformal factor $\rho\in\mathbb{R}^{+}$ so that the rescaled leaf volume is
$\mu Lvol(x),$ where $\mu=(\rho^{k_{N}})^{1/2}.$ Let $g^{\prime}_{\mu}(x,y)$
denote the new metric. One derives immediately from (2), that the mean
curvature vector field $H^{\prime}_{\mu}$ for
$(M,\mathcal{F},g^{\prime}_{\mu})$ is equal to the original mean curvature
form $H.$ By [2], every representation given by such a rescaling is basic
isospectral to the original one.

Now suppose $Q$ is not a regular Riemannian foliation222We will not need this
case for the proof of subsequent results., again with maximum leaf dimension
equal to $k_{N}$. Then we can perform the same rescaling of the leaf-wise
metric over the regular region, and extend the entire metric on $M$ so that it
is a smooth metric that yields the same transverse metric $g_{Q}$. Note that
the new metric obtained in this way only needs to be preserve $g_{Q}$, not the
leaf-wise metric, either over the regular region or the singular strata. One
derives immediately from (2), that the mean curvature vector field
$H^{\prime}_{\mu}$ for $(M,\mathcal{F},g^{\prime}_{\mu})$ is equal to the
original mean curvature form $H$. Thus, by [2], each representation
$(M,\mathcal{F},g^{\prime}_{\mu})$ is basic isospectral to the original one. ∎

######  Remark 2.20.

We note the following for use in Section 4.

  1. 1.

From the above proof, observe that the length spectrum of $(Q,g_{Q})$ is the
same for all representation in $\mathcal{R}(Q,g_{Q},H)$. However, the length
spectrum of $M$ has been altered for the closed geodesics that correspond to
non-transverse curves in $M$.

  2. 2.

For the purposes of calculating the wave invariants of the basic spectrum, we
can choose any representation from $\mathcal{R}(Q,g_{Q},H)$ and some may be
more suited to a given calculation of the localized wave trace than others for
a particular period $T$.

##  3 Proof of the Basic Wave Trace Formula Results

In this section we state and prove the results behind Theorem 1.8 which
implies Theorem 1.5 and Corollary 1.6. The basic wave trace results follow
from applying microlocal analytic techniques to analyze the singularities of
the basic wave operator. These techniques naturally take place in the
cotangent bundle of the ambient space $M.$ These singularities propagate along
certain hamiltonian curves in the cotangent bundle which project to geodesics
on $M$ which are orthogonal to the leaves of the foliation. When the trace of
the wave kernel is taken, the result is a distribution on the real line whose
singularities are located at the lengths of certain geodesics which are
relatively closed with respect to the foliation. This yields the Poisson
relation for the basic wave trace. The foliated geometric structure lifts to a
stratification of a portion of the cotangent bundle of $M$, found in [28].
This will yield a stratified singular configuration space, which will be
needed to understand the interplay of this stratification with the dynamics of
the geodesic flow.

We begin by describing the stratified singular configuration space. Next we
describe the properties of the hamiltonian flow of the transverse metric and
describe the relatively closed curves for singular Riemannian foliations with
possibly disconnected leaves. We then define the basic wave operator and
calculate the Poisson relation. We can then state and prove the basic wave
trace formula. Finally, by localizing to specific periods that either stay
confined to the various strata or pass through exceptional leaves, we show
Theorem 1.8 and its corollaries.

###  3.1 The Stratification of the Singular Configuration Space

The foliation and the associated stratification of $M$ induces a foliation and
a corresponding stratification of a subset of $T^{*}M,$ which will serve as
the configuration space and arises as follows. Let $\pi:T^{*}M\rightarrow M$
denote the usual map on the cotangent bundle $(T^{*}M,\,\omega)$, equipped
with its usual symplectic form. We will use the following notation: for any
distribution $V$ of $TM,$ let $V^{0}$ denote the subset of $T^{*}M$ defined as
follows

$V^{0}=\\{\xi_{x}\in T^{*}M\,|\,\xi_{x}(v_{x})=0\,\,\forall v_{x}\in
V_{x}\\}.$

Let $T\mathcal{F}\subset TM$ denote the distribution of variable dimension
that is defined by the tangent spaces to the leaves. Then the space
$(T\mathcal{F})^{0}$ admits a singular foliation defined as follows.

######  Definition 3.21.

The foliation of each stratum induces a foliation
$N\Sigma_{k}=(T\Sigma_{k})^{\perp}$ for each leaf dimension $k\geq 0$ as
follows:

$(N\Sigma_{k})^{0}=\\{\xi_{x}\in T^{*}M\,|\,\xi_{x}(v_{x})\,\forall v_{x}\in
N\Sigma_{k}\\}.$ (6)

Then we define

$\Sigma^{*}_{k}=(N\Sigma_{k})^{0}\cap(T\mathcal{F})^{0}$ (7)

for $0<k_{1}\leq k\leq k_{N}$. Each such stratum is foliated by
$k$-dimensional leaves arising from the null foliation for the symplectic form
$\omega$ restricted appropriately to the stratum $\Sigma^{*}_{k}$ for each
$k.$ We can also define a stratum (foliated by points) over the points that
lie in $(T\mathcal{F})^{0}$ but are not in the above strata $\Sigma^{*}_{k}$
for $k>0$.

If $k_{1}=0$ we define

$\Sigma_{0}^{*}=\Bigl{(}\bigcup_{\ell}\bigl{(}N\Sigma_{\ell})^{0}\bigr{)}^{c}\cup(N\Sigma_{0})^{0}\Bigr{)}\cap(T\mathcal{F})^{0},$
(8)

for $k_{1}\leq\ell\leq k_{N}$, and if $k_{1}>0$ we note that
$N\Sigma_{0}=\emptyset$ in the above. We consider this stratum to be foliated
trivially by its points. This yields a stratification of $(T\mathcal{F})^{0}$,
which we shall call the singular configuration space, as in [28]. Later, we
will wish to distinguish certain points in $\Sigma^{*}_{0}$, that we define to
be exceptional. Thus, let

$\Sigma^{*}_{e}=\Bigl{(}\bigcup_{\ell}\bigl{(}(N\Sigma^{*}_{\ell})^{0}\bigr{)}^{c}\Bigr{)}\cap(T\mathcal{F})^{0},$
(9)

for $k_{1}\leq\ell\leq k_{N}.$

Let $(\Sigma_{k}^{*},\widetilde{T\mathcal{F}_{k}})$ denote the foliation of
each stratum of the singular configuration space for each $k$, and let

$(T\mathcal{F})^{0}:=\bigcup_{k=0}^{k_{N}}\Sigma_{k}^{*},$ (10)

be the stratified singular configuration space.

Associated to this stratified space is the groupoid,
$\mathcal{G}^{*}(\mathcal{F}),$ with special properties, given in the
subsequent proposition. This proposition is a generalization of the one proved
in Proposition 1 of [28], Section 3; its proof is identical to that of
Proposition 1 in [28].

######  Proposition 3.22.

There exists a groupoid $\mathcal{G}^{*}(\mathcal{F})$ associated to
$(T\mathcal{F})^{0}$ such that the orbits of $\mathcal{G}^{*}(\mathcal{F})$
are $k$-dimensional leaves that foliate each stratum $\Sigma^{*}_{k}.$ For
each $k\geq 0,$ each of the $k$-dimensional leaves of $\Sigma^{*}_{k}$ is a
union of $k$-dimensional leaves induced by the null foliation on
$(N\Sigma_{k})^{0}$ when $k>0$ or the trivial foliation by points if $k=0$.

###  3.2 Properties of the Hamiltonian flow of the transverse metric on the
stratified configuration space

Next, we show that the hamiltonian flow of the transverse metric behaves well
with respect to the stratification described above. Let
$\mathcal{H}(\xi_{x})=\|\xi_{x}\|_{x}$ denote the usual hamiltonian function
which is determined by the adapted metric on $M$. As a consequence of the
metric being adapted to the singular Riemannian foliation, the metric is
bundle-like for each foliated stratum $(\Sigma_{k},\mathcal{F}_{k}).$ This
means that the metric $(g_{ij})$ and the induced metric on $T^{*}M,$
$(g^{ij}),$ can be written locally in distinguished coordinates with respect
to the foliation $(\boldsymbol{x},\boldsymbol{y})$ on
$(\Sigma_{k},\mathcal{F}_{k})$ where $\boldsymbol{x}$ and $\boldsymbol{y}$
denote the transverse and leaf-wise coordinates, respectively, as follows:

$g_{ij}(\boldsymbol{x},\boldsymbol{y})=g_{ij}^{T}(\boldsymbol{x})+g_{ij}^{L}(\boldsymbol{x},\boldsymbol{y})$
(11)

where $g_{ij}^{T}$ and $g_{ij}^{L}$ denote the transverse and leaf-wise
metrics, respectively. As in [26], the hamiltonian flow of $\mathcal{H}$
restricts to
$(T\mathcal{F})^{0}=\\{\mathcal{H}(\xi_{x})=\mathcal{H}_{\mathcal{F}^{\perp}}(\xi_{x})\\}$,
where $\mathcal{H}(\xi_{x})=\mathcal{H}_{\mathcal{F}^{\perp}}(\xi_{x}),$ the
part of the hamiltonian that arises from the transverse part of the bundle-
like metric on each $\Sigma_{k}\subset M$. This transverse flow on
$\Sigma^{*}_{k}$, which we denote by $\Phi^{t}(\xi_{x}),$ is particularly
well-behaved with respect to the singular configuration space
$(T\mathcal{F})^{0}$ and its stratification. In fact, as was shown in Section
3 of [28], the transverse flow restricts to the singular configuration space,
and sends leaves to leaves, with respect to all types of leaves in Proposition
3.22. We can generalize from Section 3 of [28] the following proposition:

######  Proposition 3.23.

The transverse hamiltonian flow of $H$ restricts to the singular configuration
space. In other words, if $\xi_{x}\in(T\mathcal{F})^{0}$ (with
$\xi_{x}\in\Sigma^{*}_{k}$) then the hamiltonian vector field
$\Xi_{\mathcal{H}}(\xi_{x})\in T\Sigma^{*}_{k}$ and

$\\{\Phi^{t}(\xi_{x})\,|\,t\in\mathbb{R}\\}\subset(T\mathcal{F})^{0}.$ (12)

Furthermore, if $k>0$ then either

$\\{\Phi^{t}(\xi_{x})\,|\,t\in\mathbb{R}\\}\subset\Sigma_{k}^{*}$ (13)

or

$\\{\Phi^{t}(\xi_{x})\,|\,t\in\mathbb{R}\\}\subset\Sigma_{k}^{*}\cup\Sigma_{e}^{*}.$
(14)

If $k=0$ then

$\\{\Phi^{t}(\xi_{x})\,|\,t\in\mathbb{R}\\}\subset\Sigma_{0}^{*}\setminus\Sigma_{e}^{*}.$
(15)

Otherwise, $\Xi_{\mathcal{H}}(\xi_{x})$ is not tangent to $\Sigma^{*}_{e}$.

It follows that such a transverse hamiltonian curve can be tangent to only one
of the $\Sigma^{*}_{k}$.

Equipped with these results we can define the notion of a curve being
relatively closed with respect to the foliation:

######  Definition 3.24.

We say that an arc of the hamiltonian flow $\Phi^{t}$ through $\xi_{x}$ will
be said to be relatively closed with respect to the singular Riemannian
foliation $\mathcal{F}$ with period $T$ if we have the following for each of
the following cases:

  1. Case 1:

$k_{N}=0$, disconnected leaves. There exists an element $f\in\Gamma$ such that
$df^{*}(\xi_{x})=\Phi^{T}(\xi_{x})$.

  2. Case 2:

$k_{N}>0$, leaves connected. Recall first that sliding along the leaves along
a path $\alpha$ that is contained in a leaf defines a local diffeomorphism of
the transversal space which we denote by $h_{\alpha}$. This local
diffeomorphism only depends on the homomtopy class of $\alpha;$ this defines
holonomy for the leaf. An arc of the transverse hamiltonian flow $\Phi^{t}$
through $\xi_{x}$ will be said to be relatively closed with respect to the
singular Riemannian foliation $\mathcal{F}$ with period $T$ if for endpoints
$\xi_{x}$ and $\eta_{y}=\Phi^{T}(\xi_{x}),$ there is a homotopy class
$[\alpha]$ of a curve $\alpha$ wholly contained in the leaf with endpoints $x$
and $y$, such that $dh^{*}_{\alpha^{-1}}(\xi_{x})=\Phi^{T}(\xi_{x}).$ This is
equivalent to the existence of a groupoid element of the form
$[\xi_{x},\Phi^{T}(\xi_{x}),dh^{*}_{\alpha^{-1}}]\in\mathcal{G}^{*}(\mathcal{F}).$
Let $\mathcal{RT}(M,\mathcal{F})$ denote the set of lengths of hamiltonian
arcs that are relatively closed with respect to the foliation of
$(M,\mathcal{F}).$ There exists a groupoid element $\gamma$ in
$\mathcal{G}^{*}(\mathcal{F})$ such that $\gamma(\xi_{x})=\Phi^{T}(\xi_{x}).$
More concisely, we say that the curve is relatively closed if the endpoints
$\xi_{x}$ and $\Phi^{T}(\xi_{x})$ belong to the same leaf.

  3. Case 3:

$k_{N}>0$, leaves disconnected. The endpoints $\xi_{x}$ and
$\Phi^{T}(\xi_{x})$ belong to the same disconnected leaf.

######  Definition 3.25.

Let $\mathcal{RT}(M,\mathcal{F})$ denote the set of lengths of hamiltonian
arcs that are relatively closed with respect to the foliation of
$(M,\mathcal{F}),$ as in the previous definition.

######  Remark 3.26.

Note that for each stratum $\Sigma_{k}$ we have
$\mathcal{RT}(\Sigma_{k},\mathcal{F}_{k})\subset\mathcal{RT}(M,\mathcal{F})$
and $Lspec(\mathcal{O}_{k})\subset\mathcal{RT}(M,\mathcal{F}).$

######  Remark 3.27.

Let $\gamma(t,x)=\pi(\Phi^{t}(x,\xi)),$ where $\pi:T^{*}M\rightarrow M;$ these
are geodesics in $M$. Let $\Xi_{\mathcal{H}}$ denote the hamiltonian vector
field of the transverse bundle-like metric. Then, in local distinguished
coordinates, it is easily seen that
$\gamma^{\prime}(0,x)=(d\pi)_{\xi_{x}}(\Xi_{\mathcal{H}})\perp
T_{x}\mathcal{F}.$ It is known that if $\gamma(t,x)$ is a geodesic passing
through $x$ that is perpendicular to the leaf at one point, then $\gamma(t,x)$
remains perpendicular to all the leaves that it meets, (see Chapter 6 of
[20]). Thus, the projections of such relatively closed hamiltonian curves are
geodesic arcs that are orthogonal to all the leaves through which the geodesic
passes.

###  3.3 The generalization of the basic wave trace for a singular Riemannian
foliation

Next we recall how the basic wave operator and its Schwartz kernel are
defined.

######  Definition 3.28.

  1. 1.

Recall from [28] that there is a basic projector $P$ from the space of smooth
functions on $M$ to the space of basic functions on $M$, given by the
averaging of functions over the leaves: If $L_{x}\Sigma_{k}$ is the leaf
containing $x\in M$ then for leaves that are not totally disconnected we have

$P(f)(x)=\frac{1}{Lvol_{k}(x)}\int_{L_{x}}f(y)dvol({L}_{y}),$ (16)

where $f$ a function on $M$, and $Lvol(x)$ is the volume of the leaf that
contains $x.$ We allow here for the leaves to have disconnected components. In
what follows, it will be convenient to indicate the dimension of the leaves
and their volumes. Let $Lvol_{k}(x)$ denote the volume of the leaf of
dimension $k$ that contains $x$.

On the other hand, if there are leaves that are totally disconnected with
cardinality $|L_{x}|$ then

$P(f)(x)=\frac{1}{|L_{x}|}\sum_{y\in L_{x}}f(y),$ (17)

where $f$ a function on $M$, and $L_{x}$ is the leaf that contains $x.$

  2. 2.

From the above, the operator $P,$ is essentially the push-forward
$(\pi_{*})|_{\Sigma_{k}}$ multiplied by the function $\frac{1}{Lvol_{k}(x)}$.
Its symbol, as an operator, is
$\frac{1}{Lvol_{k}(x)}\sigma((\pi_{*})|_{\Sigma_{k}}).$

The case of totally disconnected leaves would be handled similarly.

  3. 3.

The basic wave trace is the trace of the basic wave operator, denoted by
$U_{B}(t)$ and its Schwartz Kernel is given by

$U_{B}(t,x,y)=P_{x}P_{y}U(t,x,y)=\Sigma_{j=1}^{\infty}e^{-it\sqrt{\lambda_{j}^{B}}}e_{j}(x)e_{j}(y),$
(18)

where $U(t,x,y)$ is the wave kernel of the ordinary Laplacian on $M$,
$0\leq\lambda_{1}^{B}\leq\lambda_{2}^{B}\leq\cdots$ are the eigenvalues of the
basic Laplacian, $e_{j}$ denote the corresponding basic eigenfunctions, and
$P$ denotes the projector onto the space of basic functions. The subscripts
$x$ and $y$ on $P$ indicate which of the first two factors of $M\times
M\times\mathbb{R}$ is being acted upon.

As suggested by the previous paragraph, we will be interested in composing the
basic projector $P$ with the wave operator. Consequently, we will need the
following lemma.

######  Lemma 3.29.

The stratified canonical relation $\mathcal{C}$ of the basic projector $P$ is
given by

$\\{(\eta_{y},\xi_{x})\,|\,\xi_{x}\in(T\mathcal{F})^{0},\exists\,\boldsymbol{\alpha}\in\mathcal{G}^{*}(\mathcal{F}),\eta_{y}=\boldsymbol{\alpha}\cdot\xi_{x}\\}.$
(19)

###### Proof.

The proof is quite similar to that of the corresponding proposition ([28],
Proposition 27) of the original proof, except that over points in
$\Sigma_{0}^{*}$ the basic projector does not propagate singularities, and
thus its canonical relation is that of the identity operator. In other words,
the underlying foliation is the trivial foliation by points. Thus,

$\mathcal{C}\cap(\Sigma^{*}_{0}\times\Sigma^{*}_{0})=\\{(\xi_{x},\xi_{x})\,|\,\xi_{x}\in\Sigma^{*}_{0}\\}.$
(20)

∎

We then have the following Poisson relation for the wave front set of the
basic wave kernel:

######  Theorem 3.30.

In the notation previously established,

$WF\bigl{(}Trace(U_{B}(t,x,y)\bigr{)}\subset\\{(T,\tau)\,|\,T\in\mathcal{RT}(M,\mathcal{F}),\,\tau<0\\}.$
(21)

The proof of this result can be taken nearly verbatim from the prior result in
[28], where the result was proved for a singular Riemannian foliation that
arose as the closure of a regular Riemannian foliation or from [29]. However,
now we drop this assumption and simply assume that $(M,\mathcal{F})$ is a
singular Riemannian foliation with closed leaves that does not necessarily
arise as the closure of a regular Riemannian foliation. We note the the leaves
may be disconnected.

######  Remark 3.31.

In light of Proposition 3.23, we observe that each $Z^{T}$ can be decomposed
with respect to the stratification and establish the following notation:

$Z^{T}=\bigcup_{k=0}^{k_{N}}Z^{T}_{k},$ (22)

where for each $k$, $Z^{T}_{k}$ is the union of all the relatively closed
curves of length $T$. (For some values of $k$, $Z^{T}_{k}$ may be empty.) Let
$S(Z^{T}_{k})=S^{*}(M)\cap Z^{T}_{k};$ this set is saturated by the $k$
dimensional leaves since the transverse metric is constant on the leaves of
$Z^{T}_{k}$.

######  Remark 3.32.

We note that for a given $T$ the set $Z_{T}$ may contain both curves whose
saturation contains exceptional leaves and curves that remain confined to the
associated strata. This will be important in proving Theorem 1.8.

The following lemma from [28] still holds in this case, but we alter its
statement here to make the contribution of the function
$\frac{1}{Lvol_{k}(x)}$ that arises from the symbol of $P$ explicit from
Definition 3.28.

######  Lemma 3.33.

There exists a smooth canonical density, $d\mu_{Z^{T}_{k}}$, on each component
of the relative fixed point set $Z^{T}_{k}.$ Furthermore,
$d\mu_{Z^{T}_{k}}=\frac{1}{Lvol_{k}(x)}d\pi^{*}\chi_{\mathcal{F}_{k}}\otimes
d\mu^{\prime}_{Z^{T}_{k}}$ where $d\mu^{\prime}_{Z^{T}_{k}}$ is a canonical
measure on the orbifold $N(Z^{T}_{k})=S(Z^{T}_{k})/\widetilde{T\mathcal{F}}$
and $\chi_{\mathcal{F}_{k}}$ is the characteristic measure on
$(\Sigma_{k},\mathcal{F}_{k}).$

######  Remark 3.34.

It will be useful in what follows to note the observation in [33], p. 37, that
$\chi_{\mathcal{F}_{k}}$ is none other than $dLvol_{k}(x),$ the volume for the
$k$-dimensional leaves of $(\Sigma_{k},\mathcal{F}_{k}),$ and so
$\frac{1}{Lvol_{k}(x)}d\pi^{*}\chi_{\mathcal{F}_{k}}$ is a probability measure
on each leaf in the sense that each leaf has volume one with respect to this
measure. Let $d\eta_{k}$ denote the measure
$\frac{1}{Lvol_{k}(x)}d\pi^{*}\chi_{\mathcal{F}_{k}.}$

######  Definition 3.35.

For each foliated stratum of the configuration space
$(\Sigma^{*}_{k},\widetilde{T\mathcal{F}_{k}}),$ let
$\widetilde{T\mathcal{F}_{k}}$ denote the transverse distribution. Then the
local diffeomorphisms $dh_{\alpha}$ of Definition 3.24 define a lifted
holonomy action on the leaves of the foliation
$(\Sigma^{*}_{k},\widetilde{T\mathcal{F}_{k}}),$ which we denote by
$d\widetilde{h}_{(\alpha,\xi_{x})},$ as follows. For any
$\boldsymbol{\alpha}\in\mathcal{G}^{*}(\mathcal{F})$ with $\alpha(0)=x$,
$\alpha(1)=y$, and $\xi_{x}$ with $\eta_{y}=(dh_{\alpha^{-1}})^{*}(\xi_{x})$,
then

$d\widetilde{h}_{(\alpha,\xi_{x})}:\widetilde{N_{\xi_{x}}\mathcal{F}_{k}}\rightarrow\widetilde{N_{\eta_{y}}\mathcal{F}_{k}}.$
(23)

We say that the relatively closed set of curves $Z^{T}_{k}$ is clean if (1)
$Z^{T}_{k}$ is a smooth submanifold of $(T\mathcal{F})^{0};$ and (2) for every
$\xi_{x}\in Z^{T}_{k}$ with
$\eta_{y}=(dh_{\alpha}^{-1})^{*}\xi_{x}=\Phi^{T}(\xi_{x})$ then
$d\Phi^{T}_{\xi_{x}}(T_{\xi_{x}}Z^{T}_{k})=T_{\eta_{y}}Z^{T}_{k}$ for
$\alpha\in\mathcal{G}(\mathcal{F})$ with $\alpha(0)=x$ and $\alpha(1)=y$. The
condition that $\eta_{y}=(dh_{\alpha}^{-1})^{*}\xi_{x}$ implies that for all
$\xi_{x}\in Z^{T}_{k}$

$d\Phi^{T}_{\xi_{x}}(\widetilde{N_{\xi_{x}}\mathcal{F}_{k}})=d\widetilde{h}_{(\alpha,\xi_{x})}(\widetilde{N_{\xi_{x}}\mathcal{F}_{k}})=\widetilde{N_{\eta_{y}}\mathcal{F}_{k}}.$
(24)

We will need the following definition:

######  Definition 3.36.

Let $S$ be a saturated set with respect to the singular foliation of the
stratified singular configuration space $(T\mathcal{F})^{0}.$ The quotient
dimension of $S$ is defined as follows

$qdim(S)=max_{k}(dim(S\cap\Sigma^{*}_{k})-k).$ (25)

######  Remark 3.37.

We establish the following notation. Let
$\Gamma^{T}=\\{(T,\tau)\,|\,\tau<0\\}$ denote the ray over
$T\in\mathcal{RT}(M,\mathcal{F})$. Henceforward, we assume that the set
$Z^{T}_{k}$ of relative fixed points of the hamiltonian flow $\Phi^{T}$ on
$(T\mathcal{F})^{0}$ is clean for all $T\in\mathcal{RT}(M,\mathcal{F})$ in the
sense of Definition 3.35. Let $Z^{T}_{max}$ denote the union of the components
of $Z^{T}$ for which the quotient dimension is largest. Henceforward, let
$Z^{T}_{k}$ denote the strata that make up $Z^{T}_{max}$ to avoid excessive
subscripts. Note: the set $S(Z^{T}_{max})$ is also saturated. The relatively
closed hamiltonian arcs of a given length $T$ of maximal quotient dimension
decompose as previously noted in Remark 3.31 into conic submanifolds
$Z^{T}_{k}$ whose connected components are finite in number. Let
$\varepsilon^{T}:=dim(S(Z^{T}_{k})),$ and let $e^{T}$ be the corresponding
quotient dimension so that $\varepsilon^{T}=k+e^{T}.$

Then we have the following in the notation above:

######  Theorem 3.38.

Near $t=T\in\mathcal{RT}(M,\mathcal{F})$ and assuming that the $Z^{T}_{k}$
satisfy Definition 3.35, then, in the above notation,

$Trace\bigl{(}U_{B}(t,x,y))\bigr{)}=\sum_{T\in\mathcal{RT}(M,\mathcal{F})}\nu_{T}(t),$
(26)

where $\nu_{T}\in I^{-1/4-e_{T}/2-j}(\mathbb{R},\Gamma^{T},\mathbb{R}).$
Furthermore, $\nu_{T}$ has an expansion of the form

$\nu_{T}(t)=\sum_{k=0}^{k_{N}}e^{\frac{i\pi
m^{T}_{k}}{4}}\sum_{j=0}^{\infty}\sigma_{j}(T,k)(t-T+i0)^{-\frac{e_{T}+1}{2}-j}\,mod\,C^{\infty}(\mathbb{R}),$
(27)

where $m^{T}_{k}$ is the Maslov index of $Z^{T}_{k}.$ If
$Z^{T}_{k}\cap\Sigma^{*}_{k}=\emptyset$ then we set $\sigma_{j}(T,k)=0$.

In particular, the leading term $\sigma_{0}(T,k)$ is

$\sigma_{0}(T,k)=\Bigl{[}\int_{S(Z^{T}_{k})}\sigma(U)d\mu_{Z^{T}_{k}}\Bigr{]}\,\tau^{\frac{e_{T}-1}{2}}\sqrt{d\tau}.$
(28)

Furthermore, if $Z^{T}_{k}$ contains no curves that pass through exceptional
leaves, then the leading term $\sigma_{0}(T,k)$ is

$\sigma_{0}(T,k)=\Bigl{[}\int_{N(Z^{T}_{k})}\sigma(U)d\mu^{\prime}_{Z^{T}_{k}}\Bigr{]}\,\tau^{\frac{e_{T}-1}{2}}\sqrt{d\tau}.$
(29)

######  Remark 3.39.

The statement of this theorem is slightly different than that found in [28] to
clarify several points.

  1. 1.

This generalization of that theorem clarifies the order of the singularity. In
the previous version, the dimension of the fixed point set was maximized and
the leaf dimension was minimized, and this tacitly assumes that the
contributions come from only one stratum, but this is not necessarily the
case. The contribution of highest degree comes from the component(s) of the
fixed point set of largest quotient dimension, and there is no ostensible
reason why different strata of the fixed point set could not contribute
subsets of the same quotient dimension. For this reason we take the sum over
all the strata that might contribute to the fixed point set of highest
quotient dimension. The order of the singularity was expressed in previous
versions in terms of $\varepsilon^{T}=e^{T}+k$ in the previous remark, but
this degree is always reduced by the dimension of the leaves, $k$, and so it
makes sense to express the degree in terms of $e^{T}$ instead.

  2. 2.

We emphasize the role of the leaf volumes in light of Lemma 3.33 and Remark
3.34, which are closely related to the mean curvature vector fields.

  3. 3.

In the event that there are no curves passing through exceptional leaves, the
last part of the theorem will allow us to calculate the contribution
$\sigma_{0}(T,k)$ solely in terms of the orbifold $\mathcal{O}_{k},$ as we
shall see in the next proposition following the proof of the above theorem.

###### Proof.

Sketch of Proof. We first consider the case of totally disconnected leaves.
This is essentially the original result in [13], or in the case of orbifolds,
the result of E. Stanhope and A. Uribe in [31]. Generalizing the proof to
allow for such disconnected leaves boils down to expanding the definition of
the relatively closed geodesics and accounting for the presence of totally
disconnected leaves with Remark 3.28.

For the case of leaves with connected components, the proof of the
generalization can be taken almost verbatim from the proof of the
corresponding result in [28], aside from the clarifications noted in Remark
3.39. The prior result in [28], proved the result for a singular Riemannian
foliation that is the closure of a regular Riemannian foliation whose leaf
dimension is $p$.

We make note of the following changes in notation from the proof in [28]. Here
we denote the singular configuration space by $(T\mathcal{F})^{0}$ instead of
$(T\overline{\mathcal{F}})^{0}$ in the original proof. We note that lift of a
leaf of the form $L_{x}=\\{x\\}$ to $(T\mathcal{F})^{0}$ is just
$\\{\xi_{x}\\}$ for $\xi_{x}\in T^{*}M\setminus\\{0\\}$. Thus, the globally
defined space corresponding to the underlying foliation that was formerly
denoted by $(T\mathcal{F})^{0}$ is really just $T^{*}M\setminus\\{0\\}$ as in
the discussion preceding Definition 3.35. Even if $\Sigma_{0}=\emptyset$, the
stratified configuration space will still have leaves consisting of singleton
covectors over the exceptional points.

We make note only of the minor changes required to generalize the result. The
main modification consists of setting the leaf dimension of the underlying
foliation to $p=0$ in the prior proof and by accounting for the fact that the
canonical relation for the basic projector is slightly different in this case,
as described in Lemma 3.29. The results up to (28) follow immediately.

To show equation (29), we simply integrate over the leafwise portion via the
density $|d\eta_{k}|,$ which we may due without difficulty since every
geodesic in the set $Z^{T}$ is contained entirely in the regular foliation
$\Sigma^{*}_{k},$ by hypothesis.

∎

The rest of the result follows from the following proposition.

######  Proposition 3.40.

If $T$ is such that the $Z^{T}_{max}$ does not contain curves that pass
through exceptional leaves then for each $k$ with $Z^{T}_{k}\not=\emptyset$
(29) simplifies to

$\sigma_{0}(T,k)=\sigma_{0}(T,\mathcal{O}_{k}),$ (30)

where $\sigma_{0}(T,\mathcal{O}_{k})$ denotes the first wave invariant of the
orbifold $\mathcal{O}_{k}$ for $T\in\mathcal{RT}(\Sigma_{k},\mathcal{F}_{k})$
which is equal to $Lspec(\mathcal{O}_{k}).$

###### Proof.

We proceed by cases. First consider the curves that correspond to geodesics
over the regular region. Suppose $T$ is a relatively closed period in
$\mathcal{RT}(M,\mathcal{F})$ such that $Z^{T}_{k}=\emptyset$ for all
$k<k_{N}$. By equations (27), and (29) the expansion of the basic wave trace
in a neighborhood of $T$ is of the form

$\nu_{T}(t)=\sum_{j=0}^{\infty}\sigma_{j}(T)(t-T+i0)^{-\frac{e_{T}+1}{2}-j}$
(31)

and

$\sigma_{0}(T,k)=\Bigl{[}\int_{N(Z^{T}_{k_{N}})}\sigma(U)d\mu^{\prime}_{Z^{T}_{k_{N}}}\Bigr{]}\,\tau^{\frac{e_{T}-1}{2}}\sqrt{d\tau},$
(32)

where $\sigma(U)$ denotes the symbol of the (ordinary) wave kernel which
depends on the entire metric, including the leaf-wise part.

We will demonstrate the result by calculating the first wave invariants for an
appropriate $T$ on both $M$ and $\mathcal{O},$ and comparing them. To this
end, denote $\sigma_{0}(T,(M,\mathcal{F}))$ and $\sigma_{0}(T,\mathcal{O})$
the first wave invariants for the basic spectrum for $(M,\mathcal{F})$ and
$\mathcal{O},$ respectively. Let $\sigma(U)$ denote the symbol of the wave
operator on $M$, and let $\sigma(U_{\mathcal{O}})$ denote the symbol of the
wave operator on $\mathcal{O}.$

Now proceeding as in [13], we recall from Theorem 4.5 of [13] that $\sigma(U)$
is given by the expression

$\frac{1}{2\pi}\int_{S(Z^{T}_{k_{N}})}exp\Bigl{(}\frac{-i}{T}\int_{0}^{T}sub(\Delta_{B})(\Phi^{s}(x,\xi))\,ds\Bigr{)}\,d\mu_{Z^{T}_{k_{N}}}.$
(33)

Next we calculate of the subprincipal symbol of the basic Laplacian locally on
$\Sigma_{k_{N}}.$ Recall from [13] the following calculus of subprincipal
symbols: if $Q$ is an operator of degree $m$ with principal symbol $q(x,\xi),$
for coordinates $(x,\xi)\in T^{*}M$ then

$sub(Q)(x,\xi)=q_{m-1}(x,\xi)-\frac{1}{2i}\sum_{j=1}^{n}\frac{\partial^{2}q}{\partial
x_{j}\partial\xi_{j}}(x,\xi),$ (34)

and also:

$sub(Q^{\alpha})=\alpha p^{\alpha-1}sub(Q)$ (35)

Thus if $Q=\Delta_{B}=\Delta_{\mathcal{O}}-H$ (locally on $\Sigma_{k_{N}}$)
where $H$ is the mean curvature vector field, then locally,

$sub(\Delta_{\mathcal{O}}-H)(x,\xi)=-h(x,\xi)+sub(\Delta_{\mathcal{O}})(x,\xi)=-h(x,\xi),$
(36)

since it is well-known that $sub(\Delta_{\mathcal{O}})=0$. Hence,

$sub([\Delta_{\mathcal{O}}-H]^{1/2})(x,\xi)=\frac{1}{2}[\|\xi\|_{x}]^{-1/2}sub(\Delta_{\mathcal{O}}-H)(x,\xi).$
(37)

On the fixed point sets, which are in the cosphere bundle, $\|\xi\|_{x}=1$, it
follows that when restricted to those sets, we have:

$sub([\Delta_{\mathcal{O}}-H]^{1/2})(x,\xi)=-\frac{1}{2}h(x,\xi),$ (38)

where $h(x,\xi)$ is the principal symbol of $H$ regarded as a first-order
operator. In fact, it is just the mean curvature form, which we denote by
$\alpha$.

The proof of the result will follow from showing that

$\int_{0}^{T}h(\Phi^{s}(x,\xi))\,ds=0,$ (39)

and thus the first wave invariant for $\Delta_{B}$ is the same as that for
$\Delta_{O}$. But this follows from the fact that $H$ is conservative, by (2),
with a basic potential function and from the fact that the hamiltonian flow
$\Phi^{t}$ is dual to the geodesic flow. In other words, (39) is the integral
of an exact basic form around a geodesic curve that is closed in the quotient.
As such, its endpoints lie in the same leaf. In fact,
$\alpha=d\Bigl{(}\ln(\frac{1}{Lvol(x)})\Bigr{)}$ and so is the differential of
a basic function. More precisely, we have the following:

$\displaystyle\int_{0}^{T}h(\Phi^{s}(x,\xi))\,ds$ $\displaystyle=$
$\displaystyle\int_{0}^{T}g(H(\gamma(s)),\dot{\gamma}(s))\,ds$ (40)
$\displaystyle=$ $\displaystyle\int_{\gamma}\alpha$ (41) $\displaystyle=$
$\displaystyle\int_{0}^{T}d(\log(\frac{1}{Lvol(\gamma(s))})\,ds=0,$ (42)

since $\gamma(0)=\gamma(T)$ belong to the same leaf.

On the other hand, from [31], $\sigma_{0}(T,\mathcal{O})$ is equal to

$\Bigl{[}\int_{S(\tilde{Z}^{T}_{k_{N}})}\sigma(U_{\mathcal{O}})d\mu_{\tilde{Z}^{T}_{k_{N}}}\Bigr{]}\,\tau^{\frac{e_{T}-1}{2}}\sqrt{d\tau},$
(43)

where $\sigma(U_{\mathcal{O}})$ denotes the symbol of the (ordinary) wave
kernel on $\mathcal{O}$, and $\tilde{Z}^{T}_{k}$ are the analogously defined
sets of fixed points under the hamiltonian flow on $T^{*}\mathcal{O}$. Because
the transverse flow on $M$ projects to the hamiltonian flow on
$T^{*}\mathcal{O}$, $S(\tilde{Z}^{T}_{k_{N}})=N(Z^{T}_{k_{N}}).$ It is also
immediate that $\mu_{\tilde{Z}^{T}_{k_{N}}}=\mu^{\prime}_{Z^{T}_{k_{N}}}.$

Then, since $sub(\Delta_{\mathcal{O}})=0$, we conclude that
$\sigma_{0}(T,(M,\mathcal{F}))=\sigma_{0}(T,\mathcal{O}),$ proving the result.

The proofs of the first case is straightforward. The proof of the result over
the singular strata follows from the fact that hypothesis guarantees that the
fixed point sets under the flow are contained in the respective singular
strata, and hence depend on the underlying orbifold, $\mathcal{O}_{k}.$ We
recall from [8] that a version of equation (2) holds for each singular
stratum, not just the regular one. In other words, on each foliated stratum
$(\Sigma_{k},\mathcal{F}_{k}),$ we have a basic mean curvature vector field

$H_{k}=-\nabla\log(Lvol_{k}(x)),$ (44)

where $Lvol_{k}(x)$ is the volume of the leaf $L\subset\Sigma_{k}$ containing
$x.$

Let $\mathcal{O}_{k}$ denote the orbifold represented by the leaf space
$\Sigma_{k}/\mathcal{F}_{k}.$ Note that $\mathcal{O}_{k}$ is not necessarily
closed. Nonetheless, we can still think about the wave invariants associated
to closed geodesics on $\mathcal{O}_{k}.$

Suppose $T$ is in the length spectrum $\mathcal{RT}(M,\mathcal{F}),$ and there
exist relatively closed curves of this length that are confined to one or more
of the singular strata, and suppose that for the component of $Z^{T}$ of
largest quotient dimension, the curves of that length do not leave any of the
strata $\Sigma_{k}$. By the general wave trace formula in Theorem 3.38, the
leading term in the expansion has order $-\frac{1}{4}-\frac{e_{T}}{2},$ and
the coefficient $\sigma_{0}(T,k)$ are given by (28).

We now consider the restriction of $\sigma(U)$ to each of the strata in
$Z^{T}_{max}.$ Let $k\in\\{k_{1},\dots,k_{N}\\}$. We begin by taking suitable
coordinates in a neighborhood $V$ of a point $w$ in $\Sigma_{k}^{*}\cap
Z^{T}_{k}.$ Recall that $\Sigma_{k}$ has codimension at least 2 in $M$, [20].
Let $d_{k}:=dim(\Sigma_{k}),$ $q_{k}:=d_{k}-k.$ We can take $V$ to be a
tubular neighborhood around $\Sigma_{k}^{*}\cap Z^{T}_{k}$ that contains the
point $w$. Let $\Sigma_{k}$ be locally cut out by the $n-d_{k}$ functions
$z_{d_{k}+1},\cdots,z_{n}$. Let $(\boldsymbol{x},\boldsymbol{y})$ be
distinguished coordinates on $\Sigma_{k}$ where
$\boldsymbol{x}=(x_{1},\dots,x_{q_{k}})$ and
$\boldsymbol{y}=(y_{q_{k}+1},\dots,y_{d_{k}})$. Let
$(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z};\boldsymbol{\xi},\boldsymbol{\eta},\boldsymbol{\zeta})$
denote the corresponding local coordinates in $T^{*}M.$ Then $V\cap Z^{T}_{k}$
is given locally by $\boldsymbol{z}=0,$ $\boldsymbol{\eta}=0,$ and
$\boldsymbol{\zeta}=0$.

Next, we choose a suitable basis in which to express the metric $g^{ij}$ on
the fixed point sets. Choose a basis for the vector subspace
$(T_{w}\Sigma_{k})^{\perp}$ so that it is orthonormal with respect to the
corresponding metric (which we denote by $g^{ij}_{\perp}(w)).$ Then extend
this to a basis that is constant in the transverse directions by parallel
transporting along transverse geodesics (recall that $\Sigma_{k}$ is
transversely totally geodesic). Note that we have arranged it so that
$\frac{\partial}{\partial x_{i}}\bigl{(}g^{ij}_{\perp}\bigr{)}=0$. We can then
complete the basis of the tangent space by appending a basis of $T\Sigma_{k}$
that respects the splitting $T\Sigma_{k}=N\mathcal{F}_{k}\oplus
T\mathcal{F}_{k}.$ With respect to this basis, $g^{ij}$ is blockwise diagonal:
$g^{ij}=g^{ij}_{T}\oplus g^{ij}_{L}\oplus g^{ij}_{\perp}$ where $g_{L}^{ij}$
is the leafwise metric, and $g^{ij}_{T}$ is the transverse metric which does
not depend on $\boldsymbol{y}$ since it is bundle-like with respect to the
foliation of $\Sigma_{k}$.

Next, we examine the expression of the basic laplacian with an eye to
calculating the restriction of the top order symbol and subprincipal symbol of
$\Delta$ on $M$ to the fixed point sets $Z_{k}^{T}.$ First consider the
formula for $\Delta$ for arbitrary variables $\boldsymbol{x}$:

$\Delta=-\frac{1}{\sqrt{det(g)}}\sum_{ij}\frac{\partial}{\partial
x_{i}}\bigl{(}\sqrt{det(g)}g^{ij}\frac{\partial}{\partial x_{j}}\bigr{)}$ (45)

Let $I_{1}=\\{1,\dots q_{k}\\},$ $I_{2}=\\{q_{k}+1,\dots d_{k}\\}$, and
$I_{3}=\\{d_{k}+1,\dots n\\}.$ With respect to the coordinates previously
chosen, we can split the sum over $i$ and $j$ over the three blocks:

$\displaystyle\Delta=-\frac{1}{\sqrt{det(g)}}\Bigl{[}\sum_{i,j\in I_{1}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{\partial}{\partial
x_{i}}\bigl{(}\sqrt{det(g)}g^{ij}\frac{\partial}{\partial x_{j}}\bigr{)}$ (46)
$\displaystyle+$ $\displaystyle\sum_{i,j\in I_{2}}\frac{\partial}{\partial
y_{i}}\bigl{(}\sqrt{det(g)}g^{ij}\frac{\partial}{\partial y_{j}}\bigr{)}$ (47)
$\displaystyle+$ $\displaystyle\sum_{i,j\in I_{3}}\frac{\partial}{\partial
z_{i}}\bigl{(}\sqrt{det(g)}g^{ij}\frac{\partial}{\partial z_{j}})\Bigr{]}$
(48)

Examining this expression with an eye to computing the principal and
subprincipal parts of the symbol and restricting to $U\cap Z^{T}_{k}$, we see
that summing over the indices in $I_{2}\cup I_{3}$ corresponding to
$T\mathcal{F}\oplus(T\Sigma_{k})^{\perp}$ produces terms that vanish when
$\boldsymbol{\eta}$ and $\boldsymbol{\zeta}$ are set equal to zero.
Suppressing these terms, and expanding $g$ blockwise with respect to a similar
basis as its inverse $(g^{ij})$, then we can express $det(g)$ as the product
of the determinants of the blocks
$det(g_{ij}^{T})det(g_{ij}^{L})det(g_{ij}^{\perp}),$

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{-1}{\sqrt{det(g_{ij}^{T})det(g_{ij}^{L})det(g_{ij}^{\perp})}}\sum_{i,j\in
I_{1}}\frac{\partial}{\partial
x_{i}}\bigl{(}\sqrt{det(g_{ij}^{T})det(g_{ij}^{L})det(g_{ij}^{\perp})}g^{ij}\frac{\partial}{\partial
x_{j}}\bigr{)}$ (49) $\displaystyle+$ $\displaystyle\cdots.$

This can be re-expressed as follows

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{-1}{\sqrt{det(g_{ij}^{T})}}\sum_{i,j\in
I_{1}}\frac{\partial}{\partial
x_{i}}\bigl{(}\sqrt{det(g_{ij}^{T}})g^{ij}\frac{\partial}{\partial
x_{j}}\bigr{)}$ (50) $\displaystyle-$
$\displaystyle\frac{1}{\sqrt{det(g_{ij}^{L})}}\sum_{i,j\in
I_{1}}\frac{\partial}{\partial x_{i}}\bigl{(}\sqrt{det(g_{ij}^{L}})\bigr{)}$
(51) $\displaystyle-$
$\displaystyle\frac{1}{\sqrt{det(g_{ij}^{\perp})}}\sum_{i,j\in
I_{1}}\frac{\partial}{\partial
x_{i}}\bigl{(}\sqrt{det(g_{ij}^{\perp}}\bigr{)}\bigr{)}+\cdots,$ (52)

(again suppressing the terms that do not contribute to the restriction to
$U\cap Z^{T}_{k}$).

From here, we see that the first of the three summations yields the principal
symbol of this operator, which is the same as the principal symbol of
$\Delta_{k}$, the induced Laplacian operator on
$\mathcal{O}_{k}=\Sigma_{k}/\mathcal{F}_{k}$ restricted to basic functions on
$\Sigma_{k}$. The second sum produces $-H_{k},$ the mean curvature form for
$(\Sigma_{k},\mathcal{F}_{k}),$ and the last sum is zero when
$\boldsymbol{z}=0$ by construction.

Thus, we see that $\sigma(U)$ restricted to $Z^{T}_{k}$ equals
$sub(\Delta_{k}-H_{k}).$ By reasoning identical to that of the first case, it
follows that if all the transverse geodesic curves of length $T$ remain in
$\Sigma_{k},$ then the basic spectrum determines the wave invariant
$\sigma_{0}(T,\mathcal{O}_{k})$ for the orbifold $\mathcal{O}_{k}.$ ∎

######  Remark 3.41.

If we have a representation of an orbifold $Q$ whose stratification
$(\Sigma_{k},\mathcal{F}_{k})$ is such that for some $k,$ $\mathcal{O}_{k}$ is
an orbifold singular stratum and all the curves of length $T$ are confined to
that one orbifold stratum, then the sum in (27) collapses and we can recover
the first wave invariant of $\mathcal{O}_{k}.$ In this way, some
representations may be more suited to capture invariants of the various
orbifold strata than others.

##  4 Applications to Detecting Orbifold Singularities

Here we prove Theorem 1.10 and Corollary 1.11. Consider a Riemannian orbifold
quotient $(Q,g_{Q})$ with representation $(M,\mathcal{F}).$ Observe that the
closed geodesics that are orthogonal to the leaves on $(M,\mathcal{F})$ always
descend to closed geodesics on $Q$ of the same length, and, thus,
$Lspec^{\perp}(M,\mathcal{F})\subset Lspec(Q)$. On the other hand, if a closed
geodesic on $Q$ contains an orbifold singularity, then it is possible that the
geodesic does not arise from a smoothly closed geodesic on $M,$ that is
orthogonal to the leaves, but rather arises from a geodesic “lasso” at the
singularity $\bar{p}\in\mathcal{O}$ where some element $\gamma$ in the local
isotropy group at $\bar{p}$ corrects the angle formed by the lasso.

###  4.1 The Case of a Good Orbifold

To illustrate this more precisely, consider a very good orbifold
$Q=\Gamma\setminus M$, and suppose there exists a closed (but not smoothly
closed) geodesic $\bar{\phi}=\pi_{Q}(\phi)$ where $\phi$ is closed but not
smoothly closed geodesic on $M$ that passes through an orbifold singularity at
$p\in M$ with corresponding singularity in $Q$ denoted by $\bar{p}$, and local
isotropy group $G_{p}\subset\Gamma$. Suppose $\phi(0)=\phi(\tau)$ and
$\phi^{\prime}(\tau)=d\gamma_{*}(\phi^{\prime}(0))$ for some element $\gamma$
of $G_{p}$ acting non-trivially then $\tau\in Lspec(Q)$. If $\tau\notin
Lspec(M)$ (in other words, if there is no other smoothly closed curve of
length $\tau$) then $Lspec(M)$ is properly contained in $Lspec(Q)$ and the
existence of such a $\tau\in Lspec(Q)\setminus Lspec(M)$ indicates the
presence of non-trivial isotropy (regardless of whether or not the associated
wave trace singularities cancel). We can generalize this argument to prove
Theorem 1.10.

###### The proof of Theorem 1.10..

More generally, consider any closed orbifold. In place of the global quotient
$(M,\Gamma)$ we consider the frame bundle representation
$(\mathcal{F}r(\mathcal{O}),O(q))$ with the usual metric, which we denote by
$g_{1}.$ The main problem here is to identify the smoothly closed geodesic
curves on the ambient space $M$ that are orthogonal to the leaves. The length
spectrum of $\mathcal{F}r(\mathcal{O})$ contains the lengths of all smoothly
closed geodesics, not just the ones that are strictly orthogonal to the
leaves.

We claim that the smoothly closed orthogonal geodesics are precisely the
smoothly closed geodesics in $\mathcal{F}r(\mathcal{O})$ that are invariant
under leaf-wise rescaling in
$\mathcal{R}_{\mu}(\mathcal{F}r(\mathcal{O}),O(q),H=0)$, and hence the
corresponding length spectrum is that given by equation (5). Let $c(t)$ be any
curve on $\mathcal{F}r(\mathcal{O}),$ and let
$\dot{c}(t)=(\dot{c}_{\perp},\dot{c}_{L})$ denote its tangent decomposed with
respect to $t$ decomposed with respect to the transverse and leafwise
splitting on the tangent space. One can observe by writing down the arc length
functional

$\ell(c,g_{\mu},I)=\int^{t_{0}}_{0}(\|\dot{c}_{\perp}\|_{g_{Q}}^{2}+\rho^{2}\|\dot{c}_{L}\|_{g_{L}}^{2})^{1/2}\,dt$
(53)

for the length of $c$ over any interval $I=[0,t_{0}]$ with respect to
$g_{\mu}$ that if $\mu_{1}>\mu_{2}$ (and thus $\rho_{1}>\rho_{2}$ ) then the
only curves whose lengths over $I$ remain unchanged are those for which
$\dot{c}_{L}=0$ over $I$. It follows that the lengths of closed geodesics that
are invariant under rescaling are precisely those in
$Lspec(\mathcal{F}r(Q),g_{\mu})$ for all $\mu$.

It just remains to show that $L^{\perp}(\mathcal{F}r(\mathcal{O}),O(q))$ can
be defined as the elements of the length spectrum that are invariant under the
rescaling described in the proof of Theorem 1.4. Let $\phi$ be a closed
geodesic on $\mathcal{F}r(\mathcal{O}).$ Its tangent vector can be decomposed
as a direct sum of its orthogonal and leaf-wise parts:
$\dot{\phi}(t)=(\dot{\phi}_{\perp}(t),\dot{\phi}_{L}(t))$. ∎

Next we prove Corollary 1.11.

###### Proof of Corollary 1.11..

Consider now iterates of a closed geodesic in the orbifold $Q=\mathcal{O}$
that is closed, but not smoothly closed. Suppose we find a smallest $\tau\in
Lspec(Q)\setminus L^{\perp}(\mathcal{F}r(\mathcal{O}),O(q)).$ Let $\bar{\phi}$
be the corresponding closed geodesic in $Q$. Then
$\bar{\phi}(0)=\bar{\phi}(T)$ and by iterating we have
$\bar{\phi}(0)=\bar{\phi}(jT)$ for all $j\in\mathbb{Z}.$ Hence, we have
$\phi^{\prime}(jT)=(d\gamma_{*})^{j}(\phi^{\prime}(0)).$ Since the isotropy is
finite, for some $k$, we have $\gamma^{k}$ equal to the identity, and thus $k$
divides the order of $\gamma$ and the corresponding geodesic must be smoothly
closed. If $j$ is such that $\dot{\phi}(jT)\not=\dot{\phi}(0)$ for all
$1<j<k$, then $\gamma$ has order $k.$ ∎

###  4.2 Some Remarks

The hypotheses of Theorem 1.10 and Corollary 1.11 are admittedly hard to check
because it is hard to determine if a given $T$ lies in the set of periods that
are invariant under leaf-wise rescaling. In practice, if we are interested in
checking whether or not a particular $T$ corresponds to the length of a
smoothly closed curve, then we make the following observation. There exists a
closed non-orthogonal geodesic of smallest length $T_{min}$ which depends on
$\mu$. It is possible to choose $\mu$ so large so that $T_{min}>T+1$. Thus,
the interval $[0,T+1]$ will not contain the lengths of any closed non-
orthogonal geodesics for this choice of $\mu$, and thus, if $T\in
Lspec(\mathcal{F}r(\mathcal{O},O(q),g_{\mu})$ then $T\in
Lspec^{\perp}(\mathcal{F}r(\mathcal{O}),O(q))$.

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