# An average intersection estimate for families of diffeomorphisms

Axel Kodat  and  Michael Shub

###### Abstract.

We show that for any sufficiently rich compact family $\mathcal{H}$ of $C^{1}$
diffeomorphisms of a closed Riemannanian manifold $M$, the average geometric
intersection number over $h\in\mathcal{H}$ between $h(V)$ and $W$, for $V,W$
any complementary dimensional submanifolds of $M$, is approximately (i.e. up
to a uniform multiplicative error depending only on $\mathcal{H}$) the product
of their volumes. We also give a construction showing that such families
always exist.

##  1\. Introduction

Let $M$ be a closed Riemannian manifold of dimension $n$, and let
$\operatorname{Diff}^{r}(M)$ denote the space of $C^{r}$ diffeomorphisms of
$M$ with the $C^{r}$ topology. For $0\leq k\leq n$, let
$\operatorname{Gr}_{k}(TM)$ denote the $k$-Grassmannian bundle over $M$, i.e.
the set of $k$-planes tangent to $M$.

In this note we study conditions on compact $C^{1}$ families in
$\operatorname{Diff}^{1}(M)$ sufficient to guarantee a kind of “uniform
mixing” on average of the (smooth) incidence geometry of $M$. By a _$C^{r}$
family of diffeomorphisms of $M$_ we will mean a pair $(\mathcal{H},\psi)$,
where $\mathcal{H}$ is a smooth manifold, possibly with boundary, and
$\psi:\mathcal{H}\to\operatorname{Diff}^{r}(M)$ is such that the associated
evaluation map $ev_{\psi}:\mathcal{H}\times M\to M$ is $C^{r}$. This implies
in particular that the parametrizing map $\psi$ is continuous. When a family
$(\mathcal{H},\psi)$ is fixed we will usually suppress the map $\psi$, using
the same letter $h$ to denote both an element in $\mathcal{H}$ and its image
under $\psi$; by this mild abuse of notation we will often write, for
instance, $ev_{\psi}(h,p)=\psi(h)(p)=h(p)$.

We will also assume that any such $\mathcal{H}$ comes equipped with a
Riemannian metric, and thus a Riemannian density which we denote by
$d\mathcal{H}$. Given these assumptions, our main result is the following
estimate:

######  Theorem 1.1.

Suppose $(\mathcal{H},\psi)$ is compact $C^{1}$ family of diffeomorphisms of
$M$ such that

  1. $\operatorname{(A1)}$

$ev:\mathcal{H}\times M\to M\times M$ defined by $ev(h,p)=(p,h(p))$ is a
submersion;

  2. $\operatorname{(A2)}$

$\accentset{\circ}{ev}_{k}:\accentset{\circ}{\mathcal{H}}\times\operatorname{Gr}_{k}(TM)\to\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{k}(TM)$
defined by $ev_{k}(h,\sigma)=(\sigma,dh(\sigma))$ is surjective for all $0\leq
k\leq n$.

Then there is a constant $C=C(\mathcal{H},\psi)\geq 1$ such that

(1) $\dfrac{1}{C}\operatorname{vol}(V)\operatorname{vol}(W)\ \ \leq\ \
\displaystyle\int_{h\in\mathcal{H}}\\#(h(V)\cap W)\ d\mathcal{H}\ \ \leq\ \
C\operatorname{vol}(V)\operatorname{vol}(W).\\\ $

for all complementary dimensional submanifolds $V,W\subset M$.

Assumption (A1) says equivalently that for all $p\in M$, the map
$ev_{\psi,p}:\mathcal{H}\to M$, $h\mapsto\psi(h)(p)$ is a submersion, or
(again equivalently) that for all $(h,p)\in\mathcal{H}\times M$,

$\partial_{h}(ev_{\psi})_{(h,p)}:T_{h}\mathcal{H}\to T_{h(p)}M$

is surjective. Assumption (A2) says that for all $0\leq k\leq n$ and
$\sigma_{1},\sigma_{2}\in\operatorname{Gr}_{k}(TM)$, there is an
$h\in\accentset{\circ}{\mathcal{H}}$ such that $dh(\sigma_{1})=\sigma_{2}$; in
other words, the interior of $\mathcal{H}$ “acts transitively” on
$\operatorname{Gr}_{k}(TM)$ for every $k$.

Here $\\#(h(V)\cap W)\in{\mathbb{N}}_{0}\cup\\{\infty\\}$ simply means the
number of points in $h(V)\cap W$, i.e. the geometric intersection number
between $h(V)$ and $W$. Note that (1) implicitly includes the claim that the
function

$\phantom{.}\hskip 56.9055pt\mathcal{H}\ \ni\ h\ \ \xmapsto{\ \ \ \ \ \ \ }\ \
\\#(h(V)\cap W)\ \in\ {\mathbb{N}}_{\geq 0}\cup\\{\infty\\}$

is measurable, and integrable if both $V$ and $W$ have finite volume; in the
latter case it implies that $\\#(h(V)\cap W)<\infty$ for a.e.
$h\in\mathcal{H}$.

By the volume of a submanifold $S\subset M$ we will always mean its intrinsic
volume as a Riemannian manifold with metric inherited from $M$, i.e.
$\operatorname{vol}(S)=\int_{S}dS$, where $dS$ denotes the density induced by
the restricted metric. Alternatively, if $\dim(S)=k$ we have
$\operatorname{vol}(S)=\operatorname{Haus}_{M}^{k}(S)$, the $k$-dimensional
Hausdorff measure of $S$ in $M$.

Theorem 1.1 is actually a special case of a more general result, which applies
whenever $\dim(V)+\dim(W)\geq\dim(M)$ and correspondingly replaces the
cardinality of the intersection with its appropriate-dimensional Hausdorff
measure. We focus on the complementary dimensional case for simplicity of
exposition (and greater proximity to our original motivations), but the proof
of the general case works similarly. For the precise statement and some
remarks on its proof, see §4.

By analogy with the (exact) kinematic formulas of integral geometry (see e.g.
[Che66], [Cal20], [San04]), we call (1) a _kinematic inequality_ for the
family $(\mathcal{H},\psi)$. Taken alone, Theorem 1.1 is a relatively
straightforward adaptation of these classical formulas—which obtain when
$\mathcal{H}=G$ is a compact Lie group acting isometrically on $M$ and
transitively on every $\operatorname{Gr}_{k}(TM)$—to a specific (a priori
somewhat artificial and potentially rare) non-homogeneous context. Its
possible utility derives from the following existence result, which we prove
in §5:

######  Theorem 1.2.

Every closed manifold $M$ admits an $\mathcal{H}$ and
$\psi:\mathcal{H}\to\operatorname{Diff}^{1}(M)$ satisfying the hypotheses of
Theorem 1.1.

We emphasize that the $\mathcal{H}$ constructed here has non-empty boundary.
This is the main motivation for allowing $\partial\mathcal{H}\neq\varnothing$
in the definition of $C^{r}$ families, and in fact Theorem 1.2 fails in
general if one requires $\partial\mathcal{H}=\varnothing$. For details see
§1.3.

###  1.1. Background and motivation

Formulas of the type considered here have a long history in integral geometry
and a well-developed theory in the setting of homogeneous spaces of compact
Lie groups. From this perspective the basic antecedent to our result is

######  Theorem 1.3 (Poincaré’s formula for homogeneous spaces).

Let $M^{n}$ be a closed Riemannian manifold with $G$ a compact Lie group
acting smoothly and transitively on $M$ by isometries. Suppose the action is
also transitive on $\operatorname{Gr}_{k}(TM)$, for every $1\leq k\leq n-1$.
Then for every $k,\ell\geq 0$ satisfying $k+\ell\geq n$, there is a constant
$C_{k,\ell}>0$ such that for any submanifolds $V,W\subset M$ with $\dim(V)=k$
and $\dim(W)=\ell$, we have

(2) $\displaystyle\int_{g\in G}\operatorname{vol}(gV\cap W)\
dG=C_{k,\ell}\operatorname{vol}(V)\operatorname{vol}(W),$

where $dG$ denotes the Haar measure on $G$.

In the form given here this result is due to Howard [How93]. The basic idea
behind the proof is to view the left-hand side as a double integral over what
we will later call the _intersection manifold_ associated to $V$ and $W$, i.e.
the space of all intersections between $g(V)$ and $W$ for $g\in G$. By
changing the order of integration one then (essentially) equates this integral
with the average volume of the spaces $G_{x,y}=\\{g\in G\ :\ gx=y\\}$ over all
$(x,y)\in V\times W$. (In other words, one replaces the average volume of
intersections with the average volume of the spaces of maps realizing each
possible intersection.) Since all such spaces are isometric for any bi-
invariant metric on $G$ inducing the Haar measure, the result follows. For
details, see [How93, §3] or [BC13, §A.4]. Our adaptation of this approach
below is particularly indebted to [BCSS98, §13] and [SS94, §4]. For a classic
presentation of the foundational material, see [San04].

Observe that any action $G\curvearrowright M$ as in Theorem 1.3 can be viewed
as a compact $C^{\infty}$ family of diffeomorphisms of $M$, namely $(G,\psi)$
where $\psi:G\ni g\mapsto(x\mapsto gx)\in\operatorname{Diff}^{\infty}(M)$.
This family clearly satisfies (A1) and (A2), and thus Theorem 1.1 yields (2)
_up to a uniform multiplicative error_. The fact that this error can be
eliminated in the setting of Theorem 1.3 depends on (among other things) the
assumption that the family $(G,\psi)$ consists of isometries. For an analogous
result applicable to general manifolds, however, this is too much to ask: if
$(\mathcal{H},\psi)$ satisfies (A1) and
$\operatorname{im}(\psi)\subset\operatorname{Isom}(M)$ for some choice of
Riemannian metric on $M$, then the action of $\operatorname{Isom}(M)$ on $M$
is automatically transitive and thus $M$ is homogeneous, i.e. $M\cong G/H$ for
$G$ a compact Lie group and $H<G$ closed. An inequality incorporating some
bounded multiplicative error therefore seems to be the most one could expect
from an extension of Poincaré’s formula to non-homogeneous manifolds.

Our primary motivation for formulating such an extension comes from smooth
dynamics.111For a similar generalization from a different direction, see the
work of Fu in [Fu16], and especially his definition of an “admissible measured
family” of diffeomorphisms in §6. We regret that we became aware of this paper
too late to explore the connection further. Given a $C^{1}$ diffeomorphism
$f:M\to M$ and a pair of complementary dimensional submanifolds $V,W\subset
M$, one would like to understand how the growth of
$\operatorname{vol}(f^{n}(V))$ affects the distribution of (a large compact
piece of) the isotopy class of $f^{n}(V)$ with respect to $W$. Our results
here imply that we can fix a compact family of diffeomorphisms $\mathcal{H}$
of $M$ so that the average number of intersections between $h(f^{n}(V))$ and
$W$ has the same order of growth (in the sense of e.g. [CP23]) as
$\operatorname{vol}(f^{n}(V))$. In future work we hope to show how this could
be a useful tool in a new approach to the second author’s _$C^{r}$ entropy
conjecture_ for the open case(s) of $r<\infty$.

###  1.2. Assumptions

Theorem 1.1 is a “backwards” theorem in the sense that we started with the
estimate (1) and looked for reasonable (and in particular abundantly realized)
assumptions that would make it true. Here we collect some comments on our
eventual selections (A1) and (A2).

First, we emphasize that the statement of Theorem 1.1 is not quite optimal, in
two main ways:

  1. (i)

The upper bound in (1) actually follows from (A1) alone; the surjectivity
assumption (A2) is needed only for the lower bound. Note also that (A1), in
contrast to (A2), is a purely local property of $(\mathcal{H},\psi)$. Since
the property (A1) is also easily seen to be invariant under (either left- or
right-) translation of the family by a diffeomorphism $g$, this implies that
an upper bound as in (1) can be obtained for finite-dimensional slices of
$\operatorname{Diff}^{1}(M)$ concentrated arbitrarily close to any given
diffeomorphism $f$.

  2. (ii)

In the other direction, it follows from the proof that (A2) is somewhat
stronger than necessary for our result. What we really need is the following:

     * $\operatorname{(A2^{@itemi})}$

For any $\sigma_{p}\in\operatorname{Gr}_{k}(TM)$ and
$\sigma_{q}\in\operatorname{Gr}_{n-k}(TM)$, there is an
$h\in\accentset{\circ}{\mathcal{H}}$ such
. that $h(p)=q$ and $dh(\sigma_{p})\pitchfork\sigma_{q}$.

For further discussion of this point see the remark after Claim 2.9. We have
opted to use $\operatorname{(A2)}$ in the statement of Theorem 1.1
nonetheless, primarily to emphasize the analogy with the transitivity
assumption in Theorem 1.3, and also because it is no more difficult in
practice to construct a family satisfying $\operatorname{(A2)}$ than one
satisfying the weaker condition $\operatorname{(A2^{*})}$.

These qualifications in mind, the purpose of our assumptions is heuristically
the following. The assumption that $ev$ is a submersion implies that no
particular intersection $h(p)=q$ between $h(V)\cap W$ can be locally
persistent (since one can move $h(p)$ off itself by a small change in $h$). In
particular, any non-transversal intersection between $h(V)$ and $W$ (near
which the cardinality of $h(V)\cap W$ could explode) can be destroyed by a
small perturbation of $h$. This ensures that almost all intersections are
isolated, and moreover that the total number of such points (summed over
$\mathcal{H}$) is never too large compared to
$\operatorname{vol}(V)\cdot\operatorname{vol}(W)$. This yields the upper bound
in Theorem 1.1 using (A1) alone.

To get a lower bound, one needs to show that $\mathcal{H}$ can always
_produce_ intersections between pairs of points in $V$ and $W$. Moreover,
since the quantity we are trying to bound is a sort of average of the number
of intersections with respect to the volume measure on $\mathcal{H}$, these
intersections should persist over open (thus positive measure) subsets of
$\mathcal{H}$. In other words, $\mathcal{H}$ should be able to produce
_transverse_ intersections between any pair of points in $V$ and $W$. This
will certainly hold if $\mathcal{H}$ can move any tangential $k$-plane to any
other, as guaranteed by (A2).

The following example cleanly illustrates the main ideas in the isometric
case:

######  Example 1.4 (${\mathbb{T}}^{2}\curvearrowright{\mathbb{T}}^{2}$).

Let $M={\mathbb{T}}^{2}={\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$ be the standard
flat $2-$torus, and $\mathcal{H}={\mathbb{T}}^{2}$ with
$\psi:{\mathbb{T}}^{2}\longrightarrow\operatorname{Diff}({\mathbb{T}}^{2})$
the smooth action of ${\mathbb{T}}^{2}$ on itself by translations; explicitly,
$\psi:a\mapsto(\psi_{a}:x\mapsto x+a)$. Then the pair $(\mathcal{H},\psi)$
satisfies assumption (A1), but not (A2), since $dh=\text{id}$ for all
$h\in\text{im}(\psi)$ (using the canonical identifications
$T_{p}{\mathbb{T}}^{2}\cong{\mathbb{R}}^{2}$) and thus

$\text{im}(ev_{1})=\\{(\sigma_{1},\sigma_{2})\ :\ \sigma_{1}\text{ and
}\sigma_{2}\text{ are
parallel}\\}\neq\operatorname{Gr}_{1}(T{\mathbb{T}}^{2})\times\operatorname{Gr}_{1}(T{\mathbb{T}}^{2}).$

More strongly, the fact that every element of $\mathcal{H}$ maps any element
of $\operatorname{Gr}_{1}(T{\mathbb{T}}^{2})$ to a parallel translate means
that no element of $\operatorname{Gr}_{1}(T{\mathbb{T}}^{2})$ can be made
transverse to itself via $\mathcal{H}$.

Theorem 1.1 correspondingly fails. For instance, take $I$, $J$ two parallel
Euclidean line segments in ${\mathbb{T}}^{2}$. Then it is easy to see that
$\psi_{a}(I)\cap J=\varnothing$ for a.e. $a\in{\mathbb{T}}^{2}$, so

$\displaystyle\int_{a\in{\mathbb{T}}^{2}}\\#(\psi_{a}(I)\cap J)\
d{\mathbb{T}}^{2}=0.$

This implies that the lower bound in Theorem 1.1 cannot hold for any positive
$C$.

If the angle between $I$ and $J$ is $\theta\neq 0$, however, it is similarly
straightforward to show that

$\displaystyle\int_{a\in{\mathbb{T}}^{2}}\\#(\psi_{a}(I)\cap J)\
d{\mathbb{T}}^{2}=|\sin\theta|\operatorname{length}(I)\cdot\operatorname{length}(J).$

In particular, we have

$\displaystyle\int_{a\in{\mathbb{T}}^{2}}\\#(\psi_{a}(I)\cap J)\
d{\mathbb{T}}^{2}\leq\operatorname{length}(I)\cdot\operatorname{length}(J)$

for any Euclidean line segments $I,J\subset{\mathbb{T}}^{2}$, and by a
subdivision and approximation argument it is not difficult to see that this
holds more generally for $I,J\subset{\mathbb{T}}^{2}$ any embedded $C^{1}$
curves. Thus the upper bound in Theorem 1.1 holds here with $C=1$.

###  1.3. A digression on $\mathcal{H}$

One technical hurdle in proving Theorem 1.1 stems from our allowance of the
possibility that the parameter space $\mathcal{H}$ has boundary. If one
assumes instead that $\partial\mathcal{H}=\varnothing$, the availability of
Ehresmann’s fibration theorem (applied to the submersion $ev$) substantially
streamlines the proof; see the remark concluding §2.4. However, here
Ehresmann’s theorem cuts both ways, imposing strong topological constraints on
any boundaryless $\mathcal{H}$ satisfying $\operatorname{(A1)}$. These
constraints imply in particular that for general $M$, the condition
$\partial\mathcal{H}=\varnothing$ is too much to ask:

######  Proposition 1.5.

Suppose $M$ is a closed manifold and $(\mathcal{H},\psi)$ a compact, connected
family of $C^{1}$ diffeomorphims of $M$ with
$\partial\mathcal{H}=\varnothing$, such that $ev_{p}$ is a submersion for some
$p\in M$. Then $\pi_{1}(M)$ is a finite extension of a quotient of a subgroup
of $\pi_{1}(\operatorname{Diff}_{0}(M))$.

###### Proof.

Let $(\mathcal{H},\psi)$ be such a family. After translating and perturbing we
may assume that $\psi(\mathcal{H})\subset\operatorname{Diff}_{0}(M)$ the
identity component of $\operatorname{Diff}^{\infty}(M)$. Choose $p\in M$ such
that $ev_{p}$ a submersion.

By Ehresmann’s theorem, $ev_{p}:\mathcal{H}\to M$ is a locally trivial
fibration, say with fiber $\mathcal{F}$. The end of the homotopy long exact
sequence for this fibration reads

$\pi_{1}(\mathcal{H})\xrightarrow[]{\ (ev_{p})_{*}\ }\pi_{1}(M)\xrightarrow[\
\ \ \ ]{}\pi_{0}(\mathcal{F})\xrightarrow[\ \ \ \ ]{}1.$

Since $\mathcal{H}$ and (therefore) $\mathcal{F}$ are compact we must have
$\\#\pi_{0}(\mathcal{F})<\infty$, so this sequence implies that $(ev_{p})_{*}$
is virtually onto. On the other hand, $ev_{p}=EV_{p}\circ\psi$ where
$EV_{p}:\operatorname{Diff}_{0}(M)\to M$, $f\mapsto f(p)$. Thus $(ev_{p})_{*}$
factors through $\pi_{1}(\operatorname{Diff}_{0}(M))$, and the result follows.
∎

######  Corollary 1.6.

Suppose $M$ is a closed hyperbolic manifold with $\dim(M)\leq 3$. Then there
is no boundaryless compact family of diffeomorphisms of $M$ satisfying
$\operatorname{(A1)}$.

###### Proof.

By work of Earle-Eells (in dimension 2) and Gabai (in dimension 3),
$\operatorname{Diff}_{0}(M)$ is contractible ([EE69], [Gab01]). In particular,
$\pi_{1}(\operatorname{Diff}_{0}(M))=0$. Since (by hyperbolicity of $M$)
$\pi_{1}(M)$ is infinite, by Proposition 1.5 we are done. ∎

The upshot here is that Theorem 1.2 fails if one requires that
$\partial\mathcal{H}=\varnothing$. More strongly, these results (combined with
the restrictions on higher homotopy groups obtained by continuing the above
exact sequence) suggest that manifolds which admit a submersive compact family
of diffeomorphisms without boundary may be quite rare. Clearly homogeneous
spaces of compact Lie groups form the main class of examples. Are there
others?

######  Question 1.7.

Which closed manifolds $M$ admit a compact smooth family of diffeomorphisms
$\mathcal{H}\longrightarrow\operatorname{Diff}(M)$ with
$\partial\mathcal{H}=\varnothing$, such that the associated evaluation map
$ev:\mathcal{H}\times M\longrightarrow M\times M$ is a submersion?

We note that, on the question of existence, the choice of regularity is
irrelevant. A $C^{r}$ family of diffeomorphisms can be viewed as a map
$\Phi\in C^{r}(\mathcal{H}\times M,M)$ such that
$\Phi(h,\cdot)\in\operatorname{Diff}^{r}(M)$ for all $h\in\mathcal{H}$; with
this notation, the submersion condition simply means that
$\Phi(\cdot,p):\mathcal{H}\to M$ is a submersion for all $p\in M$. When
$\mathcal{H}$ and $M$ are compact, these properties are open in
$C^{1}(\mathcal{H}\times M,M)$, so if one obtains a boundaryless $C^{1}$
family satisfying $\operatorname{(A1)}$, by perturbing one can obtain a
$C^{\infty}$ family as well.

This intriguing diversion aside, the proof of Theorem 1.1 in the general,
possibly-with-boundary setting turns out to work in much the same way as the
Ehresmann-based proof for the boundaryless case, modulo some basic lemmas
concerning submersions from compact manifolds with boundary which we relegate
to Appendix A.222Though these straightforward results are presumably well-
known, we were unable to locate them in the literature. The content of these
lemmas is to show that our required bounds (which follow trivially from
compactness when the fibers of $ev$ move $C^{1}$-continuously) still follow
from our assumptions when we allow boundary, provided that the surjectivity
requirement in (A2) is phrased in terms of the interior of $\mathcal{H}$.

###  1.4. Acknowledgements

We would like to thank Bryce Gollobit, Charles Pugh, Enrique Pujals, and
Federico Rodriguez Hertz for many stimulating conversations and helpful
suggestions on the contents of this paper. The first author would also like to
thank Ryan Utke for pointing out the relevance of Ehresmann’s theorem to these
results.

##  2\. Preliminaries

Let $(\mathcal{H},\psi)$ satisfy the hypotheses of Theorem 1.1, and write
$d:=\dim(\mathcal{H}).$ We begin by introducing the main objects used in our
proof.

###  2.1. Normal Jacobians

Let $E_{1},E_{2}$ be finite-dimensional real inner product spaces. Then for
any linear map $A:E_{1}\to E_{2}$, one defines the _normal Jacobian_ by the
formula

$\operatorname{NJ}(A):=\det(AA^{*})^{1/2}\in{\mathbb{R}}_{\geq 0}.$

Note that $\operatorname{NJ}(A)>0$ iff $A$ is surjective. One can check that

$\operatorname{NJ}(A)=|\det(A|_{\ker(A)^{\perp}})|.$

For a $C^{1}$ map $f:M\to N$ between Riemannian manifolds, we define the
_normal Jacobian of $f$_ pointwise, by

$\operatorname{NJ}(f)(x):=\operatorname{NJ}(df_{x})=\det(df_{x}\cdot
df_{x}^{*})^{1/2}.$

Thus $\operatorname{NJ}(f):M\to{\mathbb{R}}_{\geq 0}$, and since $f$ is
$C^{1}$ it is easy to see that $NJ(f)$ is continuous. By the above,
$\operatorname{NJ}(f)(x)>0$ iff $x$ is a regular point of $f$.

###  2.2. The solution manifold $\widetilde{\mathcal{V}}$

The _solution manifold_ $\widetilde{\mathcal{V}}\subset\mathcal{H}\times
M\times M$ is the subspace consisting of all triples of the form $(h,p,h(p))$.
Equivalently, $\widetilde{\mathcal{V}}=\Gamma(ev_{\psi})$, where

$\displaystyle ev_{\psi}\ :\ $ $\displaystyle\mathcal{H}\times
M\longrightarrow M$ $\displaystyle(h,p)\ \longmapsto\ h(p).$

Since $ev_{\psi}$ is $C^{1}$, $\widetilde{\mathcal{V}}$ is a neat $C^{1}$
submanifold with boundary of $\mathcal{H}\times M\times M$, diffeomorphic to
$\mathcal{H}\times M$.333By _neat_ we mean that
$\partial\widetilde{\mathcal{V}}\subset\partial(\mathcal{H}\times M\times
M)=\partial\mathcal{H}\times M\times M$. In particular, it inherits $C^{1}$
projection maps $\widetilde{\pi}_{1},\widetilde{\pi}_{2}$:

${\widetilde{\mathcal{V}}}$${\mathcal{H}}$${M\times
M.}$$\scriptstyle{\tilde{\pi}_{1}\ \ \ }$$\scriptstyle{\tilde{\pi}_{2}}$

By (A1) and the $k=0$ case of (A2),
${\tilde{\pi}_{2}}|_{\text{int}(\widetilde{\mathcal{V}})}:\text{int}(\widetilde{\mathcal{V}})\to
M\times M$ is a surjective submersion. For $(p,q)\in M\times M$ we write

$\mathcal{V}_{p,q}:=\tilde{\pi}_{2}^{-1}(p,q)\cap\text{int}(\widetilde{\mathcal{V}})=\left({\tilde{\pi}_{2}}|_{\text{int}(\widetilde{\mathcal{V}})}\right)^{-1}(p,q).$

Then each $\mathcal{V}_{p,q}$ is an embedded $(d-n)$-dimensional submanifold
of $\text{int}(\widetilde{\mathcal{V}})$ with Riemannian metric inherited from
$\mathcal{H}\times M\times M$ and induced Riemannian density which we will
denote by $d\mathcal{V}_{p,q}$.

###  2.3. The intersection manifolds $\mathcal{V}$

Now let $V,W\subset M$ be complementary dimensional submanifolds of $M$, say
$\dim(V)=k$ and $\dim(W)=n-k$. We define the _intersection manifold_
$\mathcal{V}$ associated to $\mathcal{H}$, $V$, and $W$ by

$\displaystyle\mathcal{V}$
$\displaystyle:=\\{(h,p,q)\in\accentset{\circ}{\mathcal{H}}\times V\times W\
:\ h(p)=q\\}$
$\displaystyle=\left({\tilde{\pi}_{2}}\big{|}_{\text{int}(\widetilde{\mathcal{V}})}\right)^{-1}(V\times
W)=\tilde{\pi}_{2}^{-1}(V\times W)\cap\text{int}(\widetilde{\mathcal{V}}).$

Since
${\tilde{\pi}_{2}}\big{|}_{\text{int}(\widetilde{\mathcal{V}})}:\text{int}(\widetilde{\mathcal{V}})\to
M\times M$ is a surjective submersion, $\mathcal{V}$ is a $C^{1}$ submanifold
of $\text{int}({\widetilde{V}})$ with
$\operatorname{codim}_{\widetilde{\mathcal{V}}}(\mathcal{V})=\operatorname{codim}_{M\times
M}(V\times W)=n$, so

$\dim(\mathcal{V})=\dim(\widetilde{\mathcal{V}})-n=\dim(\mathcal{H})+n-n=\dim(\mathcal{H}).$

Again note that $\mathcal{V}$ inherits a Riemannian metric from
$\mathcal{H}\times M\times M$; as usual we will write $d\mathcal{V}$ for the
induced Riemannian density.

For $i=1,2$, define $\pi_{i}:={\tilde{\pi}_{i}}|_{\mathcal{V}}$. Then we have
the diagram

${\mathcal{V}}$${\mathcal{H}}$${V\times W}$$\scriptstyle{\pi_{1}\ \ \
}$$\scriptstyle{\pi_{2}}$

of $C^{1}$ maps, with $\pi_{2}$ again a surjective submersion.

In the first step of the proof we will apply the co-area formula to this
diagram to convert our main integral

$\displaystyle\int_{h\in\mathcal{H}}\\#(h(V)\cap W)\ d\mathcal{H}$

to an integral over $V\times W$ involving the normal Jacobians of $\pi_{1}$
and $\pi_{2}$. To estimate this latter integral, we will need a description of
the tangent spaces

$T_{(h,p,q)}\mathcal{V}\subset T_{h}\mathcal{H}\times T_{p}V\times T_{q}W.$

This is given easily by

######  Claim 2.1.

For all $(h,p,q)\in\mathcal{V}$ we have

$T_{(h,p,q)}\mathcal{V}=\\{(\dot{h},\dot{p},\dot{q})\in T_{h}\mathcal{H}\times
T_{p}V\times T_{q}W\ :\
J_{(h,T_{p}V,T_{q}W)}(\dot{p},\dot{q})=d(ev_{p})_{h}(\dot{h})\\}$

where $J_{(h,T_{p}V,T_{q}W)}:T_{p}V\times T_{q}W\to T_{q}M$ is defined by

$J_{(h,T_{p}V,T_{q}W)}(v,w)=w-dh(v).$

###### Proof.

Differentiating the defining equation $h(p)=q$ yields

$\displaystyle\dot{q}=d(ev_{p})_{h}(\dot{h})+dh(\dot{p}).$

The claim follows. ∎

Notice that, since $d(ev_{p})_{h}:T_{p}\mathcal{H}\to T_{h(p)}M$ is surjective
for every $(h,p)\in\mathcal{H}\times M$, the projection map

$\displaystyle d(\pi_{1})_{(h,p,q)}\ :\ T_{(h,p,q)}\mathcal{V}$
$\displaystyle\longrightarrow T_{h}\mathcal{H}$
$\displaystyle(\dot{h},\dot{p},\dot{q})$ $\displaystyle\longmapsto\dot{h}$

will be surjective if and only if $J_{(h,T_{p},T_{q}W)}$ is surjective, i.e.
iff $dh(T_{p}V)+T_{q}W=T_{q}M$. We summarize this observation in the
following:

######  Proposition 2.2.

The regular points of the projection map $\pi_{1}:\mathcal{V}\to\mathcal{H}$
are precisely those $(h,p,q)\in\mathcal{V}$ such that $h(V)$ intersects $W$
transversally at $q$. Thus $h\in\accentset{\circ}{\mathcal{H}}$ is a regular
value of $\pi_{1}$ iff $h(V)\pitchfork W$.

The restriction to the interior of $\mathcal{H}$ in the above statement is
necessary because by definition
$\mathcal{V}\subset\accentset{\circ}{\mathcal{H}}\times V\times W$, and thus
every $h\in\partial\mathcal{H}$ is trivially a regular value of $\pi_{1}$,
even though $\mathcal{V}$ gives no (immediate) information about the
intersections $h(V)\cap W$ for such $h$. However, since $\partial\mathcal{H}$
has measure zero in $\mathcal{H}$ and $\dim(\mathcal{H})=\dim(\mathcal{V})$,
by applying Sard’s theorem to the $C^{1}$ map
$\pi_{1}:\mathcal{V}\to\accentset{\circ}{\mathcal{H}}$ we get

######  Corollary 2.3.

$h(V)\pitchfork W$ for almost every $h\in\mathcal{H}$.

######  Remark.

The argument given here essentially follows the usual proof of the classical
transversality theorem. In low regularity, however, its appeal to Sard’s
theorem requires constraints on the dimensions of $V$ and $W$. In particular,
in the $C^{1}$ category Corollary 2.3 fails when $\dim(V)+\dim(W)>\dim(M)$.
This difficulty is the main reason for considering Hausdorff measures instead
of volumes of intersections in the generalization to Theorem 1.1; see §4 for
details.

An immediate consequence of the formula above is that the tangent space
$T_{(h,p,q)}\mathcal{V}$ depends only on $p,T_{p}V$ and $T_{q}W$. The same is
therefore true of the linear maps $(d\pi_{i})_{(h,p,q)}$ (themselves just
restrictions of projections), and consequently their normal Jacobians.

We would like to show that more is true, namely that
$\operatorname{NJ}(\pi_{1})$ and $\operatorname{NJ}(\pi_{2})$ are continuous
when viewed as functions of $h$, $T_{p}V$, and $T_{q}W$. The next section
introduces the objects necessary to make this statement precise.

###  2.4. The bundles $\mathcal{G}_{k}\to\widetilde{\mathcal{V}}$

For each $0\leq k\leq n$ let

$\operatorname{Gr}_{k}(TM)=\displaystyle\bigsqcup_{p\in
M}\operatorname{Gr}_{k}(T_{p}M)$

denote the $k$-Grassmanian bundle over $M$. This is a smooth fiber bundle over
$M$, associated to the tangent bundle $TM$ in the sense that it can be
obtained by gluing local trivializations
$U_{\alpha}\times\operatorname{Gr}_{k}({\mathbb{R}}^{n})$ according to the
smooth clutching functions $\rho_{\alpha,\beta}:U_{\alpha}\cap U_{\beta}\to
GL(n,{\mathbb{R}})$ for $TM$. For each $k$ we let
$p_{k}:\operatorname{Gr}_{k}(TM)\to M$ denote the standard (smooth) projection
map. For $\sigma\in\operatorname{Gr}_{k}(TM)$ we will typically write
$\sigma=\sigma_{x}$ for $x\in M$ to mean that $x=p_{k}(\sigma)$, i.e.
$\sigma\in\operatorname{Gr}_{k}(T_{x}M)$. In words,
$\sigma_{x}\in\operatorname{Gr}_{k}(TM)$ simply denotes a $k$-plane tangent to
$M$ at the point $x$.

Now define

$\mathcal{G}_{k}:=\left\\{(h,\sigma_{p},\sigma_{q})\in\mathcal{H}\times\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)\
:\ h(p)=q\right\\}.$

Clearly $\mathcal{G}_{k}$ is a $C^{1}$ submanifold of
$\mathcal{H}\times\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$,
and indeed the map $\text{id}\times p_{k}\times p_{n-k}$ exhibits it as a
$C^{1}$
$(\operatorname{Gr}_{k}({\mathbb{R}}^{n})\times\operatorname{Gr}_{n-k}({\mathbb{R}}^{n}))$-bundle
over $\widetilde{\mathcal{V}}$. (In particular,
$\mathcal{G}_{0}\cong\mathcal{G}_{n}\cong\widetilde{\mathcal{V}}$.)

######  Claim 2.4.

The map
$\widehat{\pi}^{k}_{2}:\mathcal{G}_{k}\to\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$,
$(h,\sigma_{1},\sigma_{2})\mapsto(\sigma_{1},\sigma_{2})$ is a $C^{1}$
submersion and surjective when restricted to
$\accentset{\circ}{\mathcal{G}}_{k}$.

###### Proof.

This follows from the observation that $\mathcal{G}_{k}$ can be obtained as
the pullback of the smooth fiber bundle
$\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)\to M\times M$ via
the $C^{1}$ submersion $\tilde{\pi}_{2}:\widetilde{\mathcal{V}}\to M\times M$.
More precisely, we have the following diagram:

${\mathcal{G}_{k}}$${\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)}$${\widetilde{\mathcal{V}}}$${M\times
M.}$$\scriptstyle{\text{id}\times p_{k}\times
p_{n-k}}$$\scriptstyle{\widehat{\pi}^{k}_{2}}$$\scriptstyle{p_{k}\times
p_{n-k}}$$\scriptstyle{\tilde{\pi}_{2}}$

In local coordinates (i.e. above any chart $U\subset\widetilde{\mathcal{V}}$),
$\hat{\pi}_{k}$ has the form

$\hat{\pi}_{k}\ \cong\ \tilde{\pi}_{2}\times\text{id}\ :\
U\times\left(\operatorname{Gr}_{k}({\mathbb{R}}^{n})\times\operatorname{Gr}_{n-k}({\mathbb{R}}^{n})\right)\
\longrightarrow\
\tilde{\pi}_{2}(U)\times\left(\operatorname{Gr}_{k}({\mathbb{R}}^{n})\times\operatorname{Gr}_{n-k}({\mathbb{R}}^{n})\right),$

which is clearly a submersion because $\tilde{\pi}_{2}$ is. The surjectivity
of $\left.\hat{\pi}_{k}\right|_{\accentset{\circ}{\mathcal{G}}_{k}}$ follows
immediately from that of $ev\big{|}_{\accentset{\circ}{\mathcal{H}}\times M}$,
i.e. the $k=0$ case of $\operatorname{(A2)}$. ∎

Now for each $(h,\sigma_{p},\sigma_{q})\in\mathcal{G}_{k}$ we can define

$\displaystyle J_{(h,\sigma_{p},\sigma_{q})}:\sigma_{p}\times\sigma_{q}$
$\displaystyle\to T_{q}M$ $\displaystyle(v,w)$ $\displaystyle\mapsto w-dh(v),$

and

$E_{k}(h,\sigma_{p},\sigma_{q}):=\\{(\dot{h},v,w)\in
T_{h}\mathcal{H}\times\sigma_{p}\times\sigma_{q}\ :\
J_{(h,\sigma_{p},\sigma_{q})}(v,w)=d(ev_{p})_{h}(\dot{h})\\}.$

Finally, set

$\displaystyle\Pi_{1}(h,\sigma_{p},\sigma_{q})\
:E_{k}(h,\sigma_{p},\sigma_{q})$ $\displaystyle\longrightarrow
T_{h}\mathcal{H}$ $\displaystyle(\dot{h},v,w)$
$\displaystyle\longmapsto\dot{h},$
$\displaystyle\Pi_{2}(h,\sigma_{p},\sigma_{q})\
:E_{k}(h,\sigma_{p},\sigma_{q})$
$\displaystyle\longrightarrow\sigma_{p}\times\sigma_{q}$
$\displaystyle(\dot{h},v,w)$ $\displaystyle\longmapsto(v,w),$

and for $i=1,2$, define

$\eta_{i}:\mathcal{G}_{k}\longrightarrow{\mathbb{R}}_{\geq 0}$

by

$\eta_{i}(h,\sigma_{p},\sigma_{q})=\operatorname{NJ}(\Pi_{i}(h,\sigma_{p},\sigma_{q})).\vspace{2mm}$

The point of these objects, immediate from the definitions and Claim 2.1, is
the following:

######  Lemma 2.5.

Let $V,W\subset M$ be complementary dimensional submanifolds with $\dim(V)=k$,
and $\mathcal{V}$ their associated intersection manifold. Then for all
$(h,p,q)\in\mathcal{V}$ and $i=1,2$ we have

$\displaystyle T_{(h,p,q)}\mathcal{V}$ $\displaystyle=E_{k}(h,T_{p}V,T_{q}W);$
$\displaystyle d(\pi_{i})_{(h,p,q)}$ $\displaystyle=\Pi_{i}(h,T_{p}V,T_{q}W);$
$\displaystyle\operatorname{NJ}(\pi_{i})_{(h,p,q)}$
$\displaystyle=\eta_{i}(h,T_{p}V,T_{q}W).$

We will also need the following properties:

######  Claim 2.6.

$E_{k}$ is a continuous vector bundle over $\mathcal{G}_{k}$ of rank $d$.

###### Proof.

Let $\Omega_{j}$ denote the tautological vector bundle over
$\operatorname{Gr}_{j}(TM)$, i.e. the smooth rank-$j$ vector bundle
$\Omega_{j}\to\operatorname{Gr}_{j}(TM)$ with fibers
$\Omega_{j}(\sigma)=\sigma$. Observe that, since each
$\sigma\in\operatorname{Gr}_{j}(TM)$ is a subspace of some tangent space of
$M$, $\Omega_{j}$ carries a natural bundle metric obtained by restricting the
Riemannian metric on $M$.

Let $\mathcal{E}_{k}$ denote the restriction of the vector bundle
$T\mathcal{H}\times\Omega_{k}\times\Omega_{n-k}\to\mathcal{H}\times\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$
to $\mathcal{G}_{k}$. Again this carries a natural bundle metric, namely the
product metric induced from its factors.

We can now view $J$ as a bundle map

$\displaystyle J:\mathcal{E}_{k}\longrightarrow\gamma^{*}TM,$

where $\gamma:\mathcal{G}_{k}\to M$ is given by
$\gamma(h,\sigma_{p},\sigma_{q})=q$. Continuity of $J$ follows from that of
our parametrization $\psi:\mathcal{H}\to\operatorname{Diff}^{1}(M)$. Define

$H:\mathcal{E}_{k}\longrightarrow\gamma^{*}TM$

by

$\displaystyle
H_{(h,\sigma_{p},\sigma_{q})}:T_{h}\mathcal{H}\times\sigma_{p}\times\sigma_{q}$
$\displaystyle\longrightarrow T_{q}M$ $\displaystyle(\dot{h},v,w)$
$\displaystyle\longmapsto d(ev_{p})_{h}(\dot{h}),$

and note that $H$ is a continuous bundle map because $ev$ is $C^{1}$. Finally
set $L:=J-H$.

Since $d(ev_{p})_{h}$ is surjective for all $(h,p)\in\mathcal{H}\times M$ and
$J_{(h,\sigma_{p},\sigma_{q})}$ does not depend on $\dot{h}$,
$L_{(h,\sigma_{p},\sigma_{q})}$ is surjective for all
$(h,\sigma_{p},\sigma_{q})\in\mathcal{G}_{k}$. In particular, $L$ is a
constant rank vector bundle map, so $\ker(L)$ is a continuous vector subbundle
of $\mathcal{E}_{k}$ (see e.g. [Lee13, Theorem 10.34]), with

$\operatorname{rank}(\ker(L))=\operatorname{rank}(\mathcal{E}_{k})-\operatorname{rank}(TM)=(d+n)-n=d.$

Since clearly $\ker(L)=E_{k}$, we are done. ∎

######  Claim 2.7.

$\eta_{1}$ and $\eta_{2}$ are continuous.

###### Proof.

$\Pi_{1}$ and $\Pi_{2}$ can be viewed as continuous bundle maps between metric
vector bundles, namely

$\displaystyle\Pi_{1}:E_{k}\to(\widehat{\pi}^{k}_{1})^{*}T\mathcal{H};$
$\displaystyle\Pi_{2}:E_{k}\to(\widehat{\pi}^{k}_{2})^{*}(\Omega_{k}\times\Omega_{n-k}),$

where $\widehat{\pi}^{k}_{1}:\mathcal{G}_{k}\to\mathcal{H}$ and
$\widehat{\pi}^{k}_{2}:\mathcal{G}_{k}\to\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$
denote the natural projections. Thus their normal Jacobians are continuous
functions of the base. ∎

######  Claim 2.8.

$\eta_{2}>0$ everywhere on $\mathcal{G}_{k}$.

###### Proof.

$\Pi_{2}(h,\sigma_{p},\sigma_{q})$ has full rank for all
$(h,\sigma_{p},\sigma_{q})\in\mathcal{G}_{k}$. ∎

Now define

$\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}}:=\left(\widehat{\pi}^{k}_{2}|_{\accentset{\circ}{\mathcal{G}}_{k}}\right)^{-1}(\sigma_{p},\sigma_{q})=(\widehat{\pi}^{k}_{2})^{-1}(\sigma_{p},\sigma_{q})\cap\accentset{\circ}{\mathcal{G}}_{k}.$

Since $\widehat{\pi}^{k}_{2}|_{\accentset{\circ}{\mathcal{G}}_{k}}$ is a
surjective $C^{1}$ submersion, $\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}}$
is an embedded $(d-n)$-dimensional submanifold of
$\accentset{\circ}{\mathcal{G}}_{k}$ for all
$(\sigma_{p},\sigma_{q})\in\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$.
As usual we let $d\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}}$ denote the
Riemannian density induced by the restriction of the metric on
$\mathcal{G}_{k}$ to $\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}}$.

######  Claim 2.9.

For all
$(\sigma_{p},\sigma_{q})\in\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$,
$\eta_{1}$ is positive somewhere on the fiber
$\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}}$.

###### Proof.

Let
$(\sigma_{p},\sigma_{q})\in\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$.
Clearly it suffices to find $h\in\accentset{\circ}{\mathcal{H}}$ such that
$h(p)=q$ and $\Pi_{1}(h,\sigma_{p},\sigma_{q})$ is surjective. Observe as in
Proposition 2.2 that $\Pi_{1}(h,\sigma_{p},\sigma_{q})$ is surjective iff
$dh(\sigma_{p})\pitchfork\sigma_{q}$. But by (A2) we can find
$h\in\accentset{\circ}{\mathcal{H}}$ such that
$dh(\sigma_{p})={\sigma_{q}}^{\perp}$, so we are done. ∎

######  Remark.

Claim 2.9 is in fact the only place where (A2) is used, and its proof makes
clear that this assumption is actually stronger than required for this result.
What we really need is the following:

  * $\operatorname{(A2^{@itemi})}$

For any $\sigma_{p}\in\operatorname{Gr}_{k}(TM)$ and
$\sigma_{q}\in\operatorname{Gr}_{n-k}(TM)$, there is an
$h\in\accentset{\circ}{\mathcal{H}}$ such that $h(p)=q$ and
$dh(\sigma_{p})\pitchfork\sigma_{q}$.

So for instance (A2), which is equivalent to the claim that for all
$\sigma\in\operatorname{Gr}_{k}(TM)$ and $q\in M$ we have

$\operatorname{Gr}_{k}(T_{q}M)\subset\\{dh(\sigma)\ :\
h\in\accentset{\circ}{\mathcal{H}}\\},$

could be replaced by the requirement that for all
$\sigma\in\operatorname{Gr}_{k}(TM)$ and $q\in M$, the set

$\operatorname{Gr}_{k}(T_{q}M)\cap\\{dh(\sigma)\ :\
h\in\accentset{\circ}{\mathcal{H}}\\}$

has nonempty interior in $\operatorname{Gr}_{k}(T_{q}M)$.444This latter
property is satisfied, in particular, if we assume that
$\accentset{\circ}{ev}_{0}$ is surjective (i.e. the interior of $\mathcal{H}$
acts transitively on points) and $ev_{k}$ is a submersion for all $k$. We
prefer (A2) here largely for simplicity, and because, in the general setting
we consider, it is no more difficult to construct a family $\mathcal{H}$
satisfying this hypothesis than $\operatorname{(A2^{*})}$; see §5. The
sufficiency of the weaker assumption may, however, be useful in applications
where $\operatorname{(A2^{*})}$ is easy to verify for a given family
$\mathcal{H}$, despite additional constraints which render (A2) unachievable.

The main result of this section is now an immediate consequence of the
following general fact. For details of the proof, see Appendix A.

######  Proposition 2.10 (Appendix A, Lemma 6.4).

Let $f:M^{m}\to N^{n}$ be a $C^{1}$ submersion between compact Riemannian
manifolds, with $\partial M\neq\varnothing$ and $\partial N=\varnothing$, and
$\phi\in C^{0}(M,{\mathbb{R}}_{\geq 0})$. Write
$\mathcal{F}_{q}:=f^{-1}(q)\cap\accentset{\circ}{M}$. Then there exists
$C_{U}^{\phi}>0$ such that

$\displaystyle\int_{\mathcal{F}_{q}}\phi\ d\mathcal{F}_{q}\leq C_{U}^{\phi}$

for all $q\in N$.

Suppose in addition that $f\big{|}_{\accentset{\circ}{M}}$ is surjective and
$\phi$ is positive somewhere on each interior fiber $\mathcal{F}_{q}$. Then
there also exists $C_{L}^{\phi}>0$ such that

$C_{L}^{\phi}\leq\displaystyle\int_{\mathcal{F}_{q}}\phi\ d\mathcal{F}_{q}$

for all $q\in N$.

###### Sketch of proof.

For the upper bound, it clearly suffices to show that
$\operatorname{vol}_{m-n}(\mathcal{F}_{q})\leq\widehat{C}_{U}$ for some
uniform $\widehat{C}_{U}<+\infty$; then one can set
$C_{U}^{\phi}:=\widehat{C}_{U}\cdot\left\lVert\phi\right\rVert_{\infty}$. By
attaching a collar to $M$ and extending the metric we obtain an open
Riemannian manifold $\widehat{M}$ isometrically containing $M$; by shrinking
the collar we can assume that $f$ extends to a submersion $F:\widehat{M}\to
N$. Now choose a finite collection $\mathcal{B}$ of submersion charts for $F$
which cover $M$, such that the plaques of elements of $\mathcal{B}$ have
volumes bounded uniformly above by some $C_{1}<+\infty$. Since every fiber
$\mathcal{F}_{q}$ intersects each submersion chart in at most one plaque, we
get $\operatorname{vol}(\mathcal{F}_{q})\leq C_{1}\cdot\\#(\mathcal{B})$.

For the lower bound, one can use surjectivity of $f|_{\accentset{\circ}{M}}$
and compactness of $N$ to find finitely many submersion charts
$B_{1},\dots,B_{\ell}\subset\accentset{\circ}{M}$ for
$f|_{\accentset{\circ}{M}}$ whose images cover $N$, with fiber plaque volumes
bounded below by some $\widehat{C}_{L}>0$ and $\phi\geq\delta>0$ on every
$B_{j}$. Then each $\mathcal{F}_{q}$ contains at least one plaque of some
$B_{j}$, so

$\displaystyle\int_{\mathcal{F}_{q}}\phi
d\mathcal{F}_{q}\geq\widehat{C}_{L}\delta>0,$

as desired. ∎

######  Corollary 2.11.

Let
$\eta=\frac{\eta_{1}}{\eta_{2}}:\mathcal{G}_{k}\longrightarrow{\mathbb{R}}_{\geq
0}$. Then there exists $C_{k}>1$ such that

$\dfrac{1}{C_{k}}\leq\displaystyle\int_{\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}}}\eta\
d\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}}\leq C_{k}$

for all
$(\sigma_{p},\sigma_{q})\in\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$.

###### Proof.

By Claims 2.4, 2.7, 2.8, and 2.9, this follows from Proposition 2.10 with
$f=\widehat{\pi}^{k}_{2}$ and $\phi=\eta$. ∎

######  Remark.

If we define

$\Phi_{k}:\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)\longrightarrow{\mathbb{R}}_{\geq
0}$

by

$\Phi_{k}(\sigma_{p},\sigma_{q})=\displaystyle\int_{\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}}}\dfrac{\eta_{1}}{\eta_{2}}\
d\widehat{\mathcal{F}}_{\sigma_{p},\sigma_{q}},$

then Corollary 2.11 simply says that $\Phi_{k}$ is bounded uniformly above and
away from zero. Since $\operatorname{Gr}_{j}(TM)$ is compact for all $j$ and
$\Phi_{k}>0$ by Claim 2.9, this would be obvious if $\Phi_{k}$ were
continuous, and indeed when $\partial\mathcal{H}=\varnothing$ this is the case
by an easy application of Ehresmann’s fibration theorem. When
$\partial\mathcal{H}\neq\varnothing$, however, Ehresmann’s theorem fails, and
in fact $\Phi_{k}$ cannot be assumed continuous in general. We were led to
Proposition 2.10 by the need to fill this gap, since it shows in particular
that the (non-)continuity of $\Phi_{k}$ is inessential for the desired
bounds.555An easy adaptation of the proof of the lower bound in Proposition
2.10 does, however, show that $\Phi_{k}$ is always lower semicontinuous.

As a final aside, we note that Poincaré’s formula (as stated in Theorem 1.3)
reduces after the results of the next two sections to the claim that the
functions $\Phi_{k}$ (and all the analogously defined functions
$\Phi_{k,\ell}$ for $n\leq k+\ell\leq 2n$, where $\Phi_{k}=:\Phi_{k,n-k}$) are
_constant_ in the homogeneous case, and moreover positive provided the induced
action on each $\operatorname{Gr}_{k}(TM)$ is transitive. This is in turn a
fairly straightforward (intuitively almost obvious) consequence of some basic
geometry of compact Lie groups.

##  3\. Proof of Main Theorem

Let $V,W\subset M$ be complementary dimensional submanifolds of $M$, say
$\dim(V)=k$ and $\dim(W)=n-k$. Our proof of Theorem 1.1 now consists of two
steps. First, we convert the integral

$\displaystyle\int_{\mathcal{H}}\\#(h(V)\cap W)\ d\mathcal{H}$

to an integral over $V\times W$. We then apply the results of §2.4 to bound
the new integrand uniformly above and away from zero, proving the
result.666For a related use of this “double fibration” method, see [ACSS22].

###  3.1. The co-area formula

The key tool in the first step is the co-area formula, which we state here for
convenience. Though we choose to give only the simplest form sufficient for
our needs, note that this is a special case of a more general result; see e.g.
[Fed69, §3.2], [Nic11], [Sim14].

######  Proposition 3.1 (The co-area formula, $C^{r}$ case).

Let $f:M^{n+k}\to N^{n}$ be a $C^{r}$ map between smooth Riemannian manifolds
(without boundary), with $r\geq k+1\geq 1$. For $q\in N$, write
$F_{q}=f^{-1}(q)$. Then for any Borel measurable function $h$ on $M$ we have

$\displaystyle\int_{M}h\cdot\operatorname{NJ}(f)\ dM=\displaystyle\int_{q\in
N}\left(\displaystyle\int_{F_{q}}h\ dF_{q}\right)dN,$

with both sides finite if one is. In particular, if $h$ is (essentially)
bounded and $\operatorname{NJ}(f)\in L^{1}(M)$, then

$\left(\Phi:q\longmapsto\displaystyle\int_{F_{q}}h\ dF_{q}\right)\in
L^{1}(N),$

and

$\displaystyle\int_{M}h\cdot\operatorname{NJ}(f)\
dM=\displaystyle\int_{N}\Phi\ dN.$

In the special case when $f$ is a submersion, the above holds for all $k\geq
0$ assuming only that $f$ is $C^{1}$.

######  Remark.

Note that the function

$\Phi:\ N\ni q\longmapsto\displaystyle\int_{F_{q}}h\
dF_{q}\in{\mathbb{R}}_{\geq 0}\cup\\{\infty\\}$

is well-defined only at regular values $q$ of $f$, for which $F_{q}$ is an
embedded $k$-dimensional submanifold of $M$ with induced Riemannian density
$dF_{q}$. But then by Sard’s theorem it is defined a.e. on $N$, so (provided
it is measurable) its integral over $N$ makes sense. This application of
Sard’s is the main reason for the required relation between the regularity of
$f$ and $k=\dim(M)-\dim(N)$ and explains why this regularity assumption is
unnecessary when $f$ is a submersion, since in that case _every_ point of $N$
is a regular value for $f$. Indeed, in the submersive case the co-area formula
is an easy consequence of Fubini’s theorem.

More general versions of the co-area formula avoid these issues entirely by
replacing $dF_{q}$ with the $k$-dimensional Hausdorff measure associated to
the metric space structure on $M$; see in particular Proposition 4.3. These
measures agree when $F_{q}$ is a $k$-dimensional submanifold of $M$.

We now apply the co-area formula to prove the following, the main result of
this section:

######  Proposition 3.2.

For $\mathcal{H},V,$ and $W$ as in Theorem 1.1 we have

$\displaystyle\int_{h\in\mathcal{H}}\\#(h(V)\cap W)\
d\mathcal{H}=\displaystyle\int_{(p,q)\in V\times
W}\left(\displaystyle\int_{\mathcal{V}_{p,q}}\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}\
d\mathcal{V}_{p,q}\right)dVdW,$

with both sides finite if one is.

###### Proof.

Recall the diagram

${\mathcal{V}}$${\accentset{\circ}{\mathcal{H}}}$${V\times
W}$$\scriptstyle{\pi_{1}\ \ \ }$$\scriptstyle{\pi_{2}}$

of Riemannian manifolds and $C^{1}$ maps. Since

$\dim(\mathcal{V})=\dim(\mathcal{H})=d$

and $\pi_{1}$ is $C^{1}$, we may apply Proposition 3.1 to $\pi_{1}$ (with
$h\equiv 1$) to obtain

(3) $\displaystyle\int_{\mathcal{V}}\ \operatorname{NJ}(\pi_{1})\
d\mathcal{V}=\displaystyle\int_{h\in\accentset{\circ}{\mathcal{H}}}\left(\displaystyle\int_{\pi_{1}^{-1}(h)}d\pi_{1}^{-1}(h)\right)d\mathcal{H}=\displaystyle\int_{h\in\mathcal{H}}\\#(h(V)\cap
W)\ d\mathcal{H}.$

The second equality follows from the fact that $\pi_{1}^{-1}(h)$ is a
$0$-dimensional submanifold of $\mathcal{V}$ for a.e. $h\in\mathcal{H}$, i.e.
a discrete set of points, in which case we have

$\displaystyle\int_{\pi_{1}^{-1}(h)}d\pi_{1}^{-1}(h)=\displaystyle\sum_{x\in\pi_{1}^{-1}(h)}1=\\#(\pi_{1}^{-1}(h))=\\#(h(V)\cap
W).$

Note in particular that the a.e. defined function

$h\longmapsto\displaystyle\int_{\pi_{1}^{-1}(h)}d\pi_{1}^{-1}(h)=\\#(h(V)\cap
W)$

will be in $L^{1}(\accentset{\circ}{\mathcal{H}})=L^{1}(\mathcal{H})$ provided
$\operatorname{NJ}(\pi_{1})\in L^{1}(\mathcal{V})$.

We now apply the co-area formula a second time, this time to the map $\pi_{2}$
with $h=\frac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}$.
Observe that this is well-defined because $\pi_{2}$ is a submersion, so
$\operatorname{NJ}(\pi_{2})>0$ everywhere on $\mathcal{V}$. We get

$\displaystyle\displaystyle\int_{\mathcal{V}}\ \operatorname{NJ}(\pi_{1})\
d\mathcal{V}$ $\displaystyle=\displaystyle\int_{\mathcal{V}}\
\left(\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}\right)\cdot\operatorname{NJ}(\pi_{2})\
d\mathcal{V}$ $\displaystyle=\displaystyle\int_{(p,q)\in V\times
W}\left(\displaystyle\int_{\mathcal{V}_{p,q}}\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}\
d\mathcal{V}_{p,q}\right)dVdW.$

Putting this together with (3) gives the desired result. ∎

###  3.2. Fiber integral bounds

By the preceding section, it now suffices to give uniform bounds for the
integrals

$\displaystyle\int_{\mathcal{V}_{p,q}}\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}\
d\mathcal{V}_{p,q}.$

Most of the necessary work has been done in §2.4. First recall Lemma 2.5,
which gives

$\displaystyle T_{(h,p,q)}\mathcal{V}$ $\displaystyle=E_{k}(h,T_{p}V,T_{q}W),$
$\displaystyle d(\pi_{i})_{(h,p,q)}$ $\displaystyle=\Pi_{i}(h,T_{p}V,T_{q}W),$
$\displaystyle\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}(h,p,q)=\dfrac{\eta_{1}}{\eta_{2}}(h,T_{p}$
$\displaystyle V,T_{q}W)=\eta(h,T_{p}V,T_{q}W).$

Here $i=1,2$ and $E_{k},$ $\Pi_{i}$, $\eta_{i}$ are as defined in §2.4. As an
immediate consequence we obtain

######  Lemma 3.3.

For all $(p,q)\in V\times W$, the fiber $\mathcal{V}_{p,q}$ is isometric to
$\widehat{\mathcal{F}}_{T_{p}V,T_{q}W}$. Moreover, we have

$\displaystyle\int_{\widehat{\mathcal{F}}_{T_{p}V,T_{q}W}}\eta\
d\widehat{\mathcal{F}}_{T_{p}V,T_{q}W}=\displaystyle\int_{\mathcal{V}_{p,q}}\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}\
d\mathcal{V}_{p,q}.$

###### Proof.

Fix $(p,q)\in V\times W$. Using the fact that the Riemannian metrics on
$\mathcal{G}_{k}$ and $\mathcal{V}$ are simply the restrictions of the product
metrics on
$\mathcal{H}\times\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$
and $\mathcal{H}\times M\times M$, respectively, it is easy to see that the
map

$\widehat{\mathcal{F}}_{T_{p}V,T_{q}W}\longrightarrow\mathcal{V}_{p,q},\ \
(h,T_{p}V,T_{q}W)\longmapsto(h,p,q)$

is an isometry.777Put more intuitively, both sides are just the set of
$h\in\accentset{\circ}{\mathcal{H}}$ such that $h(p)=q$ as a submanifold of
$\mathcal{H}$ with the induced Riemannian metric. The fact that

$\displaystyle\int_{\mathcal{V}_{p,q}}\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}(h,p,q)\
d\mathcal{V}_{p,q}=\displaystyle\int_{\widehat{\mathcal{F}}_{T_{p}V,T_{q}W}}\eta(h,T_{p}V,T_{q}W)\
d\widehat{\mathcal{F}}_{T_{p}V,T_{q}W}$

then follows by change of variables. ∎

######  Corollary 3.4.

If $C_{k}\geq 1$ is the constant given by Corollary 2.11, then

$\dfrac{1}{C_{k}}\leq\displaystyle\int_{\mathcal{V}_{p,q}}\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}\
d\mathcal{V}_{p,q}\leq C_{k}$

for all $p,q\in V\times W$.

Combining this with Proposition 3.2 and setting $C=\displaystyle\max_{0\leq
k\leq n}C_{k}$ completes the proof of Theorem 1.1.

######  Remark.

Our use of the rather abstract results of §2.4 and Appendix A to obtain the
constants $C_{k}$ leaves their geometric meaning somewhat opaque. The
advantage of this approach is that it generalizes easily to prove Theorem 4.1,
but in the complementary-dimensional case a more concrete picture is
available, which may be useful in applications where estimates of $C$ become
desirable. We sketch this approach in Appendix B.

##  4\. Generalization

We now describe the generalization of Theorem 1.1 to the case when
$\dim(V)+\dim(W)\geq\dim(M)$, where $\\#(h(V)\cap W)$ is replaced by the
appropriate-dimensional Hausdorff measure associated to the metric space
structure on $M$.

For $r\geq 0$ and any Riemannian manifold $X$, let $H_{X}^{r}$ denote the
$r$-dimensional Hausdorff measure on $X$ associated to its induced metric
space structure. When $r=\dim(X)$, $H_{X}^{r}$ agrees with the volume measure
$\nu_{X}$ induced by the metric on $X$. More generally, if $Y\subset X$ is a
$k$-dimensional submanifold with Riemannian metric inherited from $X$ and
induced volume measure $\nu_{Y}$, we have

$H_{X}^{k}\big{|}_{Y}=H_{Y}^{k}=\nu_{Y}.$

In particular, the volume of a $k$-dimensional submanifold of $X$ is equal to
its $k$-dimensional Hausdorff measure in $X$. In this sense, for
$k\in{\mathbb{N}}$ one can think of $H_{X}^{k}$ as a generalization of
$k$-dimensional volume to non-smooth subsets of $X$.

We now consider a fixed closed Riemannian manifold $M$ of dimension $n$. For
simplicity we write $H^{r}:=H^{r}_{M}$.

######  Theorem 4.1.

Suppose $(\mathcal{H},\psi)$ is compact $C^{1}$ family of diffeomorphisms of
$M$ such that

  1. $\operatorname{(A1)}$

$ev:\mathcal{H}\times M\to M\times M$ defined by $ev(h,p)=(p,h(p))$ is a
submersion;

  2. $\operatorname{(A2)}$

$\accentset{\circ}{ev}_{k}:\accentset{\circ}{\mathcal{H}}\times\operatorname{Gr}_{k}(TM)\to\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{k}(TM)$
defined by $ev_{k}(h,\sigma)=(\sigma,dh(\sigma))$ is surjective for all $0\leq
k\leq n-1$.

Then there is a constant $C=C(\mathcal{H},\psi)\geq 1$ such that for any
submanifolds $V^{k},W^{\ell}\subset M$ with $k+\ell\geq n$, we have

(4) $\dfrac{1}{C}\operatorname{vol}(V)\operatorname{vol}(W)\ \ \leq\ \
\displaystyle\int_{h\in\mathcal{H}}H^{k+\ell-n}(h(V)\cap W)\ d\mathcal{H}\ \
\leq\ \ C\operatorname{vol}(V)\operatorname{vol}(W).\\\ $

The need to work with Hausdorff measures rather than volumes here derives from
our insistence on assuming only $C^{1}$ regularity. In this context, when
$\dim(V)+\dim(W)>\dim(M)$ we can no longer assume that $h(V)\pitchfork W$ for
a.e. $h\in\mathcal{H}$; thus $\operatorname{vol}(h(V)\cap W)$ is not
necessarily a.e. well-defined. However, if we let $h(V)\cap_{T}W$ denote the
set of transversal intersection points between $h(V)$ and $W$ (which when
nonempty is always a $(k+\ell-n)$-dimensional submanifold of $M$), we will see
below that indeed

$H^{k+\ell-m}(h(V)\cap
W)=H^{k+\ell-m}(h(V)\cap_{T}W)=\operatorname{vol}(h(V)\cap_{T}W)$

for a.e. $h\in\mathcal{H}$.

Note that, since $H^{0}$ is the counting measure on $M$, Theorem 4.1 does
indeed include Theorem 1.1 as a special case. Moreover, the proof method is
essentially the same. Thus in this section we will content ourselves with some
remarks on the modifications needed to adapt our proof of Theorem 1.1 to the
general setting.

Let $V^{k},W^{\ell}\subset M$ be submanifolds with $k+\ell\geq n$. The
solution manifold $\widetilde{\mathcal{V}}\subset\mathcal{H}\times M\times M$
remains defined as before, and again (A1) implies that

$\mathcal{V}:=\left({\tilde{\pi}_{2}}|_{\text{int}(\widetilde{\mathcal{V}})}\right)^{-1}(V\times
W)$

is a $C^{1}$ submanifold of $\widetilde{\mathcal{V}}$. Now, however, we have

$\dim(\mathcal{V})=(d+n)-\operatorname{codim}(V\times
W)=(d+n)-(2n-k-\ell)=d+k+\ell-n.$

The basic observation motivating these definitions and underlying our entire
proof strategy for Theorem 1.1 was that

$\\#(h(V)\cap W)=\\#(\pi_{1}^{-1}(h)),$

or equivalently,

$H^{0}_{M}(h(V)\cap W)=H^{0}_{\mathcal{V}}(\pi_{1}^{-1}(h)).$

The reason was that there is an obvious bijection between these sets (given
explicitly by $\pi_{3}:\mathcal{V}\to M$, defined below), and bijections
between metric spaces preserve $0$-dimensional Hausdorff measure. Obviously
this fails for higher dimensional Hausdorff measures, so for the general case
we must attend to how the maps
$\pi_{3}^{h}:=\pi_{3}|_{\pi_{1}^{-1}(h)}:\pi_{1}^{-1}(h)\to h(V)\cap W$
transform $(k+\ell-n)$-dimensional Hausdorff measures, where
$\pi_{3}:\mathcal{V}\to W$ is defined as one would expect, by
$\pi_{3}(h,p,q)=q$.888In fact this issue is largely an artifact of our
original choice of set-up: it can be avoided by replacing
$\widetilde{\mathcal{V}}$ with $\mathcal{H}\times M$ and
$\widetilde{\pi}_{2}:\widetilde{\mathcal{V}}\to M\times M$ by the map
$\mathcal{H}\times M\to M\times M$, $(h,q)\mapsto(h^{-1}(q),q)$, since for
these choices the resulting identification of $\pi_{1}^{-1}(h)$ with $h(V)\cap
W$ is isometric. See [BC13] for an example of this approach.

First suppose that $h$ is a regular value of $\pi_{1}$; this is true iff
$h(V)\pitchfork W$. In this case, $\pi_{1}^{-1}(h)$ and $h(V)\cap W$ are
$(k+\ell-n)$-dimensional submanifolds of $\mathcal{V}$ and $M$, respectively,
and $\pi_{3}^{h}$ is a $C^{1}$ diffeomorphism. Their $(k+\ell-n)$-Hausdorff
measures then agree with the volume measures induced by their Riemannian
structures, and moreover we have

$\displaystyle H^{k+\ell-n}_{M}(h(V)\cap W)=\operatorname{vol}(h(V)\cap W)$
$\displaystyle=\operatorname{vol}(\pi_{3}(\pi_{1}^{-1}(h)))$
$\displaystyle=\displaystyle\int_{\pi_{1}^{-1}(h)}\left\lVert\textstyle\bigwedge^{k+\ell-n}d\pi_{3}^{h}\right\rVert
d\pi_{1}^{-1}(h)$
$\displaystyle=\displaystyle\int_{\pi_{1}^{-1}(h)}\left\lVert\textstyle\bigwedge^{k+\ell-n}d\pi_{3}^{h}\right\rVert
dH^{k+\ell-n}_{\mathcal{V}}.$

Note also that

$d\pi^{h}_{3}(h,p,q)=d\pi_{3}|_{T_{(h,p,q)}{\pi_{1}^{-1}(h)}}=d\pi_{3}|_{\ker(d\pi_{1}(h,p,q))},$

and this last expression is well-defined even when $(h,p,q)$ is not a regular
point for $\pi_{1}$. Thus we can define

$\rho:\mathcal{V}\longrightarrow{\mathbb{R}}_{\geq 0}$

by

$\rho(h,p,q)=\left\lVert\textstyle\bigwedge^{k+\ell-n}d\pi_{3}\big{|}_{\ker(d\pi_{1}(h,p,q))}\right\rVert,$

and the above equation becomes

(5) $H^{k+\ell-n}_{M}(h(V)\cap W)=\displaystyle\int_{\pi_{1}^{-1}(h)}\rho\
dH^{k+\ell-n}_{\mathcal{V}}.$

So far, (5) holds only when $h$ is a regular value of $\pi_{1}$. In the
complementary dimensional case this was enough, since when
$\dim(\mathcal{H})=\dim(\mathcal{V})$ Sard’s theorem applies and thus (5)
holds for a.e. $h\in\mathcal{H}$. When $\dim(V)+\dim(W)>\dim(M)$, however, we
have $\dim(\mathcal{V})>\dim(\mathcal{H})$, so Sard’s no longer applies (since
$\pi_{1}$ can only be assumed $C^{1}$). We appeal instead to the following
generalization:

######  Proposition 4.2 ([Sim14], Theorem 6.4).

Let $f:M^{m}\to N^{n}$ be a $C^{1}$ map between Riemannian manifolds, and let
$C\subset M$ denote the set of critical points of $f$. Set
$r=\max\\{0,m-n\\}$. Then

(6) $H^{r}_{M}(f^{-1}(y)\cap C)=0$

for Lebesgue a.e. $y\in N$.

######  Remark.

The interesting case (and the one relevant here) is when $m>n$. As Simon notes
in [Sim14], this proposition is itself a straightfoward consequence of
Federer’s general co-area formula; see [Fed69, §3.2] for details.

To apply this to the present setting, first write $C\subset\mathcal{V}$ for
the set of critical points of $\pi_{1}$, and for $h\in\mathcal{H}$, let
$h(V)\cap_{T}W$ denote the set of transversal intersection points between
$h(V)$ and $W$, and $h(V)\cap_{NT}W=(h(V)\cap W)\setminus(h(V)\cap_{T}W)$.
Then we have

$\displaystyle\pi_{3}(\pi_{1}^{-1}(h)\setminus C)$
$\displaystyle=h(V)\cap_{T}W,$ $\displaystyle\pi_{3}(\pi_{1}^{-1}(h)\cap C)$
$\displaystyle=h(V)\cap_{NT}W,$

and

$\displaystyle H^{k+\ell-n}_{M}(h(V)\cap W)$
$\displaystyle=H^{k+\ell-n}_{M}(h(V)\cap_{T}W)+H^{k+\ell-n}_{M}(h(V)\cap_{NT}W)$
$\displaystyle=\displaystyle\int_{\pi_{1}^{-1}(h)\setminus C}\rho\
dH^{k+\ell-n}_{\mathcal{V}}+H^{k+\ell-n}_{M}(\pi_{3}(\pi_{1}^{-1}(h)\cap C)).$

By Proposition 4.2,

$\displaystyle\int_{\pi_{1}^{-1}(h)\setminus C}\rho\
dH^{k+\ell-n}_{\mathcal{V}}=\displaystyle\int_{\pi_{1}^{-1}(h)}\rho\
dH^{k+\ell-n}_{\mathcal{V}}$

for a.e. $h\in\mathcal{H}$. For the second term, observe that
$\pi_{3}=\widetilde{\pi}_{3}|_{\pi_{1}^{-1}(h)}$, where
$\widetilde{\pi}_{3}:\widetilde{\pi}_{1}^{-1}(h)\to M$ is a distance non-
increasing diffeomorphism (indeed $\widetilde{\pi}_{1}^{-1}(h)$ is canonically
identified with $\Gamma(h)\subset M\times M$, with $\widetilde{\pi}_{3}$
corresponding to projection onto the second coordinate), so that

$H^{k+\ell-n}_{M}(\pi_{3}(\pi_{1}^{-1}(h)\cap C))\leq
H^{k+\ell-n}_{\mathcal{V}}(\pi_{1}^{-1}(h)\cap C).$

Since the right-hand side is zero for a.e. $h\in\mathcal{H}$ by Proposition
4.2, so is the left. We conclude that

$\displaystyle H^{k+\ell-n}_{M}(h(V)\cap W)$
$\displaystyle=\displaystyle\int_{\pi_{1}^{-1}(h)\setminus C}\rho\
dH^{k+\ell-n}_{\mathcal{V}}+H^{k+\ell-n}_{M}(\pi_{3}(\pi_{1}^{-1}(h)\cap C))$
$\displaystyle=\displaystyle\int_{\pi_{1}^{-1}(h)}\rho\
dH^{k+\ell-n}_{\mathcal{V}}$

for Lebesgue a.e. $h\in\mathcal{H}$.

To apply this formula, we need the following slightly more general form of the
co-area formula:

######  Proposition 4.3 (The co-area formula, $C^{1}$ case).

Let $f:M^{m}\to N^{n}$ be a $C^{1}$ map between Riemannian manifolds, $m\geq
n$. For $q\in N$, write $F_{q}=f^{-1}(q)$. Then for any Borel measurable
function $h$ on $M$ we have

$\displaystyle\int_{M}h\cdot\operatorname{NJ}(f)\ dM=\displaystyle\int_{q\in
N}\left(\displaystyle\int_{F_{q}}h\ dH^{m-n}_{N}\right)dN,$

with both sides finite if one is.

We can now proceed as before. First, apply Proposition 4.3 twice to deduce

$\displaystyle\displaystyle\int_{h\in\mathcal{H}}H^{k+\ell-n}(h(V)\cap W)\
d\mathcal{H}$
$\displaystyle=\displaystyle\int_{h\in\mathcal{H}}\left(\displaystyle\int_{\pi_{1}^{-1}(h)}\rho\
dH^{k+\ell-n}_{\mathcal{V}}\right)\ d\mathcal{H}$
$\displaystyle=\displaystyle\int_{\mathcal{V}}\rho\cdot\operatorname{NJ}(\pi_{1})\
d\mathcal{V}$ $\displaystyle=\displaystyle\int_{(p,q)\in V\times
W}\left(\displaystyle\int_{\mathcal{V}_{p,q}}\dfrac{\rho\cdot\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}\
d\mathcal{V}_{p,q}\right)dVdW,$

where $\mathcal{V}_{p,q}=\pi_{2}^{-1}(p,q)$ as before. The proof now reduces
to showing that the integrand
$\frac{\rho\cdot\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}$ can
be viewed as a continuous function on the appropriate Grassmannian bundle
$\mathcal{G}_{k,\ell}$ over $\widetilde{\mathcal{V}}$, which is positive
somewhere on each fiber of the submersion
$\accentset{\circ}{\mathcal{G}}_{k,\ell}\to\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{\ell}(TM).$

The space
$\mathcal{G}_{k,\ell}\subset\mathcal{H}\times\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{\ell}(TM)$,
the bundle $E_{k,\ell}$, and the functions $\eta_{1},\eta_{2}$, and $\eta$ are
defined exactly as in §2.4 (so e.g. in our original notation,
$\mathcal{G}_{k}=\mathcal{G}_{k,n-k}$). Again,
$\eta:\mathcal{G}_{k,\ell}\to{\mathbb{R}}_{\geq 0}$ is continuous, and
positive at $(h,\sigma_{p},\sigma_{q})$ iff
$dh(\sigma_{p})+\sigma_{q}=T_{q}M$. To extend $\rho$ to a function on
$\mathcal{G}_{k,\ell}$, simply define

$\displaystyle\Pi_{3}(h,\sigma_{p},\sigma_{q}):E_{k,\ell}(h,\sigma_{p},\sigma_{q})$
$\displaystyle\longrightarrow T_{q}M$ $\displaystyle(\dot{h},v,w)$
$\displaystyle\longmapsto w,$

and $\widehat{\rho}:\mathcal{G}_{k,\ell}\to{\mathbb{R}}_{\geq 0}$ by

$\widehat{\rho}(h,\sigma_{p},\sigma_{q})=\left\lVert\textstyle\bigwedge^{k+\ell-n}\Pi_{3}(h,\sigma_{p},\sigma_{q})\big{|}_{\ker(\Pi_{1}(h,\sigma_{p},\sigma_{q}))}\right\rVert.$

Clearly we have

$\widehat{\rho}(h,T_{p}V,T_{q}W)=\rho(h,p,q).$

Moreover, $\widehat{\rho}$ is continuous at all points where $\Pi_{1}$ has
full rank, that is, away from the zeros of $\eta$. Since also
$0<\widehat{\rho}\leq 1$, it follows that $\widehat{\rho}\cdot\eta$ is
continuous on $\mathcal{G}_{k,\ell}$, with the same zero set as $\eta$, and
satisfies

$(\widehat{\rho}\cdot\eta)(h,T_{p}V,T_{q}W)=\left(\rho\cdot\dfrac{NJ(\pi_{1})}{NJ(\pi_{2})}\right)(h,p,q).$

One can now apply the results of Appendix A as before to complete the proof.

##  5\. Construction of $(\mathcal{H},\psi)$

In this section we show that Theorems 1.1 and 4.1 are non-vacuous, by
constructing, for an arbitrary closed manifold $M$, a compact $C^{1}$ family
of diffeomorphisms of $M$ satisfying hypotheses (A1) and (A2). This proves
Theorem 1.2.

Actually, since it is no more difficult (and arguably more natural) to work in
the smooth category here, the family we construct will be $C^{\infty}$. In
addition, we will be unconcerned with compactness until the end. Specifically,
we first obtain a $C^{\infty}$ family $({\mathbb{R}}^{N},\psi)$ of the form

$\psi:({\mathbb{R}}^{N},0)\xrightarrow[\ \ \
]{}(\operatorname{Diff}^{\infty}(M),\operatorname{id})$

which satisfies hypothesis (A1) and (A2), such that the surjectivity
requirements of (A2) are satisfied by
$({\mathbb{B}}^{N}_{R},\psi|_{{\mathbb{B}}^{N}_{R}})$ for all $R$ sufficiently
large.999Here and below we write
${\mathbb{B}}^{N}_{R},\overline{{\mathbb{B}}}^{N}_{R}\subset{\mathbb{R}}^{N}$
for the open and closed balls, respectively, centered at $0$ of radius $R$,
with the subscript occasionally omitted in the special case $R=1$, i.e.
${\mathbb{B}}^{N}:={\mathbb{B}}^{N}_{1}$. The desired compact family is then
obtained by restricting $\psi$ to $\overline{{\mathbb{B}}}^{N}_{R}$ for some
large $R$.

We begin with some general lemmas. Recall that for $1\leq r\leq\infty$, a
_$C^{r}$ family of diffeomorphisms of $M$_ is a pair $(\mathcal{H},\psi)$,
with $\mathcal{H}$ a smooth manifold, possibly with boundary, and
$\psi:\mathcal{H}\to\operatorname{Diff}^{r}(M)$ such that the associated
evaluation map $ev_{\psi}:\mathcal{H}\to M$ is $C^{r}$; this implies in
particular that $\psi$ is continuous. Any $C^{r}$ family of diffeomorphisms of
$M$ is, of course, a $C^{s}$ family of diffeomorphisms for all $s<r$.

One motivation for interpreting smoothness in this way—instead of via the
strictly stronger requirement that $\psi\in
C^{r}(\mathcal{H},\operatorname{Diff}^{r}(M))$ for the standard Banach/Fréchet
manifold structure on $\operatorname{Diff}^{r}(M)$—is the following elementary
fact, which shows that the class of $C^{r}$ families of diffeomorphisms in our
weaker sense is in a natural sense closed under composition.101010That this
fails for the stronger notion is a consequence of the well-known fact that the
composition map
$\left[\operatorname{Diff}^{r}(M)\right]^{2}\to\operatorname{Diff}^{r}(M)$ is
only $C^{0}$ when $r<\infty$.

######  Lemma 5.1.

Let $(\mathcal{H}_{1},\psi_{1})$ and $(\mathcal{H}_{2},\psi_{2})$ be $C^{r}$
families of diffeomorphisms of $M$, with
$\partial\mathcal{H}_{1}=\partial\mathcal{H}_{2}=\varnothing$. Define

$\psi:\mathcal{H}_{1}\times\mathcal{H}_{2}\xrightarrow[\ \ \
]{}\operatorname{Diff}^{r}(M)$

by $\psi(h_{1},h_{2})=\psi_{2}(h_{2})\circ\psi_{1}(h_{1})$, i.e.
$\psi=\operatorname{comp}\circ(\psi_{1}\times\psi_{2})$.111111For notational
convenience we adopt the convention which defines the composition map
$\left[\operatorname{Diff}^{r}(M)\right]^{k}\to\operatorname{Diff}^{r}(M)$ by
$\operatorname{comp}(f_{1},\dots,f_{k})=f_{k}\circ\cdots\circ f_{1}$. Then
$(\mathcal{H}_{1}\times\mathcal{H}_{2},\psi)$ is a $C^{r}$ family of
diffeomorphisms of $M$.

###### Proof.

It suffices to check that the evaluation map

$\displaystyle ev_{\psi}:(\mathcal{H}_{1}\times\mathcal{H}_{2})\times M$
$\displaystyle\xrightarrow[\ \ \ \ \ \ ]{}M$ $\displaystyle(h_{1},h_{2},p)$
$\displaystyle\xmapsto[\ \ \ \ \ \ ]{}\psi(h_{1},h_{2})(p)$

is $C^{r}$. We have

$\displaystyle ev_{\psi}(h_{1},h_{2},p)$ $\displaystyle=\psi(h_{1},h_{2})(p)$
$\displaystyle=\psi_{1}(h_{1})(\psi_{2}(h_{2})(p))$
$\displaystyle=ev_{\psi_{1}}(h_{1},\psi_{2}(h_{2})(p))=ev_{\psi_{1}}(h_{1},ev_{\psi_{2}}(h_{2},p)).$

Since $ev_{\psi_{1}}$ and $ev_{\psi_{2}}$ are both $C^{r}$ by assumption, so
is $ev_{\psi}$. ∎

This lemma means that we can assemble our $C^{\infty}$ family in the obvious
way, namely by composing a finite collection of smaller $C^{\infty}$ families
supported in (and defined via) charts. Moreover, working within charts it is
easy to produce families which _locally_ satisfy the desired properties (A1)
and (A2); see Proposition 5.3 below. The next lemma gives us a criterion for
ensuring that a composition of such families satisfies (A1) globally.

######  Lemma 5.2.

Let $(\mathcal{H}_{1},\psi_{1}),\dots,(\mathcal{H}_{k},\psi_{k})$ be $C^{r}$
families of diffeomorphisms of $M$, with $\partial\mathcal{H}_{i}=\varnothing$
for all $1\leq i\leq k$. Suppose that for all $i=1,\dots,k$,
$h_{i}\in\mathcal{H}_{i}$, and $p\in M$, we have

$\psi_{i}(h_{i})(p)\neq p\ \ \Longrightarrow\ \
d(ev_{\psi_{i},p})_{h_{i}}\text{ is surjective, }$

where
$ev_{\psi_{i},p}:=ev_{\psi_{i}}(\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},p):\mathcal{H}_{i}\to
M$. Finally, suppose that for all $p\in M$, there exists $j$ such that
$ev_{\psi_{j},p}$ is a submersion, i.e. such that $d(ev_{\psi_{j},p})_{h_{j}}$
is surjective for all $h_{j}\in\mathcal{H}_{j}$. Then letting
$\mathcal{H}:=\mathcal{H}_{1}\times\cdots\mathcal{H}_{k}$, the map

$\displaystyle\psi:\mathcal{H}$
$\displaystyle\longrightarrow\operatorname{Diff}^{r}(M)$
$\displaystyle(h_{1},\dots,h_{k})$
$\displaystyle\longmapsto\psi_{k}(h_{k})\circ\cdots\circ\psi_{1}(h_{1})$

defines a $C^{r}$ family of diffeomorphisms of $M$, and
$ev_{\psi,p}:=ev_{\psi}(\cdot,p):\mathcal{H}\to M$ is a submersion for all
$p\in M$.

###### Proof.

The fact that $(\mathcal{H},\psi)$ is a $C^{r}$ family of diffeomorphisms of
$M$ follows inductively from Lemma 5.1. Now let $p\in M$ and
$h=(h_{1},\dots,h_{k})\in\mathcal{H}$. We claim that

$d(ev_{\psi,p})_{h}:T_{h}\mathcal{H}\xrightarrow[]{\ \ \ \ }T_{q}M$

is surjective, where
$q=ev_{\psi,p}(h)=\psi(h)(p)=\psi_{k}(h_{k})\circ\cdots\circ\psi_{1}(h_{1})(p).$
Clearly it suffices to show that

$\partial_{h_{j}}(ev_{\psi,p})_{h}:T_{h_{j}}\mathcal{H}_{j}\xrightarrow[]{\ \
\ \ }T_{q}M$

is surjective for some $1\leq j\leq k$.

To compute these derivatives, write $p_{0}:=p$ and
$p_{i}=\psi_{i}(h_{i})\circ\cdots\circ\psi_{1}(h_{1})(p)$ for $i\geq 1$; in
particular $p_{k}=q$. Setting
$F_{j}=\psi_{k}(h_{k})\circ\cdots\circ\psi_{j+1}(h_{j+1})$, we then have

$\displaystyle\partial_{h_{j}}(ev_{\psi,p})_{h}=(dF_{j})_{p_{j}}\circ
d(ev_{\psi_{j},p_{j-1}})_{h_{j}}.$

Now set $\ell=\min\ \\{1\leq i\leq k\ |\ d(ev_{\psi_{i},p})_{h_{i}}\text{ is
surjective}\\}$. Then by our assumption we must have $\psi_{i}(h_{i})(p)=p$,
and thus inductively $p_{i}=p$, for all $i<\ell$. Applying the above formula
gives

$\displaystyle\partial_{h_{\ell}}(ev_{\psi,p})_{h}=(dF_{\ell})_{p_{\ell}}\circ
d(ev_{\psi_{\ell},p_{\ell-1}})_{h_{j}}=(dF_{\ell})_{p_{\ell}}\circ
d(ev_{\psi_{\ell},p})_{h_{\ell}}.$

By the definition of $\ell$, $d(ev_{\psi_{\ell},p})_{h_{\ell}}$ is surjective;
since $F_{\ell}$ is a diffeomorphism, it follows that
$\partial_{h_{\ell}}(ev_{\psi,p})_{h}:T_{h_{\ell}}\mathcal{H}_{\ell}\to
T_{p_{k}}M$ is surjective as well, so we are done. ∎

We can now proceed with the construction of $\mathcal{H}$. Our approach is the
“obvious” one, beginning with the observation that
$\text{Isom}^{+}({\mathbb{R}}^{n})\cong
SO(n,{\mathbb{R}})\ltimes{\mathbb{R}}^{n}$ acts smoothly and transitively on
$\operatorname{Gr}_{k}(T{\mathbb{R}}^{n})\cong\operatorname{Gr}_{k}({\mathbb{R}}^{n})\times{\mathbb{R}}^{n}$
for all $0\leq k\leq n-1$. In particular, for every
$\sigma\in\operatorname{Gr}_{k}(T{\mathbb{R}}^{n})$,
$ev_{\sigma}:SO(n,{\mathbb{R}})\times{\mathbb{R}}^{n}\to\operatorname{Gr}_{k}(T{\mathbb{R}}^{n})$
is a surjective submersion. To realize these properties inside an arbitrary
manifold $M$, we first “cut off” this action to a smooth family of (uniformly)
compactly supported diffeomorphisms of ${\mathbb{R}}^{n}$, which satisfies the
submersion property (A1) on the interior of its support, and “acts
transitively” on the restrictions of the Grassmann bundles to some slightly
smaller ball. Using charts this family can be be realized inside any
coordinate ball in $M$ by a locally supported family of diffeomorphisms. The
lemmas above then allow us to obtain the desired “action” of
${\mathbb{R}}^{N}$ on $M$ by composing sufficiently many such families.

###  5.1. Step 1: Local construction.

Choose $\beta\in C^{\infty}({\mathbb{R}}_{\geq 0},[0,1])$ such that

  * •

$\beta|_{[0,2]}\equiv 1$

  * •

$\beta|_{[3,+\infty)}\equiv 0$

  * •

$\beta^{\prime}(x)<0$ for all $x\in(2,3)$.

Define

$L:\mathfrak{so}_{n}\longrightarrow\operatorname{Diff}_{c}^{\infty}(\mathbb{R}^{n},0)$

by

$L(v)(p)=\exp(\beta(|p|)v)(p),$

where $\exp:\mathfrak{so}_{n}\to SO(n,{\mathbb{R}})$ denotes the Lie
exponential. Thus $L(v)$ preserves each sphere
$\mathbb{S}^{n-1}_{r}:=\\{x\in{\mathbb{R}}^{n}\ :\ |x|=r\\}$, and acts on
$\mathbb{S}^{n-1}_{r}$ by $\exp(\beta(r)v)\in SO(n,{\mathbb{R}})$. In
particular, $L(v)|_{{\mathbb{B}}^{n}_{2}}=\exp(v)$, and
$\operatorname{supp}(L(v))=\overline{{\mathbb{B}}}^{n}_{3}$ for all nonzero
$v\in\mathfrak{so}_{n}$.

Define smooth vector fields
$V_{1},\dots,V_{n}\in\Gamma^{\infty}_{c}({\mathbb{R}}^{n})=C^{\infty}_{c}({\mathbb{R}}^{n},{\mathbb{R}}^{n})$
by

$V_{i}(p)=\beta(|p|)\mathbf{e}_{i},\ 1\leq i\leq n.$

For each $i$ let $h_{i}^{t}$ denote the flow associated to $V_{i}$. By
composing these flows, we obtain a map

$\displaystyle\tau:{\mathbb{R}}^{n}$
$\displaystyle\longrightarrow\operatorname{Diff}_{c}^{\infty}({\mathbb{R}}^{n})$
$\displaystyle(t_{1},\dots,t_{n})$ $\displaystyle\longmapsto
h_{n}^{t_{n}}\circ\cdots\circ h_{1}^{t_{1}}.$

Finally, we define

$\psi:{\mathbb{R}}^{n}\times\mathfrak{so}_{n}\longrightarrow\operatorname{Diff}_{c}^{\infty}({\mathbb{R}}^{n})$

by

$\psi(t,v)=\tau(t)\circ L(v).$

######  Proposition 5.3.

The map

$\displaystyle
ev_{\psi}:({\mathbb{R}}^{n}\times\mathfrak{so}_{n})\times{\mathbb{R}}^{n}$
$\displaystyle\longrightarrow{\mathbb{R}}^{n}$ $\displaystyle(t,v,p)$
$\displaystyle\longmapsto\psi(t,v)(p)$

is $C^{\infty}$, and for all $p\in{\mathbb{B}}^{n}_{3}$, the map
$ev_{\psi,p}=ev_{\psi}(\cdot,p)$ is a submersion. Moreover, for all $0\leq
k\leq n-1$ and
$\sigma_{p},\sigma_{q}\in\operatorname{Gr}_{k}(T{\mathbb{B}}^{n})$, we have

$d(\psi(t,v))(\sigma_{p})=\sigma_{q}.$

for some $(t,v)\in{\mathbb{B}}^{n}_{2}\times B_{\mathfrak{so}_{n}}(0,R)$,
where $R>0$ is such that
$\exp(B_{\mathfrak{so}_{n}}(0,R))=SO(n,{\mathbb{R}})$.

###### Proof.

To see that $ev_{\psi}$ is smooth, first observe that smoothness of

$\displaystyle ev_{L}:\mathfrak{so}_{n}\times{\mathbb{R}}^{n}$
$\displaystyle\longrightarrow{\mathbb{R}}^{n}$ $\displaystyle(v,p)$
$\displaystyle\longmapsto L(v)(p)=\exp(\beta(|p|)v)(p)$

follows from that of the Lie exponential $\exp$ and $\beta$. Also, since each
$V_{i}$ is smooth,

$\displaystyle ev_{i}:{\mathbb{R}}\times{\mathbb{R}}^{n}$
$\displaystyle\longrightarrow{\mathbb{R}}^{n}$ $\displaystyle(t,p)$
$\displaystyle\longmapsto h_{i}^{t}(p)$

is smooth for all $1\leq i\leq n$. Smoothness of $ev_{\psi}$ then follows
inductively from Lemma 5.1.

Now let $p\in{\mathbb{B}}^{n}_{3}$ and
$(t,v)\in{\mathbb{R}}^{n}\times\mathfrak{so}_{n}$. Writing
$t=(t_{1},\dots,t_{n})$, we set $q_{0}=L(v)(p)$ and

$q_{j}=(h_{j}^{t_{j}}\circ\cdots\circ h_{1}^{t_{1}})(q)$

for $1\leq j\leq n$; in particular, $q_{n}=\psi(t,v)(p)$. Observe that since
every $L(v)$ and $h_{j}^{t_{j}}$ preserves ${\mathbb{B}}^{n}_{3}$,
$q_{i}\in{\mathbb{B}}^{n}_{3}$ for all $i$.

We would like to show that that

$d(ev_{\psi,p})_{(t,v)}:T_{t}{\mathbb{R}}^{n}\times
T_{v}\mathfrak{so}_{n}\longrightarrow T_{q_{n}}{\mathbb{R}}^{n}$

is surjective. Clearly it suffices to show that this is true for

$\partial_{t}(ev_{\psi,p})_{(t,v)}:T_{t}{\mathbb{R}}^{n}\longrightarrow
T_{q_{n}}{\mathbb{R}}^{n}.$

Fix $1\leq j\leq n$. We compute

$\displaystyle\partial_{t_{j}}(\psi(t,v)(p))=\partial_{t_{j}}(\tau(t)(q_{0}))$
$\displaystyle=\partial_{t_{j}}(h_{n}^{t_{n}}\circ\cdots\circ
h_{1}^{t_{1}}(q_{0}))$
$\displaystyle=\partial_{t_{j}}(h_{n}^{t_{n}}\circ\cdots\circ
h_{j}^{t_{j}}(q_{j-1}))$ $\displaystyle=d(h_{n}^{t_{n}}\circ\cdots\circ
h_{j+1}^{t_{j+1}})\left(\partial_{t_{j}}(h_{j}^{t_{j}}(q_{j-1}))\right)$
$\displaystyle=d(h_{n}^{t_{n}}\circ\cdots\circ
h_{j+1}^{t_{j+1}})\left(V_{j}(q_{j})\right)$
$\displaystyle=d(h_{n}^{t_{n}}\circ\cdots\circ
h_{j+1}^{t_{j+1}})\left(\beta(|q_{j}|)\mathbf{e}_{j}\right)$
$\displaystyle=\beta(|q_{j}|)\cdot d(h_{n}^{t_{n}}\circ\cdots\circ
h_{j+1}^{t_{j+1}})\left(\mathbf{e}_{j}\right)$
$\displaystyle=\beta(|q_{j}|)\left(\mathbf{e}_{j}+\alpha_{j+1,j}\mathbf{e}_{j+1}+\cdots+\alpha_{n,j}\mathbf{e}_{n}\right),$

for some $\alpha_{i,j}\in{\mathbb{R}}$. In particular,
$\partial_{t}(ev_{\psi,p})_{(t,v)}$ is lower triangular, with diagonal entries
$\beta(|q_{1}|),\dots,\beta(|q_{n}|)$. Since $|q_{j}|<3$ and thus
$\beta(|q_{j}|)>0$ for all $1\leq j\leq n$, it follows that
$\partial_{t}(ev_{\psi,p})_{(t,v)}$ is surjective.

For the final statement, let
$\sigma_{p},\sigma_{q}\in\operatorname{Gr}_{k}(T{\mathbb{B}}^{n})={\mathbb{B}}^{n}\times\operatorname{Gr}_{k}({\mathbb{R}}^{n})$,
say $\sigma_{p}=(p,\sigma_{1}),\sigma_{q}=(q,\sigma_{2})$. Pick $A\in
SO(n,{\mathbb{R}})$ such that $A\sigma_{1}=\sigma_{2}$ and $v\in
B_{\mathfrak{so}_{n}}(0,R)$ such that $\exp(v)=A$. Since
$L(v)|_{{\mathbb{B}}^{n}}=A$ we have

$d(L(v))(\sigma_{p})=(Ap,A\sigma_{1})=(Ap,\sigma_{2}).$

Observe that, for all $1\leq j\leq n$, if $b$ and
$h_{j}^{t_{j}}(b)\in{\mathbb{B}}^{n}_{2}$, then $h_{j}^{t_{j}}:x\mapsto
x+t_{j}\mathbf{e}_{j}$ in a neighborhood of $b$, so that
$d(h_{j}^{t_{j}})_{b}=\operatorname{id}$. Writing $q=(q_{1},\dots,q_{n})$,
$r=(r_{1},\dots,r_{n})=Ap$, and
$t=(t_{1},\dots,t_{n})=q-r\in{\mathbb{B}}^{n}_{2}$, for all $0\leq j\leq n$ we
have

$h_{j}^{t_{j}}\circ\cdots\circ
h_{1}^{t_{1}}(r)=(q_{1},\dots,q_{j},r_{j+1},\dots,r_{n}).$

Thus $|h_{j}^{t_{j}}\circ\cdots\circ h_{1}^{t_{1}}(Ap)|\leq|q|+|r|<2$. By the
comment above, we inductively conclude that

$d(h_{n}^{t_{n}}\circ\cdots\circ h_{1}^{t_{1}})_{Ap}=\operatorname{id}.$

It follows that

$\displaystyle d(\psi(t,v))(\sigma_{p})$
$\displaystyle=d(h_{n}^{t_{n}}\circ\cdots\circ h_{1}^{t_{1}})\circ
d(L(v))(\sigma_{p})$ $\displaystyle=d(h_{n}^{t_{n}}\circ\cdots\circ
h_{1}^{t_{1}})(Ap,\sigma_{2})$ $\displaystyle=(q,\sigma_{2})=\sigma_{q},$

so we are done. ∎

###  5.2. Step 2: Patching together.

Choose a finite collection $\\{(U_{i},\phi_{i})\\}_{i=1}^{L}$ of smooth charts
$\phi_{i}:U_{i}\xrightarrow[]{\ \ \cong\ \ }{\mathbb{B}}^{n}_{4}$ for $M$ such
that

$M=\displaystyle\bigcup_{i=1}^{L}\phi_{i}^{-1}({\mathbb{B}}^{n}_{1}).$

Set $W_{i}=\phi_{i}^{-1}({\mathbb{B}}^{n}_{1})$, and let $D$ denote the
diameter of the Čech $1$-complex associated to the cover
$\\{W_{i}\\}_{i=1}^{L}$ of $M$.121212Explicitly, $D$ is the smallest number
such that for all $1\leq k,\ell\leq L$, there exist
$i_{0},i_{1},\dots,i_{D-1},i_{D}\in\\{1,\dots,L\\}$ such that $i_{0}=k$,
$i_{D}=\ell$, and $W_{i_{j}}\cap W_{i_{j+1}}\neq\varnothing$ for all $0\leq
j\leq D-1$. Since $M$ is connected clearly we have $D\leq L-1<\infty$.

Now for each $i=1,\dots,L$ we can define

$\psi_{i}:{\mathbb{R}}^{n}\times\mathfrak{so}_{n}\longrightarrow\operatorname{Diff}^{\infty}(M)$

by

$\psi_{i}(t,v)=\begin{cases}\phi_{i}^{-1}\circ\psi(t,v)\circ\phi_{i}&\text{on
}U_{i}\\\ \text{id}&\text{on }M\setminus U_{i}.\end{cases}$

Then define

$\widehat{\psi}:\left({\mathbb{R}}^{n}\times\mathfrak{so}_{n}\right)^{L}\longrightarrow\operatorname{Diff}^{\infty}(M)$

by
$\widehat{\psi}=\operatorname{comp}\circ(\psi_{1}\times\cdots\times\psi_{L})$,
i.e.

$\widehat{\psi}((t_{1},v_{1}),\dots,(t_{L},v_{L}))=\psi_{L}(t_{L},v_{L})\circ\cdots\circ\psi_{1}(t_{1},v_{1}).$

Finally, define

$\Psi:(({\mathbb{R}}^{n}\times\mathfrak{so}_{n})^{L})^{D+1}\to\operatorname{Diff}^{\infty}(M)$

by

$\Psi(q_{1},\dots,q_{D+1})=\widehat{\psi}(q_{D+1})\circ\cdots\circ\widehat{\psi}(q_{1}).$

For convenience, let $N=\left(2n+{n\choose 2}\right)(D+1)L$, and fix some
isomorphism
$(({\mathbb{R}}^{2n}\times\mathfrak{so}_{n})^{L})^{D+1}\cong{\mathbb{R}}^{N}$.
Then we have

######  Proposition 5.4.

The evaluation map

$ev_{\Psi}:{\mathbb{R}}^{N}\times M\longrightarrow M\times M$

associated to $\Psi$ is a $C^{\infty}$ submersion. Moreover, if for all $0\leq
k\leq n$ we define

$\displaystyle\widehat{ev}^{k}_{\Psi}:{\mathbb{R}}^{n}\times\operatorname{Gr}_{k}(TM)$
$\displaystyle\longrightarrow\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{k}(TM)$
$\displaystyle(w,\sigma)$
$\displaystyle\longmapsto(\sigma,d(\Psi(w))(\sigma)),$

then
$\widehat{ev}^{k}_{\Psi}\big{|}_{{\mathbb{B}}^{N}_{R}\times\operatorname{Gr}_{k}(TM)}$
is surjective for all $R$ sufficiently large.

###### Proof.

The fact that $ev_{\Psi}$ is a smooth submersion follows immediately from
Lemma 5.2 and Proposition 5.3. The surjectivity of
$\widehat{ev}^{k}_{\Psi}\big{|}_{{\mathbb{B}}^{N}_{R}\times\operatorname{Gr}_{k}(TM)}$
for $R\gg 0$ follows from the second half of Proposition 5.3 and the
definition of $D$. ∎

######  Corollary 5.5.

The hypotheses of Theorem 1.1 are satisfied by the family
$(\overline{{\mathbb{B}}}^{N}_{R},\Psi)$.

##  6\. Appendix A: Submersions from Manifolds with Boundary

Our inclusion of the case $\partial\mathcal{H}\neq\varnothing$ necessitates a
discussion of some basic properties of submersions from compact Riemannian
manifolds with boundary. These maps are messier than submersions from compact
manifolds without boundary, essentially because their fibers can intersect the
boundary badly. Most dramatically, the fibers need not be manifolds (even with
boundary) in general. For similar reasons, submersions from manifolds with
boundary need not be open maps at boundary points, and (most relevant for our
purposes) the locally trivial fibration property enjoyed by proper submersions
between manifolds without boundary (known as Ehresmann’s theorem) fails
completely in this setting.131313All these pathologies are easily illustrated
by maps of the form $\pi:D\to{\mathbb{R}}$ where $D\subset{\mathbb{R}}^{2}$ is
a compact planar domain with smooth boundary curves and $\pi$ denotes (the
restriction of) projection onto some one-dimensional subspace. One can of
course also obtain examples with closed codomain by post-composing $\pi$ with
a smooth covering map ${\mathbb{R}}\to\mathbb{S}^{1}$.

Here we circumvent these issues rather naively, by restricting our maps to the
interiors of the manifolds in question. An obvious drawback of this approach
is the loss of compactness. Nevertheless, this trick turns out to be
sufficient for our purposes because the geometry of such maps is still tamer
than that of submersions from general open manifolds. The purpose of this
section is to state and prove the needed constraints.

Let $f:M^{m}\to N^{n}$ be a $C^{1}$ map between smooth manifolds without
boundary. Recall that a _box chart_ (or _submersion chart_) adapted to $f$ on
$M$ is a triple $(B,\phi,\psi)$ consisting of an open set $B\subset M$ and a
pair of $C^{1}$ diffeomorphisms $\phi:B\to(-r,r)^{m}$ and
$\psi:f(B)\to(-r,r)^{n}$ such that $\psi\circ
f\circ\phi^{-1}:(-r,r)^{m}\to(-r,r)^{n}$ is given by projection onto the first
$n$ coordinates. For convenience we will use the following nominal refinement
of this notion:

######  Definition 6.1.

We say that a box chart $(B,\phi_{B},\psi_{B})$ is _nice_ if $B$ is
precompact, and there is another box chart
$(B^{\prime},\phi_{B^{\prime}},\psi_{B^{\prime}})$ such that
$\overline{B}\subset B^{\prime}$, $\phi_{B}=\phi_{B^{\prime}}|_{B}$, and
$\psi_{B}=\psi_{B^{\prime}}|_{f(B)}$. In this case we say that the box chart
$B^{\prime}$ _cocoons_ $B$.

The fundamental fact about submersions from manifolds without boundary is that
their domains can be covered by box charts. Observe that nothing is lost in
this statement by restricting our attention to nice box charts, since for
$f:M\to N$ a smooth map with $\partial M=\varnothing$ it is immediate that

$\displaystyle f\text{ is a submersion }$ $\displaystyle\Longleftrightarrow
f\text{ can be covered by box charts }$ $\displaystyle\Longleftrightarrow
f\text{ can be covered by nice box charts.}$

On the other hand, we have the following straightforward fact:

######  Lemma 6.2.

Let $f:M^{m}\to N^{n}$ be a $C^{1}$ submersion between smooth Riemannian
manifolds without boundary and $(B,\phi_{B},\psi_{B})$ a nice box chart
adapted to $f$. Then there are constants $0<C_{L}^{B}<C_{U}^{B}$ such that

$C_{L}^{B}\leq\operatorname{vol}_{m-n}(f^{-1}(q)\cap B)\leq C_{U}^{B}$

for all $q\in f(B)$.

###### Proof.

Such an inequality holds trivially for the standard projection
$\pi_{n}:{\mathbb{R}}^{m}\to{\mathbb{R}}^{n}$ and any (precompact) box
$(-r,r)^{m}\subset{\mathbb{R}}^{m}$, with $C_{L}=C_{U}=r^{m-n}$. For the
general case, let $B$ be a nice box chart for $f$ and
$(B^{\prime},\phi_{B^{\prime}},\psi_{B^{\prime}})$ a box chart cocooning $B$.
By definition $\overline{B}\subset B^{\prime}$ is compact and
$\phi_{B}^{\prime}$ is $C^{1}$; it follows that
$\phi_{B^{\prime}}\big{|}_{B}:B\hookrightarrow{\mathbb{R}}^{m}$ distorts
Riemannian metrics by a uniformly bounded amount, i.e. for some $C>1$ we have
$C^{-1}g\leq(\phi_{B^{\prime}})^{*}\eta\leq Cg$ on $B$, where $g$ denotes the
metric on $M$ and $\eta$ the (Euclidean) metric on ${\mathbb{R}}^{m}$.
Moreover, for all $q\in f(B)$ we have

$f^{-1}(q)\cap
B=\phi_{B^{\prime}}^{-1}\big{(}\pi_{n}^{-1}(\psi_{B^{\prime}}(q))\cap\phi_{B^{\prime}}(B)\big{)}.$

The general result then follows trivially from the Euclidean case. ∎

######  Lemma 6.3.

Let $f:M^{m}\to N^{n}$ be a $C^{1}$ submersion between compact Riemannian
manifolds, with $\partial M\neq\varnothing$ and $\partial N=\varnothing$. Then
$\operatorname{vol}_{m-n}(f^{-1}(q)\cap\accentset{\circ}{M})$ is finite for
all $q\in N$, and in fact uniformly bounded above. If
$f\big{|}_{\accentset{\circ}{M}}$ is also surjective, then
$\operatorname{vol}_{m-n}(f^{-1}(q)\cap\accentset{\circ}{M})$ is bounded away
from zero.

###### Proof.

Let
$\mathcal{F}_{q}:=f^{-1}(q)\cap\accentset{\circ}{M}=(f|_{\accentset{\circ}{M}})^{-1}(q)$,
and note that, since $\accentset{\circ}{M}$ is a manifold without boundary and
$f\big{|}_{\accentset{\circ}{M}}$ is a submersion, each $\mathcal{F}_{q}$ is
either empty or an embedded $(m-n)$-dimensional submanifold of
$\accentset{\circ}{M}$. Now define

$\nu(q):=\operatorname{vol}_{m-n}(\mathcal{F}_{q})=\displaystyle\int_{\mathcal{F}_{q}}d\mathcal{F}_{q}.$

A priori we have $\nu:N\to[0,+\infty]$. We first show that in fact
$\sup\nu<+\infty$.

Upper bound: By attaching a collar $C_{\epsilon}\cong\partial
M\times[0,\epsilon)$ to $M$ and extending the metric, we obtain an open
Riemannian manifold without boundary $\widehat{M}$ containing $M$ as an
isometrically embedded compact submanifold. Choose an extension of $f$ to a
$C^{1}$ map $F:\widehat{M}\to N$; since the set $\\{p\in\widehat{M}\ :\
dF_{p}\text{ is surjective}\\}$ is an open set containing $M$, by shrinking
the collar width $\epsilon$ we may assume that $F$ is a submersion as well.

Now pick a cover $\mathcal{B}$ of $\widehat{M}$ by nice box charts adapted to
$F$; by compactness of $M$ we can find a finite subset
$\\{B_{1},\dots,B_{k}\\}\subset\mathcal{B}$ such that $M\subset
B_{1}\cup\cdots\cup B_{k}$. By Lemma 6.2 there are $C_{U}^{i}<+\infty$, $1\leq
i\leq k$, such that $\operatorname{vol}(F^{-1}(q)\cap B_{i})<C_{U}^{i}$ for
all $q\in N$. Since clearly every fiber $F^{-1}(q)$ intersects each $B_{j}$ in
at most one plaque, for all $q$ we have

$\displaystyle\nu(q)=\operatorname{vol}(f^{-1}(q)\cap\accentset{\circ}{M})$
$\displaystyle\leq\operatorname{vol}\left(F^{-1}(q)\cap\left(\displaystyle\bigcup_{j=1}^{k}B_{j}\right)\right)$
$\displaystyle\leq\displaystyle\sum_{j=1}^{k}\operatorname{vol}\left(F^{-1}(q)\cap
B_{j}\right)$ $\displaystyle\leq k\cdot\displaystyle\max_{1\leq j\leq
k}C_{U}^{j}<+\infty.$

Lower bound: It remains to show that $\inf\nu>0$ if
$\accentset{\circ}{f}:=f|_{\accentset{\circ}{M}}$ is surjective. This
assumption implies that for all $q\in N$, there is a nice box chart
$E_{q}\subset\accentset{\circ}{M}$ adapted to $\accentset{\circ}{f}$ such that
$E_{q}\cap\accentset{\circ}{f}^{-1}(q)\neq\varnothing$. For each such $E_{q}$,
$\accentset{\circ}{f}(E_{q})=f(E_{q})\subset N$ is an open neighborhood of
$q$, and

$N=\displaystyle\bigcup_{q\in N}f(E_{q}).$

Since $N$ is compact we may extract a finite subcover
$\\{f(E_{1}),\dots,f(E_{\ell})\\}$; in particular, every fiber
$\mathcal{F}_{q}=\accentset{\circ}{f}^{-1}(q)$ intersects and thus contains a
fiber plaque in some $E_{j}$, $1\leq j\leq\ell$. On the other hand, the lower
bound in Lemma 6.2 implies that every plaque in any $E_{j}$ has volume at
least $C_{L}:=\min\\{C_{L}^{E_{1}},\dots,C_{L}^{E_{\ell}}\\}>0$, so

$\nu(q)=\operatorname{vol}(\accentset{\circ}{f}^{-1}(q))\geq C_{L}>0$

for all $q\in N$, as desired. ∎

######  Lemma 6.4.

Let $f,M,N$ be as in Lemma 6.3. Suppose $\phi\in C^{0}(M,{\mathbb{R}}_{\geq
0})$ is such that for all $q\in N$, we have $\phi(p)>0$ for some
$p\in\mathcal{F}_{q}=f^{-1}(q)\cap\accentset{\circ}{M}$. Then there exists
$C_{\phi}>1$ such that

$\dfrac{1}{C_{\phi}}\leq\displaystyle\int_{\mathcal{F}_{q}}\phi\
d\mathcal{F}_{q}\leq C_{\phi}$

for all $q\in N$.

###### Proof.

Let $C_{1}=\displaystyle\sup_{p\in M}\phi<+\infty$. Then for all $q\in N$ we
have

$\mathcal{I}_{\phi}(q):=\displaystyle\int_{\mathcal{F}_{q}}\phi\
d\mathcal{F}_{q}\leq C_{1}\operatorname{vol}(\mathcal{F}_{q})\leq CC_{1},$

where $C$ is the upper bound on fiber volumes guaranteed by Lemma 6.3.

Thus it remains only to show that $\mathcal{I}_{\phi}(q)$ is uniformly bounded
away from $0$. This follows from an easy adaptation of the proof of the fiber
volume lower bound. Consider the set $\mathcal{B}$ of all nice box charts
$B_{i}$ for $f|_{\accentset{\circ}{M}}$ such that

$\displaystyle\inf_{p\in B_{i}}\phi(p)>0.$

Since $\phi$ is continuous and positive somewhere on each fiber of
$f|_{\accentset{\circ}{M}}$, the image of $\mathcal{B}$ under $f$ forms an
open cover of $N$. Extract a finite subcover
$\\{f(B_{1}),\dots,f(B_{k})\\}\subset f(\mathcal{B})$. Then for some
$\delta>0$ we have $\phi(p)>\delta$ for all $p\in B_{1}\cup\dots\cup B_{k}$,
and by Lemma 6.2 there exists $C_{2}>0$ such that the volumes of the plaques
of each $B_{i}$ are uniformly bounded below by $C_{2}$. Since
$\\{f(B_{i})\\}_{i=1}^{k}$ covers $N$, each $\mathcal{F}_{q}$ contains at
least one full plaque in some $B_{i}$; thus

$\displaystyle\int_{\mathcal{F}_{q}}\phi\ d\mathcal{F}_{q}\geq\delta C_{2}$

for all $q\in N$. Setting

$C_{\phi}=\max\left\\{CC_{1},\dfrac{1}{\delta C_{2}}\right\\}$

we are done. ∎

##  7\. Appendix B: Normal Jacobians formula

Recall that our proof of Theorem 1.1 hinges on the existence of uniform bounds
for the fiber integrals

$\displaystyle\int_{\mathcal{V}_{p,q}}\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}\
d\mathcal{V}_{p,q}.$

The key point is that these integrals can be identified with fiber integrals
of a _universally defined_ function $\eta$ on a slightly larger compact
manifold with boundary which submerses onto
$\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$. By the results
of Appendix A, it then suffices to show that $\eta$ is (i) continuous, and
(ii) positive somewhere on the interior of each fiber.

There are two main benefits to this approach. First, by using only what is
required to apply Lemma 6.4, it avoids obscuring the logic of the argument.
Second, and largely as a consequence, it generalizes easily to the case when
$\dim(V)+\dim(W)>M$.

The downside is that the constant $C$ remains somewhat opaque. In particular,
this method gives no further analysis of the function $\eta$, on which $C$
depends, or equivalently of the function(s)

$\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}:\mathcal{V}\xrightarrow[\
\ \ ]{}{\mathbb{R}}_{\geq 0}.$

This is unsatisfying, especially given the fairly straightforward geometric
intuition motivating the bounds. In addition, the mere existence of some
constant is clearly insufficient for applications where more precise control
over the constant is needed.

In the complementary dimensional case, a more transparent expression for
$\frac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}$ is available,
which in particular allows us to decompose $C$ into more tractable factors:

######  Proposition 7.1.

For $(\mathcal{H},\psi)$ satisfying the hypotheses of Theorem 1.1 and any
complementary dimensional submanifolds $V,W\subset M$, we have

$\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}(h,p,q)=\dfrac{|\det(J_{(h,T_{p}V,T_{q}W)})|}{\operatorname{NJ}(d(ev_{p})_{h})}$

for all $(h,p,q)\in\mathcal{V}$. Here $J$ is defined as in Claim 2.1, and
$ev_{p}$ as usual denotes the map

$ev_{p}:=ev_{\psi}(\cdot,p):\mathcal{H}\longrightarrow M$

given by $ev_{p}(h)=h(p)$.

The purpose of this section is to prove this statement. We begin with

######  Lemma 7.2.

Let $V$, $W$ be finite dimensional inner product spaces and suppose $S\subset
V\times W$ is a linear subspace such that $\pi_{1}|_{S}$ is an isomorphism.
Then if

$G:=\pi_{2}\circ(\pi_{1}|_{S})^{-1}:V\to W,$

we have $S=\Gamma(G)$. If in addition $\pi_{2}|_{S}$ is surjective and $S$ is
endowed with the inner product inherited from the product structure on
$V\times W$, then

(7)
$\det(GG^{*})^{-1/2}=\dfrac{|\det(\pi_{1}|_{S})|}{\det((\pi_{2}|_{S})\cdot(\pi_{2}|_{S})^{*})^{1/2}}.$

###### Proof.

The first statement is obvious. The second follows from some elementary linear
algebra computations; see [BCSS98, pp. 241-243] for details. ∎

######  Remark.

In terms of normal Jacobians, this result specifies a context in which the
innocent-looking formula

(8)
$\dfrac{1}{\operatorname{NJ}(AB^{-1})}=\dfrac{\operatorname{NJ}(B)}{\operatorname{NJ}(A)}$

holds. Note however that this formula fails in general: normal Jacobians of
general (not necessarily invertible) linear maps between inner product spaces
are not typically multiplicative under composition. In particular, the fact
that (8) holds in the setting of Lemma 7.2 depends on the assumption that the
metric on $V\times W$ (and thus the metric on $S$) is the product metric
induced by the metrics on $V$ and $W$.

_Proof of Proposition 7.1._ As in the discussion following Claim 2.1, observe
that

$\operatorname{NJ}(\pi_{1})(h,p,q)=|\det((d\pi_{1})_{(h,p,q)})|=0\ \
\Longleftrightarrow\ \ \det(J_{(h,T_{p}V,T_{q}W)})=0.$

Thus it suffices to consider the case when both $d\pi_{1}$ and $J$ are
invertible. Let $(h,p,q)$ be a regular point for $\pi_{1}$. Then we can define

$G\ :=\
(d\pi_{2})_{(h,p,q)}\circ(d\pi_{1})_{(h,p,q)}^{-1}:T_{h}\mathcal{H}\longrightarrow
T_{p}V\times T_{q}W.$

Since $\Gamma(G)=T_{(h,p,q)}\mathcal{V},$ with metric induced by the product
metric on $T_{h}\mathcal{H}\times T_{p}V\times T_{q}W,$ Lemma 7.2 applies to
give

(9)
$\det(GG^{*})^{-1/2}=\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}(h,p,q).$

Moreover, the description of $T_{(h,p,q)}\mathcal{V}$ given in Claim 2.1
immediately implies that the following diagram commutes:

${T_{(h,p,q)}\mathcal{V}}$${T_{h}\mathcal{H}}$${T_{p}V\times
T_{q}W}$${T_{q}M}$$\scriptstyle{d\pi_{1}\ \ \
}$$\scriptstyle{d\pi_{2}}$$\scriptstyle{d(ev_{p})_{h}\ \ \ \ \
}$$\scriptstyle{G}$$\scriptstyle{\ \ \ \ \ \ \ \ \ \ \ \ \
J_{(h,T_{p}V,T_{q}W)}}$

In particular,

$G=J_{(h,T_{p}V,T_{q}W)}^{-1}\circ d(ev_{p})_{h},$

so by (9) we compute

$\displaystyle\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}(h,p,q)$
$\displaystyle=\det(GG^{*})^{-1/2}$ $\displaystyle=\det((J^{-1}\circ
d(ev_{p})_{h})\cdot(J^{-1}\circ d(ev_{p})_{h})^{*})^{-1/2}$ (10)
$\displaystyle=\det(J^{-1}\cdot d(ev_{p})_{h}\cdot
d(ev_{p})_{h}^{*}\cdot(J^{-1})^{*})^{-1/2}$
$\displaystyle=\left[\det(J^{-1})\cdot\det(d(ev_{p})_{h}\cdot
d(ev_{p})_{h}^{*})\cdot\det((J^{-1})^{*})\right]^{-1/2}$
$\displaystyle=\left[\det(J^{-1})^{2}\cdot\det(d(ev_{p})_{h})\cdot
d(ev_{p})_{h}^{*})\right]^{-1/2}$
$\displaystyle=\dfrac{|\det(J)|}{\det(d(ev_{p})_{h})\cdot
d(ev_{p})_{h}^{*})^{1/2}}.\qed$

######  Remark.

Up to a uniformly bounded factor, we have

$|\det\left(J_{(h,\sigma_{p},\sigma_{q})}\right)|\approx\sin\measuredangle(dh(\sigma_{p}),\sigma_{q}).$

The multiplicative error involved is at most $\eta^{k}$, where $\eta$ is the
supremum of $\max\left\\{\left\lVert dh_{q}\right\rVert,\left\lVert
dh_{q}^{-1}\right\rVert\right\\}$ over all $h\in\mathcal{H}$ and $q\in M$.

Similarly, the fact that $ev$ is a $C^{1}$ submersion implies that
$\det(d(ev_{p})_{h})\cdot d(ev_{p})_{h}^{*})^{1/2}$ is uniformly bounded above
and away from $0$, i.e. constant up to a uniform factor.

Combining both errors we conclude that

$\dfrac{\operatorname{NJ}(\pi_{1})}{\operatorname{NJ}(\pi_{2})}(h,p,q)\approx\sin\measuredangle(dh(\sigma_{p}),\sigma_{q})$

up to a bounded multiplicative constant (which is itself a function of the
constants bounding $\operatorname{NJ}(ev_{p})_{h},\left\lVert
dh_{p}\right\rVert$, and $\left\lVert dh_{p}^{-1}\right\rVert$ for all
$(h,p)\in\mathcal{H}\times M$).

This means that one can view Theorem 1.1 as reducing to the following claim:

######  Proposition 7.3.

For all $0\leq k\leq n$ there is a constant $C_{k}>1$ such that for all
$(\sigma_{p},\sigma_{q})\in\operatorname{Gr}_{k}(TM)\times\operatorname{Gr}_{n-k}(TM)$,

$\dfrac{1}{C_{k}}\leq\displaystyle\int_{h\in\mathcal{H}_{p,q}}\sin\measuredangle(dh(\sigma_{p}),\sigma_{q})\
|d\mathcal{H}_{p,q}|\leq C_{k}.$

Here $\mathcal{H}_{p,q}=\\{h\in\accentset{\circ}{\mathcal{H}}:h(p)=q\\}$; this
is a submanifold of $\accentset{\circ}{\mathcal{H}}$, with Riemannian metric
induced from $\mathcal{H}$, which can be canonically (and isometrically)
identified with $\mathcal{V}_{p,q}$ (indeed
$\mathcal{V}_{p,q}=\mathcal{H}_{p,q}\times\\{p\\}\times\\{q\\}$).

Using the results above, our original constant $C$ can be estimated as a
function of the constants $C_{k}$ and the bounds on
$\operatorname{NJ}(ev_{p})_{h},\left\lVert dh_{p}\right\rVert$, and
$\left\lVert dh_{p}^{-1}\right\rVert$.

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