Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary ()
1. Introduction
This paper is the first in a series that investigates the Morse Theory and gradient flows on smooth compact manifolds with boundary, a special case of the well-developed Morse theory on stratified spaces (see [2] -[4] ). For us, however, the starting point and the source of inspiration is the 1929 paper of Morse [1] .
We intend to present to the reader a version of the Morse Theory in which the critical points remain behind the scene, while shaping the geometry of the boundary! Some of the concepts that animate our approach can be found in [5] , where they are adopted to the special environment a 3 D-gradient flows. These notions include stratified convexity or concavity of traversing flows in connection to the boundary of the manifold. That concavity serves as a measure of intrinsic complexity of a given manifold
with respect to any traversing flow. Both convexity and concavity have strong topological implications.
Another central theme that will make its first brief appearance in this paper is the holographic properties of traversing flows on manifolds with boundary. The ultimate aim here is to reconstruct (perhaps, only partially) the bulk of the manifold and the dynamics of the flow on it from some residual structures on the boundary, thus the name “holography”.
In Section 2, for so-called boundary generic fields
on
(see Definition 2.1), we explore the Morse stratification
of the boundary
(see Formula (1) and [1] ), induced by the vector field
on
.
In Section 3, we investigate the degrees of freedom to change this stratification by deforming a given vector field within the space of gradient-like fields (Theorem 3.2, Corollary 3.2, and Corollary 3.3).
In Section 4, for vector fields on compact manifolds, we introduce the pivotal notion of boundary
-convexity/
-concavity,
(see Definition 4.1). Then we explore some topological implications of the existence of a boundary
-convex/
-concave traversing field on
(see Lemma 4.2, Corollary 4.2, Corollary 4.3, and Corollary 4.4).
Let
denote the orthogonal projection of the field
on the bundle
tangent to the boundary. Occasionally, we can determine whether a given field
is convex/concave just by observing the behavior of the
-trajectories on the boundary
(Theorem 4.1, Theorem 4.2). We view the possibility of such determination as an instance of a more general phenomenon, which we call “holography”. This phenomenon will occupy us fully in a different paper.
The Eliashberg surgery theory of folding maps [6] [7] helps us to describe the patterns of Morse stratifications for traversing 3-concave and 3-convex fields (Theorem 5.1, Conjecture 5.1, and Corollary 5.1).
2. The Morse Stratification ![]()
Inspired by [1] , we start by introducing some basic notions and constructions that describe the way in which generic vector fields on a compact smooth manifold interact with its boundary.
Let
be a compact smooth
-dimensional manifold with a boundary
. Let v be a smooth vec- tor field on
which does not vanish on the boundary
. As a rule, we assume that X is properly contained in a
-dimensional manifold
and that the field v extends to a field
on
so that
. In fact, we always treat the pair
as a germ of a space and a field in the vicinity of the given pair
.
Often we will consider vector fields only with the isolated Morse-type singularities (zeros) located away from the boundary. This means that v, viewed as a section of the tangent bundle
, is transversal its zero section. In other words, in the vicinity of each singular point, there is a local system of coordinates
such that the field v can be represented as
, where all
.
To achieve some uniformity in our notations, let
and
.
The vector field v gives rise to a partition
of the boundary
into two sets: the locus
, where the field is directed inward of
, and
, where it is directed outwards. We assume that
, viewed as a section of the quotient line bundle
over
, is transversal to its zero section. This assumption implies that both sets
are compact manifolds which share a common boundary
. Evidently,
is the locus where
is tangent to the boundary
.
Morse has noticed that, for a generic vector field ν, the tangent locus
inherits a similar structure in con- nection to
, as
has in connection to
(see [1] ). That is,
gives rise to a partition
of
into two sets: the locus
, where the field is directed inward of
, and
, where it is directed outward of
. Again, let us assume that v, viewed as a section of the quotient line bundle
over
, is transversal to its zero section.
For generic fields, this structure replicates itself: the cuspidal locus
is defined as the locus where v is tangent to
;
is divided into two manifolds,
and
. In
, the field is directed inward of
, in
, outward of
. We can repeat this construction until we reach the zero-dimensional stratum
(see Figure 1 and Figure 2 which depict these strata for fields in
).
These considerations motivate
Definition 2.1 We say that a smooth field v on
is boundary generic if:
・
,
・
, viewed as a section of the tangent bundle
, is transversal to its zero section,
・ for each
, the
-generated stratum
is a smooth submanifold of
,
・ the field
, viewed as section of the quotient 1-bundle
,
is transversal to the zero section of
for all
.
We denote the space of smooth boundarygeneric vector fields on
by the symbol
. ![]()
Thus a boundary generic vector field
on
gives rise to two stratifications:
(1)
, the first one by closed submanifolds, the second one―by compact ones. Here
.
For simplicity, the notations “
” do not reflect the dependence of these strata on the vector field
. When the field varies, we use a more accurate notation “
”.
Remark 2.1. Replacing
with
affects the Morse stratification according to the formula:
![]()
where
when
, and
otherwise. ![]()
We will postpone the proof of the theorem below until the second paper in this series of articles (see [8] , Theorem 3.4, an extension of Theorem 2.1 below). There we will develop the needed analytical tools.
Theorem 2.1 Boundary generic vector fields form an open and dense subset
in the space
of all smooth fields on
. ![]()
Definition 2.2 We say that a smooth vector field
on
is of the gradient type (or gradient-like) for a smooth function
if:
・ the differential
and the field
vanish on the same locus
,
・ the function
in
,
・ in the vicinity of
, there exist a Rimannian metric
on
so that
, the gradient field of
in the metric
. ![]()
Definition 2.3 A smooth function
is called Morse function if its differential
, viewed as a section of the cotangent bundle
, is transversal to the zero section. ![]()
Recall that, for a Morse function
on a compact
-manifold X, the critical set ![]()
is finite, and each point
has special local coordinates
such that
, where
for all
(for example, see [9] ).
Definition 2.4 Let
be a smooth function and
its gradient-like vector field. We say that the pair
is boundary generic if the field
is boundary generic in the sense of Definition 2.1 and the res- trictions of
to each stratum
are Morse functions for all
. ![]()
Lemma 2.1 Let
be a compact smooth manifold, and
a smooth manifold which is stratified by sub- manifolds
. Let
be the space of smooth maps
which are transversal to each stratum
. Put
. Next consider the space
of pairs
such that ![]()
and
has the property:
are Morse functions for all
. Then
is open and dense in the
space
.
Proof. Consider the space
, where
denotes the cotangent bundle of
. The property
is equivalent to the property of the section
of the bundle
![]()
to be transversal to each (transversal) intersection of the
-graph
with each stratum
. The latter property defines a open set in
.
In order to validate density of
in
, we first perturb a given map
to make it transversal to each stratum
, and then perturb a given function
to make the section
of
transversal to each manifold
. ![]()
Theorem 2.2 The boundary generic1 Morse pairs
on a compact manifold
form an open and dense subset in the space of all smooth functions
and their gradient-like fields
.
Proof. By Theorem 2.1, the boundary generic fields
form an open and dense set in the space of all fields.
Let
be a complete flag in
, formed by subspaces
of codimension j. In the proof of Theorem 3.4 [8] , for every field
, we will construct a smooth map
such that
. Moreover,
is transversal to each
, if and only if,
is a boundary generic field. The construction of the map
utilizes high order Lie derivatives
of an auxiliary function
as in Lemma 3.1 [8] .
Now the property of boundary generic Morse pairs
to be open and dense in the space of all pairs follows from Lemma 2.1: just let
,
,
, and
in that lemma.
For the reader convenience, let us sketch now an alternative argument that establishes just the density of boundary generic Morse pairs
in the space of all pairs. It does not rely on the construction of the map
from [8] .
We start with a pair
where
and
at the points of the set where
. By a small perturbation of
, we can assume the
is a Morse function on
and
its gradient-like field.
Let
be a compact regular neighborhood of
in
so small that
. By Theorem 2.1, we can perturb
to a new field
so that
is boundary generic in the sense of Definition 2.1 and still
.
For a given
, the condition
defines an open cone in the space of all fields
, subject to the constraint
. Therefore
can be chosen both boundary generic and gradient-like for
. When
is fixed, so are the stratifications
.
Next, with
being fixed, we perturb
again to a new function
so that
and
are Morse functions for all j. The perturbation will be supported in the compact
. We start
constructing
inductively first from adjusting it on the 1-manifold
and then moving sequentially to the strata
with lower indices j. We pick each perturbation
so small that the open condition
is not violated. The existence of the desired
-th perturbation is based on the fact that Morse functions on a compact manifold
(in this case, on
) form an open and dense subset in
, the space of all smooth functions on
, being equipped with the Whitney topology. Note that since
is tangent
to
along
and
, the restriction
has no critical points in the vi-
cinity of
. Thus we need to perturb
only on a compact subset
which has an
empty intersection with
. This perturbation extends smoothly from
to
. Eventually, we reach the upper stratum
, thus constructing a boundary generic approximation of the given pair
.
All the changes
of
, but the first one, we have introduced so far are supported in
, where
and
. This proves that the boundary generic pairs form a dense set in the space of all pairs
, where
being a
-gradient-like field, subject to the constraints:
, and
being a Morse function. ![]()
For a given Morse pair
, we denote by
the set of critical points of the function
. For a boundary generic Morse pair
, the finite critical set
is divided into two complementary sets: the set
of positive critical points and the set
of negative ones (see Figure 3).
Remark 2.2. Note that when
, it may happen that
. However, if a component
of
is a closed manifold, then
must have local extrema, in which case
. ![]()
Consider a generic field
and a Riemannian metric
on
. We denote by
the orthogonal projection of the field
on the tangent space
. Note that if
is a gradient field for a function
in metric
, then
is automatically a gradient field for the restrictions
and
.
Take a smooth vector field
on a compact
-manifold
with isolated singularities
. We denote by
the localized index of
at its typical singular point
. In a
local chart,
is defined as the degree of a map
from a small
-centered
-sphere
to the unit
-sphere. The map takes each point
to the point
.
We define the “global” index
as the sum
.
For a generic field
and a Riemannian metric
on
, we form the fields
on
and define the global index of
by the formula:
![]()
Figure 3. Positive (the left diagram) and negative (the right diagram) singularities of
on the boundary of a solid
.
![]()
Let us revisit the beautiful Morse formulas [1] :
Theorem 2.3 (The Morse Law of Vector Fields) For a boundary generic vector field
and a Riemannian metric on a
-manifold
, such that the singularities of the fields
are isolated for all
, the following two equivalent sets of formulas hold:
![]()
(2)
where
stands for the Euler number of the appropriate space2. ![]()
For vector fields with symmetry, the Morse Law of Vector Fields has an equivariant generalization [10] . Here is its brief description: for a compact Lie group
acting on a compact manifold
, equipped with a ![]()
-equivariant field
, we prove that the invariants
can be interpreted as taking values in the
Burnside ring
of the group
(see [11] for the definitions). With this interpretation in place, the appearance of Formula (2) does not change.
Morse Formula (2) has an instant, but significant implication:
Corollary 2.1 Let
be a smooth neighborhood of the zero set of a vector field
on a compact
-manifold
. Assume that
is boundary generic with respect to both boundaries,
and
. Then
![]()
Remark 2.3. Therefore, the numbers
![]()
can serve as “more and less localized” definitions of the index invariant
. ![]()
An interesting discussion, connected to Theorem 2.3, its topological and geometrical implications, can be found in the paper of Gotlieb [12] . The “Topological Gauss-Bonnet Theorem” below is a sample of these re- sults.
Theorem 2.4 (Gotlieb) Let
be a compact smooth
-dimensional manifold and
a smooth map which is a immersion in the vicinity of the boundary
. Let
be a Riemannian metric on
which, in the vicinity of
, is the pull-back
of the Euclidean metric on
. Consider a generic linear function
such that the composite function
has only isolated singularities in the interior of
. Let
be the gradient field of
3. Assume that
is boundary generic.
Then the degree of the Gauss map
![]()
can be calculated either by integrating over
the normal curvature
(in the metric
) of the hyper- surface
, or in terms of the
-induced stratification
![]()
by the formula
(3)4
Example 2.1. Let X be an orientable surface of genus g with a single boundary component. Let
be an immersion, and let
,
and
be as in Theorem 2.4.
Since Ф is an immersion everywhere (and not only in the vicinity of ∂X as Theorem 2.4 presumes), we get that
. Thus
. Then Theorem 2.4 claims that the degree of the Gauss map
is equal to
![]()
Thus, the topological Gauss-Bonnet theorem, for immersions
, reduces to the equation
.
So the number of
-trajectories
in
that are tangent to
, but are not singletons (they correspond to points of
), as a function of genus
, grows at least as fast as
.
On the other hand, by the Whitney index formula [13] , the degree of
can be also calculated as
, where
denotes the number of positive/negative self-intersections of the curve
, and
.
By a theorem of L. Guth [14] , the total number of self-intersections
. Moreover, this lower bound is realized by an immersion
! Therefore, for any immersion
, the total number of self-intersections of the curve
can be estimated in terms of the boundary-tangent
-trajectories:
![]()
and for some special immersion
, we get
![]()
Corollary 2.2 Let
be a compact
-manifold with boundary, which is properly contained in an
open
-manifold
. Let
be a smooth map which is a immersion in the vicinity of the
boundary
. Let
be a Riemannian metric on
which, in the vicinity of
, is the pull-back
of the Euclidean metric on
.
Let
be a linear function, and
its composition with the map
. Form the gradient field
in
. Assume that the pair
is boundary generic in the sense of Definition 2.4.
For each
, consider a
-small tubular neighborhood
of the manifold
in
. Then
is an immersion. This setting gives rise to the Gauss map
, defined by the
formula
, where
and
is the unit vector inward normal to
at
.
Then the degree of the Gauss map
can be calculated either by integrating (with respect to the ![]()
-measure
) over
the normal curvature
of the hypersurface
, or in terms of the ![]()
-induced stratum
:
(4)
Proof. We will apply Theorem 2.4 to the field
in
to conclude that
![]()
Since
in
,
, and the last term of this equation reduces to
. ![]()
Remark 2.4. Of course, for an odd-dimensional
, the Euler number
, and so is
. When
is even-dimensional (i.e.,
), the integral in Equation (4) can be expressed in terms of intrinsic Riemannian geometry of the manifold
, namely, in terms of the Pfaffian
. The Pfafian is a
-differential form, whose construction utilizes the curvature tensor on the manifold (see [15] ). So, when
,
![]()
Given a boundary generic field
on
, we introduce a sequence of basic degree-type invariants ![]()
which are intimately linked, via the Morse Formula (2), to the invariants
.
We use a Riemannian metric
on
to produce the orthogonal projection
of the field
on the tangent subspace
.
Let
be the bundle of unit
-spheres associated with the tangent bundle of the manifold
. We denote by
the restriction of the bundle
to the subspace
.
For each k, consider two fields, the inward normal field
to
in
and
, as sections of the sphere bundle
(remember,
is tangent to
along
so that
along
!). Assume that the sections ν and
are transversal in the space
. This transversality can be achieved by a perturbation of
(equivalently, by a perturbation of the metric
), supported in the vicinity of the singularity locus
. Indeed, the intersections occur where the field
is positively proportional to
, that is, where
. The later locus is exactly the locus
. The perturbation does not affect the stratification
. Assuming the transversality of the intersection, the locus
is zero- dimensional.
We define the integer
as the algebraic intersection number of two
-cycles,
and
, in the ambient manifold
of dimension
.
Lemma 2.2 For a boundary generic field
on a Riemannian manifold
, the following formula holds:
![]()
Proof. We already have noticed that the intersection set
projects bijectively under the map
onto the locus
, where the component
of
vanishes and
points in- ward of
. It takes more work to see that the sign attached to the transversal intersection point
is
, where
is the index (the localized degree) of the field
in the vicinity of its singularity
. Thus
. By the Morse Formula (2), the claim of the lemma follows. ![]()
Corollary 2.3 The integer
depends only on the singular locus
of
and on the local indices of its points. ![]()
Question 2.1. How to compute
in the terms of Riemannian geometry and in the spirit of Theorem 2.4 and Corollary 2.2? ![]()
For a boundary generic field
and a fixed metric
on
, each manifold
comes equipped with a preferred normal framing
of the normal bundle
: just consider the unitary inward normal field
of
in
, then the unitary inward normal field
of
in
, being restricted to
, then the unitary inward normal field
of
in
, being restricted to
, and so on.
Via the Pontryagin construction [16] , this framing
generates a continuous map
. Its homotopy class
is an element of the cohomotopy set
. If
, then we define
to be the trivial map that takes
to the base point in
.
Unfortunately, as we will see soon,
! However, when
, each of the two loci
is a closed manifold. Then we can apply the Pontryagin construction only to, say,
to get a map
. This application leads directly to the following proposition.
Corollary 2.4 Consider a boundary generic vector field
such that
and a metric
, defined in the vicinity of
in
. Then these data give rise to continuous map
.
The homotopy class
is independent of the choice of
and a homotopy of ![]()
within the open subspace of
, defined by the constraint
.
In particular, when
, we get an element
![]()
and when
, an element
![]()
If
, we can interpret
also as an element of the homotopy group
. ![]()
The elements
and
have another classical interpretation as elements of oriented framed
cobordism set
. In fact, the pair
defines the trivial element in
. In
contrast, if
, then the bordism class
may be nontrivial.
Let us recall the definition of framed cobordisms (for example, see [17] ). Let
be oriented closed smooth
-dimensional submanifolds of a compact
-manifold
, whose normal bundles
and
are equipped with framings
and
, respectively.
We say that two pairs
and
define the same element in
, if there is a compact
-dimensional oriented submanifold
whose normal bundle
admits a framing
so that:
1)
,
2) the restriction of
to
coincides with
, and the restriction of
to
coincides with
.
Then the Pontryagin construction establishes a bijection
, where
. If
both sets admit a structure of abelian groups and the bijection
becomes a group isomorphism.
Now we are in position to explain why
. Consider the obvious embedding
![]()
We can isotop
in
to a regular embedding
![]()
such that:
1)
, and
2) the inward normal field
is parallel to the factor
in the product
.
Note that for
, all the normal fields
are preserved under the imbedding
. So,
for any
, the normal framing
of
in
extends to a normal framing ![]()
of
in
. Therefore
as an element of the framed bordisms of
. As a
result, when
, we get
in
(equivalently, in
).
3. Deforming the Morse Stratification
Let
be a smooth compact
-manifold with boundary
. A boundary generic field
(see Definition 2.1) gives rise to two stratifications (1).
We are going to investigate how the stratification
changes as a result of deforming the vector field
.
Lemma 3.1 Let
be a closed submanifold of a manifold
and
a closed manifold. Consider a family of maps
such that each
is transversal to
. All the manifolds, maps, and families of maps are assumed to be smooth.
Then all the submanifolds
are isotopic in
. In particular, the intersections
and
are diffeomorphic.
Proof. Let
be the map defined by the family
. Thanks to the transversality hypothesis,
is transversal to
and
is a submanifold of
whose boundary is
![]()
Let
be a vector field on
, normal to each codimension 1 submanifold
in
. In the construction of
, we evidently rely on the property of each
being transversal to
. Since
and
, each w-trajectory that originates at a point of
must reach
in finite time. Therefore, employing the w-flow,
is diffeomorphic to
, and the
-image of that product structure in
defines a smooth isotopy between
and
in
. This isotopy extends to an ambient isotopy of
itself [18] .
Note that these arguments fail in general if ether M or N have boundaries. However, under additional assumptions (such as
being t-independent and
), the relative versions of the lemma are valid. ![]()
Theorem 3.1 The diffeomorphism type of each stratum
is constant within each path-connected component of the space
of boundary generic fields.
Proof. If two generic fields,
and
, are connected by a continuous path
, then they can be connected by a path
such that the dependence of the field
on
is smooth. The argument is based on the property of generic fields to form an open set in the space of all fields (Theorem 2.1), the smooth partition of unity technique (which utilizes the compactness of manifold
), and the standard techniques of approximating continuos functions with the smooth ones.
Thus it suffices to consider a smooth 1-parameter family of vector fields
, connecting
to
. Since any generic field
, viewed as a section of the vector bundle
, is transversal its zero section, we may apply Lemma 3.1 (with
,
being the zero section of
,
, and
) to conclude that all the submanifolds
are isotopic in
.
Since each
divides
into a pair of complementary domains,
and
, and since their polarity ±is determined by the inward/outward direction of
, which changes continuously with
, the ambient isotopy of
(which takes
to
) must take
to
. The isotopy
extends to an isotopy
.
A similar argument applies to lower strata
. Indeed, with the isotopy
that takes
to
in place, consider the two sections,
and
, of the bundle
, both sections being transversal to the zero section of
. Applying again Lemma 3.1, we conclude that the loci
and
are isotopic in
(recall that these
loci are exactly the transversal intersections of two sections
and
of
with its zero section).
Again, an isotopy
that takes
to
must take
to
. The isotopy
extends to an isotopy
which preserves the pair
. So, the pairs
and
are diffeomorphic via the composite isotopy
.
This reasoning can be recycled to prove that all the pairs
and
are diffeomorphic via a single isotopy of
. This argument is carried explicitly in the proof of Theorem 3.4 from [8] . ![]()
Corollary 3.1 Let
be a
-dimensional compact smooth manifold with boundary.
Within each path-connected component of the space
of generic fields, the numbers
, as
well as the numbers
, are constant.
Proof. The claim follows instantly from Theorem 3.1 and Lemma 2.2. ![]()
For a manifold
with nonempty boundary, by deforming any given function
and its gradient-like field
, we can expel the isolated
-singularities from
. This can be achieved by the appropriate “finger moves” which originate at points of the boundary
and engulf the isolated singularities of
. The result of these manipulations leads to
Lemma 3.2 Any connected
-manifold
with a non-empty boundary admits a Morse function
with no critical points in the interior of
and such that
is a Morse function. Such functions form an open nonempty set in the space
of all smooth functions on
.
As a result, the gradient-like vector fields
on
form an open nonempty set in the space
of all vector fields on
.
Proof. Let us sketch the main idea of the argument. Start with a Morse function
. Connect each critical point in the interior of
by a smooth path to a point on the boundary in such a way that a system of non-intersecting paths is generated. Then delete from
small regular neighborhoods of those paths (“dig a system of dead-end tunnels”) and restrict
to the remaining portion
of
. Smoothen the entrances of the tunnels so that the boundary of
will be a smooth manifold which is diffeomorphic to
. We got a nonsingular function
on
. A slight perturbation of
on
will not introduce critical points in the interior of
and will deliver a Morse function on its boundary. Indeed, recall that the sets of Morse functions on
and
are open and dense in the spaces
and
of all smooth functions, respectively (for example, see [9] ). Of course
is an open condition imposed on a vector field on a compact manifold. On the other hand, if
, then any field
, sufficiently close to
, will have the property
. The previous arguments show that the set of gradient-like non-vanishing fields is nonempty. So it is an open nonempty subspace in the space
of all all vector fields on
. ![]()
Eliminating isolated critical points of a given function
on a manifold with boundary is not “a free lunch”: the elimination introduces new critical points of the restricted function
. This is a persistent theme throughout our program:
Expelling critical points of gradient flows from a manifold
leaves crucial residual geometry on its boundary.
This boundary-confined geometry allows for a reconstruction of the topology of
.
Ideas like these will be developed in the future papers from this series. Meanwhile, the following lemma gives a taste of things to come.
Lemma 3.3 Let
be a Morse function with no local extrema in the interior of a
-manifold
. Then an elimination by a finger move5 of each
-critical point
of the Morse index
results in the introduction of
new critical points of positive type and
new critical points of negative type for the modified function
.
Proof. Let
be a Morse singularity of
in the interior of
. Denote by
a sphere which bounds a
small disk
centered on
and such that
is a Morse function. Without loss of generality, we can
assume that, in the Morse coordinates
,
is given by
, while
with all the
being distinct. Then
has only Morse-type singularities at the points where the coordinate axes pierce the sphere
. With respect to the pair
, these points come in two flavors: positive and negative. The two types are separated by the hypersurface of the cone
![]()
In the vicinity of
, the intersection
is exactly the locus
![]()
so that the
-gradient field
(tangent to
along
) is transversal to
, the product of two spheres. Therefore, in the vicinity of
,
!
The function
has exactly
critical points of the positive type and exactly ![]()
critical points of the negative type. We shall denote these sets by
and the two domains in which
divides
―by
.
Let
be a local maximum of
. Note that it is possible to connect
to a non-singular (for
) point
by a smooth path
along which f is increasing. Indeed, any non-extendable path
such that
either approaches a critical point or reaches the boundary
. By a small perturbation, we can insure that
avoids all the (hyperbolic) critical points in the interior of X (by the hypothesis, f has no local maxima/minima in the interior of X). Thus
can be extended until it reaches the boundary
at a point
.
Drilling a narrow tunnel
, diffeomorphic to the product
, along
does not change the topology of X; the function
retains almost the same list of singularities at the boundary as the function
has: more accurately, the local maximum at
disappears in
and a negative critical point of index 1 of
appears near the
-end of the tunnel
. Thus we have modified f and have eliminated the critical point
in the interior of X at the cost of introducing on the boundary
critical points of positive type and
critical points of negative type. ![]()
Soon, motivated by Lemma 3.2, we will restrict our attention to nonsingular functions
and their gradient-like fields
―an open subset in the space of all gradient-like pairs
; but for now, let us investigate a more general case.
Consider Morse data
, where the field
is nonsingular along the boundary
. Extend
to
and
, where C is some external collar of
so that the extension
is nonsingular in C. At each point
, the
-flow defines a projection
of the germ of
into the germ of the hypersurface
.
Let
and
denote the pure strata
and
, respectively. At the points
,
is a surjection; at the points of
, it is a folding map; at the points
, it is a cuspidal map. Often we will refer to points
by the smooth types of their
-projections.
As the theorem and the corollary below testify, for a given function
, we enjoy a considerable freedom in changing the given Morse stratification
by deforming the
-gradient-like field
(cf. Section 3 in [5] ).
Theorem 3.2 Let
be a compact smooth
-manifold with nonempty boundary. Take a smooth function
with no singularities along
, and let
be its gradient-like field. Consider a stratification
![]()
of X by compact smooth manifolds
, and let
and
denote the critical sets of the restrictions
and
, respectively. Assume that the following properties are satisfied:
・
,
・
and
are regular embeddings for all
,
・ for each
the functions
and
have Morse-type critical points at the loci
and
, respectively,
・ at the points of
,
and, at the points of
,
, where
is the inward normal to
in
6.
Then, within the space of
-gradient-like fields, there is a deformation of
into a new boundary generic gradient-like field
, such that the stratification
, defined by
, coincides with the given
stratification
.
Proof. We pick a Riemannian metric g in a collar U of
in X so that
becomes the gradient field of f. Consider auxiliary vector fields
, where
denotes the orthogonal projection of
on the tangent spaces of closed manifold
.
The construction of the desired field
is inductive in nature, the induction being executed in increasing values of the index
. Figure 4 illustrates a typical inductive step.
Assume that
has been already constructed so that
and
for all
. This assumption implies that
is tangent to
exactly along its boundary
for all
. Along
(and thus along
), we decompose
as
, where
.
The idea is to modify
in the direction normal to
in
, while keeping the rest of its components
unchanged.
Denote by
the tangent space of
at
. Let
be the open half-space of
positively
spanned by the vectors that point inside of
. Let
be half of the tangent space
, defined by
, where
. We introduce the complementary to
and
open half-spaces
and
.
At each point
, consider the open cone
and, at each point
, the open cone
(see Figure 4). These cones are non-empty, except perhaps at the points of
, where
is anti-parallel to the inward normal
of
. However, at
,
, and at
,
due to the last bullet in the hypotheses of the theorem. Thus, for each
, there is a number h so that the vector
(this conclusion uses the the property
on the set
). Similarly, for each
, there is a number h so that
. By the partition of unity argument, which employes convexity of the cones
, there is a smooth function
which delivers the desired field uk along
. In order to insure the continuity of h and
across the boundary
, we require
. Thus
on
.
Put
. Now,
for all
(these strata depend on the
’s only),
and
by the construction of
. Moreover,
for all
. In fact,
is tangent to
along
. Note that this inductive argument should be modified for
since
is 0-dimensional.
We smoothly extend
into a regular neighborhood V of
in X. Abusing notations, we denote this extension by
as well. The neighborhood
is chosen so that there
.
To complete the proof of the inductive step
, we form the field
, where the functions
deliver a smooth partition of unity subordinate to the cover
of X. Since
defines a convex cone in the space of vector fields,
is a f-gradient-like field with the desired Morse stratification. ![]()
Theorem 3.2 has an immediate implication:
Corollary 3.2 Let
be a Morse function and
its boundary generic gradient-like field with the
Morse stratification
. Assume that compact codimension zero submanifolds
are chosen so that, for each
,
and
.
Then, within the space of f-gradient-like fields, it is possible to deform
into a new gradient-like boundary generic field
, such that the stratification
coincides with the given stratification
.
Moreover,
.
In particular, if
, the claim is valid for any stratification
as above that terminates with
. ![]()
The next proposition (based on Corollary 3.2) shows that, for a given Morse function
, by an appropriate choice of gradient-like field
, the Morse stratification
can be made topologically very simple and regular: namely, each stratum
is a disjoint union of
-dimensional disks. Moreover, when the boundary
is connected and
, each stratum
is a just a single disk.
Corollary 3.3 Let
be a Morse function on a compact
-manifold
,
being nonsingular along the boundary
. We divide the connected components
of the boundary into two types, A and B. By definition, for type A, the singularity set
, and for type B,
.
Then any f-gradient-like field
can be deformed, within the space of f-gradient-like fields, into a boundary generic field
so that, for each component
of type A and all
, the stratum
is diffeomorphic to a disk
. At the same time, for the components of type
and all
, the stratum
.
For the components of type
, in contrast, the 1-dimensional stratum
is a finite union of arcs residing in the circle
. Moreover,
, the number or arcs in
, and the number of points in
are linked via the formula
![]()
, where
is the index of the field
, and
is the number of boundary components of type
.
Proof. If
, for each type A connected component
of
, the singularity set
can be included in a disk
. By Corollary 3.2, we can deform
to a new
-gradient-like field
, so that the new stratum
. If
, then the singularity set
can be incapsulated in a disk
. By the same token, after still another deformation
of
, we can arrange for
and
. This process repeats itself, unless the dimension of
becomes one. At its final stage,
consists of several arcs which are contained in the circle
.
For each type
connected component
of
, by a similar reasoning, we can arrange for
. Thus,
for all
and
.
Therefore, letting
for all
in Corollary 3.2, we have established all the claims of the corollary, but the last one.
Since
and
both are the gradient-like fields for the same Morse function
, their indexes,
and
, are equal. Thus we get
![]()
, where
is the contribution of all the disk-shaped strata
to the Morse Formula (2).
Recall that, by Corollary 4.4 [5] , for any 3-fold
and a boundary generic field
on it, we get
, provided
. Thus, as a positive
increases, the boundary of the disk
becomes more “wavily”.
For example, by attaching many 2-handles to the boundary of a 2-disk, we get a 2-fold X with a big value
of
. On such
,
for any gradient-like field
such that
.
These examples motivate the following question.
Question 3.1 For boundary generic gradient-like fields
with a fixed value
of the index
and a
disk-shaped stratification
as in Corollary 3.3, what is the minimum
of
? ![]()
Evidently, such number
is an invariant of the diffeomorphism type of
. It seems that
is semi-additive under the connected sum operation: that is,
![]()
4. Boundary Convexity and Concavity of Vector Fields
We are ready to introduce pivotal concepts of the stratified convexity and concavity for smooth vector fields on manifolds with boundary.
Definition 4.1 Given a boundary generic vector field
(see Definition 2.1), we say that
is boundary
-convex, if
. In particular, if
, we say that
is boundary
-convex, or just boundary convex.
We say that
is boundary s-concave, if
. In particular, if
, we say that
is boundary 2-concave, or just boundary concave. ![]()
Example 4.1. Assume that a compact manifold X is defined as a 0-dimensional submanifold in the interior of a Riemannian manifold
, given by an inequality
, where
is a smooth function with 0 being a regular value. Then the boundary convexity of a gradient field
in X can be expressed in terms of the Hessian matrix
by the inequality
![]()
at all points
, where
is tangent to
. If
![]()
, where
is tangent to
, then the field
is boundary concave. ![]()
Example 4.2. According to the argument in Lemma 3.3, the complement to a small convex (in the Morse coordinates) disk, centered at a Morse type
-critical point, is boundary concave with respect to the gradient field
. In fact, the field
is both boundary 3-concave and 3-convex! So, if
is a Morse function on a closed manifold Y with a critical set
, then the complement X in Y to a small locally convex neighborhood of
admits a boundary concave
-gradient-like field (with
)! ![]()
Theorem 4.1 below belongs to a family of results which we call “holographic” (see also and Theorem 4.2). The intension in such results is to reconstruct some structures on the “bulk”
(or even the space
itself) from the appropriate flow-generated structures (“observables”) on its boundary
. A paper from this series will be devoted entirely to the phenomenon of holography for nonsingular gradient flows.
In Theorem 4.1, we describe how some boundary-confined interactions between the critical points of a given function
of opposite polarities can serve as an indicator of the convexity/concavity of the gradient field
in X (recall that the convexity/concavity properties of the
-flow do require knowing the field in the vicinity of
in X!).
Theorem 4.1 Let
,
be Morse functions and
and
their gradient fields with respect to a Riemannian metric
on X and its restriction to
, respectively. Assume that
is boundary generic.
If
, then there is no ascending
-trajectory
, such that
![]()
(both critical sets
depend only on
).
Conversely, if for a given
-gradient pair
, no such
-trajectory
exists, then one can deform
to a new boundary generic pair
of the
-gradient type so that
. Moreover, the fields
and
on
can be chosen to be arbitrary close in the
-topology.
In particular, if
(as sets), then X admits a boundary generic and convex f-gradient-like field
; similarly, if
, then X admits a boundary generic and concave f-gradient-like field
.
Proof. First consider the convex case, that is, the relation between the property
and the absence of an ascending
-trajectory
which connects
to
.
Consider the function
, defined via the formula
, where
denotes a unitary field inward normal to
in
. Since
is boundary generic, zero is a regular value of
. Then
![]()
and
.
If an ascending
-trajectory
, which links
with
, does exist, it must cross somewhere the boundary
of
. Since the field
is an orthogonal projection of
on
, the two fields must agree at any point
―the locus where
is tangent to
. Thus,
is the gradient of
at
. Therefore, as
crosses from
into
at
, in its vicinity, the arc
lies below the arc
(see Figure 5). By the definition of the locus
, such crossing
belongs to
. Therefore,
, contrary to the theorem hypothesis.
On the other hand, if no such
-trajectory
exists, then we claim the existence of a codimension one closed submanifold
, which separates
in two manifolds,
and
, such that the vector field
, or rather its perturbation
, is transversal to
and points outward of
. Indeed, for each critical point
, in the local Morse coordinates
on
, consider a small closed ![]()
-disk
centered on the critical point x. Denote by
the closure of the union of downward trajectories of the
-flow passing through the points of
(see Figure 6, the left diagram). Let
be the union
(see Figure 6, the right diagram).
Since we assume that no descending
-trajectory
links a point of
to a point of
, we can choose
the disks
so small that the set
belongs to the complement
.
For each
, the zero cone
of the Morse function
separates the sphere
into two handles,
and
(each being a product of a sphere with a disk). We denote by
the handle in
whose spherical core is formed by the intersection of the unstable disk through
with the sphere
. Then, by definition, the set
is a collection of downward trajectories through the points of
union with
. Note that the downward trajectories from a different set
could enter the disk
only through the complementary handle
in its boundary. As a result,
is a manifold whose piecewise smooth boundary could have corners (see Figure 6, the right diagram). Similarly,
is a domain in
whose boundary is piecewise smooth manifold with corners.
Since
consists of the downward trajectories of
, if
, then any point
which can be reached from
following the field
(for short, “is below
”) belongs to
as well. Therefore the boundary
is assembled either from downward trajectories or from singletons; the singletons are contributed by some portions of
where
points outside of the relevant disk
. Thus either
is tangent to
, or it points outside
.
![]()
Figure 6. The set
(the left diagram) and the set
(the right diagram).
Away from
,
is of the
-gradient type. Thus, in each tangent space
, where
, there is an open cone
comprised of
-gradient type vectors, and
. Therefore, in the vicinity of
, we can perturb
to a new field
of the
-gradient type, so that
points strictly outside
and still
for all
. It is possible to smoothen the boundary
so that, with respect to a new smooth boundary
, the field
still points outside
, the new domain bounded by
, and
for all
.
Note that if
, then
can serve as a separator.
Let
and
. With the separator
in place, consider a smooth function
with the properties:
1) zero is a regular value of
, and
,
2)
,
,
3)
in a neighborhood of
,
4)
, where
is the inward normal to
in
.
Note that the field
points inside of
along
and outside of
along
. It also points outside of
along
. As a result, we conclude that
and
; in other words,
is boundary convex. Note that
can be chosen arbitrary close to
. Ineeded, employing Theorem 3.2, we can perturb
to insure its genericity with respect to the pair
, and thus the boundary genericity of
itself.
The argument in the concave case, which deals with the relation between the property
and the absence of an ascending
-trajectory
, connecting
to
, is analogous. We just need to switch the polarities of the relevant sets. ![]()
Now we need to introduce a number basic notions to which we will return on many occasions in the future.
Definition 4.2 Let
be a differential
-form on a manifold
.
We say that a path
is
-positive (
-negative), if,
for all values of the parameter
. ![]()
Definition 4.3 Let
be a closed differential 1-form on a manifold
, equipped with a Riemannian metric
. We say that a vector field
on
is the gradient of
(and denote it “
”), if
for any vector field
on
. ![]()
Definition 4.4 Let
be a differential 1-form on a manifold Y and let
be the set of points
, where
is the zero map. Assume that
for some smooth function
in the vicinity of
.
We say that a vector field
is of
-gradient type if
on
and
in the vicinity of
. Here
is some Riemannian metric in the vicinity of
(cf. Definition 2.2). ![]()
We are in position to formulate a generalization of Theorem 4.1 for closed differential
-forms―another instance of somewhat weaker “holographic phenomenon”, this time for fields which may not be gradient-like globally.
Theorem 4.2 Let
be a closed 1-form on a compact manifold
, equipped with a Riemannian metric
. Assume that
and
have only Morse-type singularities. Let the gradient
be a boundary
generic field, and let
.
If
, then there is no
-ascending
-trajectory
, such that
![]()
Assume that there exists a codimension one submanifold
, which separates
and
and such that the field
is transversal to
and points outwards/inwards of the domain in
that is bounded by
and contains
. Then one can deform the
-gradient vector fields
to a new boundary generic pair
of the
-gradient type so that
.
Proof. The
-gradient fields
on
are characterized by the property
, valid on the
locus where
. Usually, in this setting, we do not have a natural choice for the wall
which would separate the singularities of opposite polarities
and
and such that the field
would be transversal to N. It seems unlikely that the absence of an ascending
-trajectory which links
with
is sufficient to guarantee the existence of a separator N. However, in the presence of such separator N, the arguments are identical with the ones employed in the proof of Theorem 4.1. ![]()
Remark 4.1. In Theorem 4.1 and Theorem 4.2, the partition
of the singular set
must satisfy some basic relations:
![]()
![]()
These relations reflect the fact that
when
, and
when
. ![]()
Given a metric
on a Riemannian
-manifold
, let us recall a definition of the Hodge Star Operator
(see [19] ).
Pick a local basis
of 1-forms in
and consider the associated basis
![]()
of
, where
and the symbol “
“ stands for omitting the
-th form
from the product
.
Assume that, in the dual to
basis
of
, the metric
is locally given by a matrix
. Then the matrix
of the
-operator in the bases
,
is given by the formula
(5)
whence
.
Definition 4.5 A closed differential 1-form
on a compact manifold Y is called intrinsically harmonic if there exists a Riemannian metric g on Y such that the form
is closed. ![]()
Example 4.3. Let
be a closed smooth manifold and
a smooth map with isolated Morse-type singularities. Consider the closed
-form
, the pull-back of the canonic 1-form
on the circle
. Assume that one of the
-fibers,
, is connected. Then
is intrinsically harmonic [20] . ![]()
Let
denote the singularity set of a closed 1-form
on a compact manifold Y. We assume that
.
By Calabi’s Proposition 1 [19] ,
is intrinsically harmonic if and only if through every point
there is a
-positive path
which either is a loop, or which starts and terminates at the boundary
.
Theorem 4.3 Let
be a closed
-form on a Riemannian manifold
, such that
. Assume that
, the restriction of
to
, is a harmonic form7.
Then the gradient field
is not boundary convex or boundary concave (that is,
and
). Thus, if
is connected, then
.
Proof. We abbreviate
to
and
to
. Here
is the *-operator on the boundary of X
with respect to the given Riemannian metric
on
.
If
is a closed
-form on
, then by the Stokes Theorem,
![]()
However, for a concave/convex gradient field
, the
-form
, being restricted to
, is proportional to the volume form of
with negative/positive functional coefficient. Indeed, at the points of
, the angle between
and the normal
to
in
is acute, while it is obtuse at the
points of
. Therefore,
when either
or
. The resulting contradic-
tion proves that
and
.
Therefore, when
is connected, then
and
must share the common nonempty boundary
―the gradient field
must have cuspidal points. ![]()
Example 4.4. Let
be a compact smooth manifold and
a smooth map with isolated Morse- type singularities. Consider the closed
-form
, the pull-back of the canonic 1-form
on the circle
. Assume that one of the fibers of the map
is connected. Then there exists a metric ![]()
on
such that the form
is harmonic ([19] [20] ). Consider the gradient field
.
Then by Theorem 4.3,
and
for any metric
that “harmonizes”
. ![]()
Definition 4.6 A non-vanishing vector field
on a compact manifold
is called traversing if each
-trajectory is either a closed segment or a singleton which belongs to
. ![]()
Remark 4.2. The definition excludes fields with zeros in
(they will generate trajectories that are homeomorphic to open or semi-open intervals) and fields with closed trajectories. Note that all gradient-like fields of nonsingular functions are traversing, but the gradient-like fields of nonsingular closed 1-forms may not be traversing! ![]()
Lemma 4.1 Any traversing vector field is of the gradient type.
Proof. Let
be a traversing field on X. We extend the pair
to a pair
so that X is properly contained in
and
.
Then every
-trajectory
has a local transversal compact section
of the
-flow. We can choose
to be diffeomorphic to a n-dimensional ball with its center at the singleton
. We denote by
the union of
-trajectories through
.
For each
-trajectory
, there exists a section
so that the set
contains a compact cylinder
, where
are positive constants (which depend on
), with the properties:
1)
,
2) for any
-trajectory
through
, the intersection
is a segment,
3) the point
belongs to the interior of segment
.
Then the collection
forms a cover of
. Since
is compact, we can choose
a finite subcover
of
.
For each
and the corresponding section
, we produce a smooth function
by integrating the vector field
and using
as the initial location for the integration.
More accurately, let
![]()
be the parametrization of a typical trajectory
, such that
![]()
for all
and
. This bijective parametrization introduces a smooth product structure
![]()
by the formula
.
We define a smooth function
by the formula
and denote it (quite appropriately)
by the symbol
.
Let
be a smooth non-negative function that vanishes only on the boundary
. Let
denote the composition of the
-directed projection
with the function
. Since
vanishes on
, the function extends smoothly on X to produce a smooth function
with the support in
.
Now consider the smooth function
(6)
It is well-defined on X. Let us compute its
-directional derivative:
(7)
Let us explain Formula (7). By the very definition of
, it is constant on each
-trajectory, so that
. Also,
in
. At the same time,
, since
and
increases in the direction of
. Finally, each
belongs to the interior of some set
.
Therefore,
, so that
is a gradient-like field for
. ![]()
Corollary 4.1 Let X be a smooth compact manifold with boundary. Then
―the space of traversing vector fields on X―is nonempty and coincides with the intersection
, where
denotes the space of gradient-like fields, and
the space of all non-vanishing fields on X.
Proof. By definition, any traversing field
on X does not vanish. By Lemma 4.1,
must be of the gradient type. Thus
![]()
On the other hand, for a compact X with a gradient-like
, each
-trajectory
through
must reach the boundary in both finite positive and negative times (since it is controlled by some Lyapunov function f).
As a result,
![]()
It remains to show that
. By Lemma 3.2,
, which implies that
. ![]()
There are simple topological obstructions to boundary convexity of any gradient-like nonvanishing field on a given manifold X. The next lemma testifies that the existence of boundary convex traversing fields
imposes severe restrictions on the topology of the manifold X.
Lemma 4.2 A connected
-manifold X admits a boundary convex traversing8 field
, if and only if, X is diffeomorphic to a product of a connected compact n-manifold and a segment, the corners of the product being smoothly rounded.
Proof. Indeed, if such convex
exists,
must be a deformation retract of X: just use the down flow to produce the retraction. Therefore, when
, then X is homeomorphic to the quotient space
, where the equivalence relation “
” is defined by collapsing each segment ![]()
to a point. If we round the corners generated in the collapse, we will get a diffeomorphism between X and the “lens”
(see Figure 7).
On the other hand, any product
, whose conners
being rounded, admits a field of the desired boundary convex type. ![]()
Corollary 4.2 For all
, any smooth compact contractible
-manifold X, which admits a boundary convex traversing field, is diffeomorphic to the standard
-disk.
![]()
Figure 7. The existence of a traversing boundary convex field
(the constant vertical field) on a (n + 1)-manifold X (the ellipsoid-bounded solid) implies that topologically it is a product of a compact n-manifold Y (the elliptical shadow) with an interval.
Proof. By Lemma 4.2, X is diffeomorphic to a product of a fake
-disk Y with
, the corners of the product being rounded.
For
, by Perelman’s results [21] [22] , Y is diffeomorphic to the standard 3-disk. Thus X is diffeomorphic to the standart 4-disk.
For
, we do not know whether Y is a standard 4-disk.
For
, the h-cobordism theorem [23] implies that any fake n-disk is diffeomorphic to the standard disk.
This leaves only the case of 5-dimensional X wide open. ![]()
We notice that
is an obstruction to finding boundary convex traversing
on a
-dimensional manifold
with a connected boundary.
Corollary 4.3 Let X be a smooth connected compact
-manifold with boundary, which admits a boundary convex traversing field.
If
, then X is diffeomorphic to the product
, where Y is a closed manifold.
In particular, no connected X with boundary
, whose number of connected components differs from two, and with the property
admits a boundary convex traversing field.
Proof. If such boundary convex traversing field
exists,
must be a deformation retract of X. Therefore, for a connected X,
must be connected as well.
On the other hand, if
, then the connected
must be of a homotopy type of a
-dimensional complex. In such a case, the groups
must vanish.
Thus when
and
is boundary convex, the only remaining option is
, which implies that
―the manifold
is closed. In such a case, X is a product of a connected closed
-manifold with an interval; so the boundary
must be the union of two diffeomorphic components. ![]()
As with the boundary convex traversing fields, perhaps, there are topological obstructions to the existence of a boundary concave traversing field on a given manifold? At the present time, the contours of the universe of such obstructions are murky. We know only that the disk
does not admit a non-vanishing boundary concave field (see Example 4.4).
Lemma 4.3 If a boundary generic vector field
on an even-dimensional compact orientable manifold
is boundary concave, then its index
![]()
If a boundary generic vector field
on an odd-dimensional compact orientable manifold X is boundary concave, then its index
![]()
Thus, for all boundary concave fields
with a fixed value of index
, the Euler number
is a topological invariant.
Proof. For a boundary concave field
,
. Therefore, the Morse Formula (2) reduces to the equation
(8)
Recall that, for any orientable odd-dimensional manifold Y,
. Therefore, when
, we get
. Thus formula (8) transforms into
![]()
For an odd-dimensional X, the closed manifold
is odd-dimensional, so
. Therefore
. ![]()
Corollary 4.4 Let X be a
-dimensional oriented smooth and compact manifold with boundary.
If
, then for any boundary generic concave vector field
on X of index 0, the locus
contains at least
two-dimensional spheres.
Proof. Since
is a closed orientable 2-manifold, its Euler number is positive only if
contains sufficiently many 2-spheres. By Lemma 4.3,
. Therefore
contains at least
two-dimensional spheres. ![]()
Example 4.4. Let
, the 2-dimensional ball. If
on X, then by the Morse formula,
![]()
If
consists of k arcs, then by this formula,
. At the same time,
. Therefore,
. So we conclude that
does not admit a non-vanishing field with
, that is, a boundary concave field.
At the same time, if we delete any number of disjoint open disks from
, the remaining surface X admits a concave non-vanishing gradient-like field: indeed, consider the radial field in an annulus A and delete from A any non-negative number of small round disks. The radial field
on A, being restricted to X, is evidently of the gradient type and concave with respect to
.
Note that, if a connected compact surface X admits a generic traversing concave field
, then X is homeomorphic either to a thickening of a finite graph
whose vertexes all have valency 3, or to an annulus. ![]()
In the previous example, we have seen that the disk
does not admit a non-vanishing concave field. In contrast,
does admit a boundary generic concave non-vanishing field: just consider the restriction of the Hopf field
on
to the northern hemisphere
. For the unitary disk
centered at the origin, informally, we can describe
as the sum of the velocity field of the solid
, spinning around the
-axis, with the solenoidal field of the loop
. However, this field
is not of the traversing type: it has closed trajectories (residing in the solid torus
).
These observations encourage us to formulate
Conjecture 4.1 The standard
-disk
does not admit a traversing boundary concave vector field
.
The construction of a boundary concave field on a 2-disk with holes (see Example 4.4) admits a simple generalization.
Example 4.5. Consider a closed
-manifold Y. Let
be compact submanifolds also of dimension
. Let
. We pick
disjointed close intervals
in the interval
. Then we form the product
. By rounding the corners of
, we get a
-manifold
so that each segment
, where
, hits
along a closed segment, and each segment
, where
, hits
along a singleton.
Form the manifold
. Its boundary consists of two copies of Y together with the disjoint union of
(they are the doubles of
’s). The obvious vertical field
on W, being restricted to X, is boundary
concave. In fact,
, where
, and
. ![]()
These examples lead to few interesting questions:
Question 4.1. Which compact manifolds admit boundary concave non-vanishing vector fields? Which compact manifolds admit boundary concave non-vanishing gradient-like fields? ![]()
Despite the “natural” flavor of these questions, we have a limited understanding of the general answers. Nevertheless, feeling a bit adventurous, let us state briefly what kind of answer one might anticipate. This anticipation is based on a better understanding of boundary concave traversing fields on 3-folds (see [5] [24] ).
We conjecture that an
-dimensional X admits a traversing concave field
such that
if (perhaps, if and only if) X has a “special trivalent” simple
-dimensional spine
, where
denotes a smooth triangulation of X (see [25] for the definitions of simple spines and for the description of their local topology). Here “special trivalent” means that each
-simplex from the singular set SK of K is adjacent to exactly three
-simplexes from K. Moreover, the vicinity of SK in K admits an oriented branching as in [24] .
When the
-manifold in question is specially manufactured from a closed
-manifold by removing a number of
-disks, another paper from this series will provide us with a wast gallery of manifolds which admit traversing concave fields.
5. Morse Stratifications of the Boundary 3-Convex and 3-Concave Fields
We have seen that the boundary 2-convexity of traversing fields on X has strong implications for the topology of X (for example, see Lemmas 4.2 - 4.3, and Corollaries 4.2 - 4.4).
By itself, the boundary 3-convexity and 3-concavity of traversing fields has no topological significance for the topology of 3-folds: we have proved in Theorem 9.5 from [5] that, for every 3-fold X, any boundary generic
of the gradient type can be deformed into new such field
with
. However, in conjunction with certain topological constraints on
(like being connected), the boundary 3-convexity has topological implications (see [5] , Corollary 2.3 and Corollary 2.5).
These observations suggest two general questions:
Question 5.1.
・ Given a manifold X, which patterns of the stratifications
are realizable by boundary generic traversing fields
on X?9
・ Given two such fields,
and
, can we find a linking path
in the space
that avoids certain types of singularities?10 Specifically, if for some
,
, is there a linking path so that
for all
? ![]()
Remark 5.1. The property of the field
in Question 5.1 being traversing (equivalently, boundary generic and of the gradient type) is the essence of the question. For just boundary generic fields, there are no known restrictions on the patterns of
.
Let us illustrate this remark for the fields
such that
. We divide the boundary
into two complementary domains,
and
, which share a common boundary
―a closed manifold of dimension
. It may have several connected components. Next, we divide the manifold
into two com- plementary closed manifolds
and
.
We claim that it is possible to find a boundary generic field
with the properties:
,
, and
. The construction of such
is quite familiar (see the arguments in Theorem 3.2).
We start with a field
which is normal to
and points outside of
along
and inside of
along
. We extend
to a field
tangent to the boundary
so that
has only isolated zeros. Let
be the outward normal field of
in X and
a smooth function such that 0 is its regular value and
![]()
Along
, form the field
and extend it to a field
on X with isolated singularities in
. By its construction,
has all the desired properties. Note that here we do not insist on the property
. ![]()
In our inquiry, we are inspired by the Eliashberg surgery theory of folding maps [6] [7] . In many cases, Eliashberg’s results give criteria for realizing given patterns of
, provided that
, thus answering Question 5.1. Let us state one such result, Theorem 5.3 from [7] .
Theorem 5.1 (Eliashberg) Let
,
, be a compact connected smooth submanifold of dimension
. Consider two disjoint closed and nonempty
-submanifolds
and
of
whose union separates
into two complementary
-manifolds,
and
. Let
be the outward normal field of
in X, and denote by
the degree of the Gauss map
. Let
be a linear surjection.
Then the topological constraints
・
, when ![]()
・
, when ![]()
are necessary and sufficient for the existence of an orientation-preserving diffeomorphism
with the following properties:
・
is the fold locus of the map
,
・
, being restricted to
, is a immersion, and the image
has only transversal self-intersections in
,
・ the differential
takes the normal field
to the field inward normal to
in
,
・ the differential
takes the normal field
to the field outward normal to
in
. ![]()
Considering a traversing field
which is tangent to the fibers of the map
from Theorem 5.1, leads instantly to
Corollary 5.1 Under the hypotheses and notations from Theorem 5.1, there exists a boundary generic traversing field
on
so that:
・
,
・
,
・
. ![]()
Thus, at least for smooth domains
and for boundary generic traversing fields
, which are both 3-convex and 3-concave, the patterns for the strata
![]()
are indeed very flexible. However, the requirement that both
and
puts breaks on any applcation of Corollary 5.1 to boundary concave and boundary convex traversing fields on
!
Example 5.1. Let us illustrate how non-trivial the conclusions of Theorem 5.1 and Corollary 5.1 are. Let
,
. When
is odd, take any codimension one submanifold
such that
,
, and
. Then
admits a boundary generic traversing field
such that
and
.
For instance,
admits a a boundary concave traversing field
such that
, the orientable surface of genus 2, and
, the 2-torus.
When
is even, take any codimension one submanifold
such that
,
, and
. Then
admits a boundary generic traversing field
such that
and
.
For example, for any collection of loops
,
,
, the disk
admits a boun-
dary generic traversing field
such that
and
. ![]()
We suspect that an important for our program generalization of Theorem 5.1 is valid and can be established by the methods as in [6] [7] .
Conjecture 5.1 Let
be a compact connected smooth manifold of dimension
, equipped with a traversing vector field
. Let
and
be two disjoint closed and nonempty
-submanifolds of
whose union separates
into two
-manifolds,
and
.
Then the topological constraints
(9)
(10)
are necessary and sufficient for the existence of an orientation-preserving diffeomorphism
with the following properties:
・ the restriction of
to the image
is boundary generic in the sense of Definition 2.111,
・
,
・
,
・
.
Moreover, in a given collar
of
in
, there is a
-supported diffeomorphism
as above which is arbitrary close in the
-topology to the identity map. ![]()
To prove the necessity of the topological constraints (9) and (10) is straightforward. By the Morse Formula (2) (see also Corollary 5.1), a necessary condition for the existence of a diffeomorphism
with the desired properties, described in the bullets, is the constraint
![]()
Since
is a homeomorphism, this equation is equivalent to
(11)
If
, then
![]()
Therefore, using Formula (11), the constraint becomes
―Formula (10).
When
, since
are closed odd-dimensional manifolds,
, Formula (11) reduces to
―Formula (9).
Therefore the topological constraints (9) and (10) imposed on the “candidates”
,
and
and
are necessary for the existence of the desired diffeomorphism
.
To prove the sufficiency of these conditions may require a clever application of the
-principle in the spirit of [6] [7] .
Corollary 5.2 Assuming the validity of Conjecture 5.1, any compact smooth manifold
with boundary admits a boundary generic traversing field
with the property
. ![]()
Proof. By Corollary 4.1,
, we can start with a traversing field
and apply Conjecture 5.1 to it to get the pull-back field
with the desired properties.
Conjecture 5.2 Given two vector fields
and
as in Corollary 5.2, there is a 1-parameter family of traversing fields
which connects
to
and such that only for finitely many instances
,
. For those exceptional
’s,
. ![]()
NOTES
1in the sense of Definition 2.4
![]()
2 By definition,
and
.
3Thus
is a transfer by
of the constant field
.
![]()
4Recall that
.
![]()
5as in the proof of Lemma 3.2.
![]()
6This condition is metric-independent: it does not depend on the choice of
.
![]()
7This assumption implies that
, provided
.
![]()
8equivalently, a non-vanishing gradient-like field.
![]()
9Theorem 5.1 and Corollary 5.1 below give just a taste of a possible answer.
10When
, Theorem 9.5 in [5] addresses some of these questions.
![]()
11and even traversally generic in the sense of Definition 3.2 from [8]