Stratified convexity&concavity of gradient flows on manifolds with boundary

As has been observed by Morse \cite{Mo}, any generic vector field $v$ on a compact smooth manifold $X$ with boundary gives rise to a stratification of the boundary $\d X$ by compact submanifolds $\{\d_j^\pm X(v)\}_{1 \leq j \leq \dim(X)}$, where $\textup{codim}(\d_j^\pm X(v))= j$. Our main observation is that this stratification reflects the stratified convexity/concavity of the boundary $\d X$ with respect to the $v$-flow. We study the behavior of this stratification under deformations of the vector field $v$. We also investigate the restrictions that the existence of a convex/concave traversing $v$-flow imposes on the topology of $X$. Let $v_1$ be the orthogonal projection of $v$ on the tangent bundle of $\d X$. We link the dynamics of the $v_1$-flow on the boundary with the property of $v$ in $X$ being convex/concave. This linkage is an instance of more general phenomenon that we call"holography of traversing fields"---a subject of a different paper to follow.


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 [GM], [GM1], and [GM2]). For us, however, the starting starting point and the source of inspiration is the 1929 paper of Morse [Mo].
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 [K], where they are adopted to the special environment a 3D-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 X 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 v on X (see Definition 2.1), we explore the Morse stratification {∂ ± j X(v)} j of the boundary ∂X (see formula 2.1 and [Mo], induced by the vector field v on X. 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).
Let v 1 denote the orthogonal projection of the field v| ∂X on the bundle T (∂X) tangent to the boundary. Occasionally, we can determine whether a given field v is convex/concave just by observing the behavior of the v 1 -trajectories on the boundary ∂ 1 X (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 Morse Stratification {∂ +
j X(v)} Inspired by [Mo], 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 X be a compact smooth (n + 1)-dimensional manifold with a boundary ∂X. Let v be a smooth vector field on X which does not vanish on the boundary ∂X. As a rule, we assume that X is properly contained in a (n + 1)-dimensional manifoldX and that the field v extends to a fieldv onX so that v|X \X = 0. In fact, we always treat the pair (X,v) as a germ of a space and a field in the vicinity of the given pair (X, v). 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 T (X), is transversal its zero section. In other words, in the vicinity of each singular Figure 2. A generic field v in the vicinity of a cusp point on the boundary of a solid X generates the Morse stratification ∂ + 3 X ⊂ ∂ + 2 X ⊂ ∂ + 1 X (the left diagram) or the Morse stratification ∂ − 3 X ⊂ ∂ + 2 X ⊂ ∂ + 1 X (the right diagram). point, there is a local system of coordinates (x 1 , . . . , x n+1 ) such that the field v can be represented as v = (a 1 x 1 , . . . , a n+1 x n+1 ), where all a i = 0.
To achieve some uniformity in our notations, let ∂ 0 X := X and ∂ 1 X := ∂X. The vector field v gives rise to a partition ∂ + 1 X ∪ ∂ − 1 X of the boundary ∂ 1 X into two sets: the locus ∂ + 1 X, where the field is directed inward of X, and ∂ − 1 X, where it is directed outwards. We assume that v, viewed as a section of the quotient line bundle T (X)/T (∂X) over ∂X, is transversal to its zero section. This assumption implies that both sets ∂ ± 1 X are compact manifolds which share a common boundary ∂ 2 X := ∂(∂ + 1 X) = ∂(∂ − 1 X). Evidently, ∂ 2 X is the locus where v is tangent to the boundary ∂ 1 X.
Morse has noticed that, for a generic vector field v, the tangent locus ∂ 2 X inherits a similar structure in connection to ∂ + 1 X, as ∂ 1 X has in connection to X (see [Mo]). That is, v gives rise to a partition ∂ + 2 X ∪ ∂ − 2 X of ∂ 2 X into two sets: the locus ∂ + 2 X, where the field is directed inward of ∂ + 1 X, and ∂ − 2 X, where it is directed outward of ∂ + 1 X. Again, let us assume that v, viewed as a section of the quotient line bundle T (∂ 1 X)/T (∂ 2 X) over ∂ 2 X, is transversal to its zero section.
For generic fields, this structure replicates itself: the cuspidal locus ∂ 3 X is defined as the locus where v is tangent to ∂ 2 X; ∂ 3 X is divided into two manifolds, ∂ + 3 X and ∂ − 3 X. In ∂ + 3 X, the field is directed inward of ∂ + 2 X, in ∂ − 3 X, outward of ∂ + 2 X. We can repeat this construction until we reach the zero-dimensional stratum ∂ n+1 X = ∂ + n+1 X ∪ ∂ − n+1 X. These considerations motivate Definition 2.1. We say that a smooth field v on X is boundary generic if: • v| ∂X = 0, • v, viewed as a section of the tangent bundle T (X), is transversal to its zero section, • for each j = 1, . . . , n + 1, the v-generated stratum ∂ j X is a smooth submanifold of ∂ j−1 X, • the field v, viewed as section of the quotient 1-bundle T ν j := T (∂ j−1 X)/T (∂ j X) → ∂ j X, is transversal to the zero section of T ν j for all j > 0. We denote the space of smooth generic vector fields on X by the symbol V † (X).
We will postpone the proof of the theorem below until the second paper in this series of articles (see [K3], Theorem 6.6, 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 V † (X) in the space V(X) of all smooth fields on X.
Definition 2.2. We say that a smooth vector field v on X is of the gradient type (or gradient-like) for a smooth function f : X → R if: • the differential df and the field v vanish on the same locus Z ⊂ X, • in the vicinity of Z, there exist a Rimannian metric g on X so that v = ∇ g f , the gradient field of f in the metric g.
Definition 2.3. A smooth function f : X → R is called Morse function if its differential df , viewed as a section of the cotangent bundle T * (X), is transversal to the zero section.
Recall that, for a Morse function f on a compact (n + 1)-manifold X, the critical set Z := {x ∈ X| df x = 0} is finite and each point x ∈ Z has special local coordinates (x 1 , . . . , x n+1 ) such that df = 1≤i≤n+1 a i x i dx i , where a i = 0 for all i (for example, see [GG]).
Definition 2.4. Let f : X → R be a smooth function and v its gradient-like vector field. We say that the pair (f, v) is boundary generic if the field v is boundary generic in the sense of Definition 2.1 and the restrictions of f to each stratum ∂ j X := ∂ j X(v) are Morse functions for all 0 ≤ j ≤ n.
Lemma 2.1. Let V be a compact smooth manifold, and Y a smooth manifold which is stratified by submanifolds The property (f, Ψ) ∈ X is equivalent to the property of the section df of the bundle In order to validate density of X in C ∞ (V, Y ×R), we first perturb a given map Ψ : V → Y to make it transversal to each stratum Y j ⊂ Y , and then perturb a given function f : V → R to make the section df of T * V transversal to each manifold V Ψ j := Ψ −1 (Y j ). Theorem 2.2. The boundary generic 1 Morse pairs (f, v) on a compact manifold X form an open and dense subset in the space of all smooth functions f : X → R and their gradient-like fields v.
Proof. By Theorem 2.1, the boundary generic fields v form an open and dense set in the space of all fields.
Let F n be a complete flag in R n , formed by subspaces F j of codimension j. In the proof of Theorem 3.4 [K3], for every field v, we will construct a smooth map Ψ ∂ (v) : if and only if, v is a boundary generic field. The construction of the map Ψ ∂ (v) utilizes high order Lie derivatives {L j v } 0≤j≤n of an auxiliary function z : X → R as in Lemma 3.1 [K3]. Now the property of boundary generic Morse pairs (f, v) to be open and dense in the space of all pairs follows from Lemma 2.1: just let V = ∂ 1 X, Y = R n , Y j = F j , and Ψ = Ψ ∂ (v) in that lemma.
For the reader convenience, let us sketch now an alternative argument that establishes just the density of boundary generic Morse pairs (f, v) in the space of all pairs. It does not rely on the construction of the map Ψ ∂ (v) from [K3].
We start with a pair (f, v) where v| ∂X = 0 and df (v) > 0 at the points of the set where v = 0. By a small perturbation of f , we can assume the f is a Morse function on X and v its gradient-like field.
Let K ⊃ ∂X be a compact regular neighborhood of ∂X in X so small that v K = 0. By Theorem 2.1, we can perturb v to a new fieldṽ so thatṽ is boundary generic in the sense of Definition 2.1 and stillṽ| K = 0.
For a given f , the condition df (u)| K > 0 defines an open cone in the space of all fields u, subject to the constraint u| K = 0. Thereforeṽ can be chosen both boundary generic and gradient-like for f | K . Whenṽ| K is fixed, so are the stratifications {∂ + j X(ṽ) ⊂ ∂ j X(ṽ)} j .
Next, withṽ| K being fixed, we perturb f again to a new functionf so that df (ṽ)| K > 0 and {f | ∂ j X(ṽ) } are Morse function for all j. The perturbation will be supported in the compact K. We start constructingf inductively first from adjusting it on the 1-manifold ∂ n X(ṽ) and then moving sequentially to the strata ∂ j X with lower indices j. We pick each perturbationf so small that the open condition df (ṽ)| K > 0 is not violated. The existence of the desired j-th perturbation is based on the fact that Morse functions on a compact manifold Y (in this case, on ∂ j X(ṽ)) form an open and dense subset in C ∞ (Y ), the space of all smooth functions on Y , being equipped with the Whitney topology. Note that sincẽ v is tangent to ∂ j X(ṽ) along ∂ j+1 X(ṽ) and df (ṽ)| ∂ j+1 X(ṽ) > 0, the restrictionf | ∂ j X(ṽ) has no critical points in the vicinity of ∂ j+1 X(ṽ). Thus we need to perturbf | ∂ j X(ṽ) only on a compact subset Q j ⊂ ∂ j X(ṽ) which has an empty intersection with ∂ j+1 X(ṽ). This perturbation extends smoothly from Q j to X. Eventually, we reach the upper stratum ∂ 0 X := X, thus constructing a boundary generic approximation of the given pair (f, v).
All the changes (f ,ṽ) of (f, v), but the first one, we have introduced so far are supported in K, whereṽ = 0 and df (ṽ) > 0. This proves that the boundary generic pairs form a dense set in the space of all pairs (f, v), where v being a f -gradient-like field, subject to the constraints: v| ∂X = 0, and f : X → R being a Morse function.  For a given Morse pair (f, v), we denote by Σ j ⊂ ∂ j X(v) the set of critical points of the function f | : ∂ j X → R. For a boundary generic Morse pair (f, v), the finite critical set Σ j is divided into two complementary sets: the set Σ + j ⊂ ∂ + j X of positive critical points and the set Σ − j ⊂ ∂ − j X of negative ones (see Fig. 3). Remark 2.2. Note that when ∂ + j X = ∅, it may happen that Σ Consider a generic field v and a Riemannian metric g on X. We denote by v j the orthogonal projection of the field v on the tangent space T (∂ j X). Note that if v is a gradient field for a function f : X → R in metric g, then v j is automatically a gradient field for the restrictions f | ∂ j X and g| ∂ j X .
Take a smooth vector field v on a compact (m + 1)-manifold Y with isolated singularities {y ⋆ ∈ Σ(v) ⊂ int(Y )}. We denote by ind y⋆ (v) the localized index of v at its typical singular point y ⋆ . In a local chart, ind y⋆ (v) is defined as the degree of a map G v : S m y⋆ → S m from a small y ⋆ -centered m-sphere to the unit m-sphere. The map takes each point a ∈ S y⋆ to the point v(a)/ v(a) ∈ S m . We define the "global" index Ind(v) as the sum y⋆∈Σ(v) ind y⋆ (v). For a generic field v and a Riemannian metric g on X, we form the fields {v j } on {∂ j X(v)} and define the global index of v j by the formula: Let us revisit the beautiful Morse formulas [Mo]: For a boundary generic vector field v and a Riemannian metric on a (n + 1)-manifold X, such that the singularities of the fields v j are isolated for all j ∈ [0, n + 1], the following two equivalent sets of formulas hold: , where χ(∼) stands for the Euler number of the appropriate space 2 .
For vector fields with symmetry, the Morse Law of Vector Fields has an equivariant generalization [K1]. Here is its brief description: for a compact Lie group G acting on a compact manifold X, equipped with a G-equivariant field v, we prove that the invariants {χ(∂ + k X)} can be interpreted as taking values in the Burnside ring B(G) of the group G (see [D] for the definitions). With this interpretation in place, the appearance of formula 2.2 does not change.
Morse formula 2.2 has an instant, but significant implication: Corollary 2.1. Let N be a smooth neighborhood of the zero set of a vector field v on a compact (n + 1)-manifold X. Assume that v is boundary generic with respect to both boundaries, ∂X and ∂N . Then Remark 2.3. Therefore, the numbers can serve as "more and less localized" definitions of the index invariant Ind(v). An interesting discussion, connected to Theorem 2.3, its topological and geometrical implications, can be found in the paper of Gotlieb [Go]. The "Topological Gauss-Bonnet Theorem" below is a sample of these results.
Theorem 2.4 (Gotlieb). Let X be a compact smooth (n + 1)-dimensional manifold and Φ : X → R n+1 a smooth map which is a immersion in the vicinity of the boundary ∂ 1 X. Let g be a Riemannian metric on X which, in the vicinity of ∂X, is the pull-back Φ * (g E ) of the Euclidean metric on R n+1 . Consider a generic linear function l : R n+1 → R such that the composite function f := l • Φ has only isolated singularities in the interior of X.
Let v := ∇ g f be the gradient field of f 3 . Assume that (f, v) is boundary generic.
Then the degree of the Gauss map can be calculated either by integrating over ∂ 1 X the normal curvature K ν (in the metric g) of the hypersurface ∂ 1 X ⊂ X, or in terms of the v-induced stratification Example 2.1. Let X be an orientable surface of genus g with a single boundary component. Let Φ : X → R 2 be an immersion, and let l : R 2 → R, f : X → R and v := ∇f be as in Theorem 2.4. 3 Thus v is a transfer by Φ of the constant field ∇g E l. 4 Recall that vol(S n ) = 2π Since Φ is an immersion everywhere (and not only in the vicinity of ∂X as Theorem 2.4 presumes), we get that v = 0. Thus Ind(v) = 0. Then Theorem 2.4 claims that the degree of the Gauss map G : ∂X → Φ(∂X) → S 1 is equal to ) . Thus, the topological Gauss-Bonnet theorem, for immersions Φ : X → R 2 , reduces to the equation So the number of v-trajectories γ in X that are tangent to ∂X, but are not singletons (they correspond to points of ∂ + 2 X(v)), as a function of genus g, grows at least as fast as 4g − 2.
On the other hand, by the Whitney index formula [W1], the degree of G : ∂X → S 1 can be also calculated as µ + N + − N − , where N ± denotes the number of positive/negative self-intersections of the curve Φ(∂X) ⊂ R 2 , and µ = ±1.
By a theorem of L. Guth [Gu], the total number of self-intersections N + + N − ≥ 2g + 2. Moreover, this lower bound is realized by an immersion Φ : X → R 2 ! Therefore, for any immersion Φ : X → R 2 , the total number of self-intersections of the curve Φ(∂X) can be estimated in terms of the boundary-tangent v-trajectories: , and for some special immersion Φ, we get Corollary 2.2. Let X be a compact (n + 1)-manifold with boundary, which is properly contained in an open (n + 1)-manifoldX. Let Φ :X → R n+1 be a smooth map which is a immersion in the vicinity of the boundary ∂ 1 X. Let g be a Riemannian metric onX which, in the vicinity of ∂ 1 X, is the pull-back Φ * (g E ) of the Euclidean metric on R n+1 . Let l : R n+1 → R be a linear function, and f := l • Φ its composition with the map Φ. Form the gradient field v := ∇ g f inX. Assume that the pair (f, v) is boundary generic in the sense of Definition 2.4.
For each j > 0, consider a ǫ-small tubular neighborhood U j of the manifold ∂ j X(v) in X. Then Φ : U j → R n+1 is an immersion. This setting gives rise to the Gauss map , where x ∈ ∂U j and ν x is the unit vector inward normal to ∂U j at x.
Then the degree of the Gauss map G j can be calculated either by integrating (with respect to the n-measure µ g ) over ∂U j the normal curvature K ν of the hypersurface ∂U j ⊂X, or in terms of the v-induced stratum ∂ j X(v): Proof. We will apply Theorem 2.4 to the field v in U j to conclude that Since v = 0 in U j , Ind(v) = 0, and the last term of this equation reduces to χ(U j ) = χ(∂ j X(v)).
Remark 2.4. Of course, for an odd-dimensional ∂ j X(v), the Euler number χ(∂ j X(v)) = 0, and so is deg(G j ). When ∂ j X(v) is even-dimensional (i.e., n + 1 − j = 2l), the integral in equation 2.4 can be expressed in terms of intrinsic Riemannian geometry of the manifold ∂ j X(v), namely, in terms of the Pfaffian P f (Ω). The Pfafian is a 2l-differential form, whose construction utilizes the curvature tensor on the manifold (see [MiS]). So, when Given a boundary generic field v on X, we introduce a sequence of basic degree-type invariants {d k (v)} which are intimately linked, via the Morse formula 2.2, to the invariants {χ(∂ + j X(v))}. We use a Riemannian metric g on X to produce the orthogonal projection v j of the field v on the tangent subspace T (∂ j X(v)) ⊂ T (X).
Let S(∂ k−1 X) be the bundle of unit (n + 1 − k)-spheres associated with the tangent bundle of the manifold ∂ k−1 X. We denote by S(∂ k−1 X) the restriction of the bundle For each k, consider two fields, the inward normal field ν k to ∂ k X in ∂ + k−1 X and v, as sections of the sphere bundle p k : . Assume that the sections v and ν k are transversal in the space S(∂ k−1 X). This transversality can be achieved by a perturbation of ν k (equivalently, by a perturbation of the metric g), supported in the vicinity of the singularity locus Σ + k . Indeed, the intersections occur where the field v k−1 is positively proportional to ν k , that is, where v k = 0. The later locus is exactly the locus Σ + k . The perturbation that does not affect the stratification {∂ + j X} j . Assuming the transversality of the intersection, the locus We define the integer d k (v) := v•ν k as the algebraic intersection number of two (n+1−k)cycles, v(∂ k X) and ν k (∂ k X), in the ambient manifold S(∂ k−1 X) of dimension 2(n + 1 − k).
Lemma 2.2. For a boundary generic field v on a Riemannian manifold X, the following formula holds: Proof. We already have noticed that the intersection set . By the Morse Formula 2.2, the claim of the lemma follows.
depends only on the singular locus Σ + k of v k and on the local indices of its points.
Question 2.1. How to compute d j (v) in the terms of Riemannian geometry and in the spirit of Theorem 2.4 and Corollary 2.2? For a boundary generic field v and a fixed metric g on X, each manifold ∂ j X(v) comes equipped with a preferred normal framing f r j of the normal bundle ν ∂ j X(v), ∂ 1 X : just consider the unitary inward normal field , and so on... Via the Pontryagin construction [Po], this framing f r j generates a continuous map G j (v, g) : ∂ 1 X → S j−1 . Its homotopy class [G j (v, g)] is an element of the cohomotopy set π j−1 (∂ 1 X). If ∂ j X(v) = ∅, then we define G j (v, g) : ∂ 1 X → S j−1 to be the trivial map that takes ∂ 1 X to the base point in S j−1 .
Unfortunately, as we will see soon, [G j (v, g)] = 0! However, when ∂ j+1 X(v) = ∅, each of the two loci ∂ ± j X(v) is a closed manifold. Then we can apply the Pontryagin construction only to, say, ∂ + j X(v) to get a map G + j (v, g) : ∂ 1 X → S j−1 . This application leads directly to the following proposition.
Corollary 2.4. Consider a boundary generic vector field v such that ∂ j+1 X(v) = ∅ and a metric g, defined in the vicinity of ∂ 1 X in X. Then these data give rise to continuous In particular, when ∂ 3 X(v) = ∅, we get an element also as an element of the homotopy group π n (S j−1 ).
have another classical interpretation as elements of oriented framed cobordism set Ω fr n−j+1 (∂ 1 X). In fact, the pair (∂ j X(v), f r j ) defines the trivial element in Ω fr Let us recall the definition of framed cobordisms (for example, see [Kos]). Let M 0 , M 1 ⊂ Y be oriented closed smooth m-dimensional submanifolds of a compact (m + k)-manifold Y , whose normal bundles ν(M 0 , Y ) and ν(M 1 , Y ) are equipped with framings f r 0 and f r 1 , respectively.
We say that two pairs (M 0 , f r 0 ) and (M 1 , f r 1 ) define the same element in Ω fr the restriction of F r to M 1 × {1} coincides with f r 1 , and the restriction of F r to M 0 × {0} coincides with f r 0 .
Then the Pontryagin construction establishes a bijection P : If m < k −1 both sets admit a structure of abelian groups and the bijection P becomes a group isomorphism. Now we are in position to explain why [G j (v)] = 0. Consider the obvious embedding Note that for j > 2, all the normal fields ν(∂ j X(v), ∂ + j−1 X(v) are preserved under the imbedding β. So, for any j ≥ 2, the normal framing f r

Deforming the Morse Stratification
Let X be a smooth compact (n + 1)-manifold with boundary ∂X. A boundary generic field v (see Definition 2.1) gives rise to two stratifications 2.1.
We are going to investigate how the stratification {∂ ± j X(v)} j changes as a result of deforming the vector field v.
Lemma 3.1. Let N ⊂ Y be a closed submanifold of a manifold Y and M a closed manifold. Consider a family of maps {f t : M → Y } t∈ [0,1] such that each f t is transversal to N . All the manifolds, maps, and families of maps are assumed to be smooth.
Then all the submanifolds {f t (M )∩N } are isotopic in N . In particular, the intersections In the construction of w, we evidently rely on the property of each f t being transversal to N .
This isotopy extends to an ambient isotopy of N itself [Thom].
Note that these arguments fail in general if ether M or N have boundaries. However, under additional assumptions (such as f t | ∂M being t-independent and f t (M ) ∩ ∂N = ∅), the relative versions of the lemma are valid.
Theorem 3.1. The diffeomorphism type of each stratum ∂ ± j X(v) is constant within each path-connected component of the space V † (X) of boundary generic fields.
Proof. If two generic fields, v 0 and v 1 , are connected by a continuous path v : [0, 1] → V † (X), then they can be connected by a pathṽ : [0, 1] → V † (X) such that the dependence of the fieldṽ(t) on t ∈ [0, 1] 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 X × [0, 1]), 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 v t ∈ V † (X), connecting v 0 to v 1 . Since any generic field v t | ∂ 1 X , viewed as a section of the vector bundle η 1 : T X| ∂ 1 X → ∂ 1 X, is transversal its zero section, we may apply Lemma 3.1 (with , and since their polarity ± is determined by the inward/outward direction of v t , which changes continuously with t, the ambient isotopy of A similar argument applies to lower strata ∂ ± j X(v t ). Indeed, with the isotopy h t : , both sections being transversal to the zero section of η 2 . Applying again Lemma 3.1, we conclude that the loci This reasoning can be recycled to prove that all the pairs ∂ + j X(v 0 ) and ∂ + j X(v t ) are diffeomorphic via a single isotopy of X. This argument will be carried explicitly in the proof of Theorem 3.4 from [K3].
Corollary 3.1. Let X be a (n + 1)-dimensional compact smooth manifold with boundary.
Within each path-connected component of the space V † (X) of generic fields, the numbers {d k (v)} 0≤k≤n , as well as the numbers {χ(∂ ± k X(v))} 1≤k≤n+1 , are constant. Proof. The claim follows instantly from Theorem 3.1 and Lemma 2.2.
For a manifold X with nonempty boundary, by deforming any given function f : X → R and its gradient-like field v, we can expel the isolated v-singularities from X. This can be achieved by the appropriate "finger moves" which originate at points of the boundary ∂X and engulf the isolated singularities of v. The result of these manipulations lead to Lemma 3.2. Any (n+1)-manifold X with a non-empty boundary admits a Morse function f : X → R with no critical points in the interior of X and such that f | : ∂X → R is a Morse function. Such functions form an open nonempty set in the space C ∞ (X) of all smooth functions on X.
As a result, the gradient-like vector fields v = 0 on X form an open nonempty set in the space V(X) of all all vector fields on X.
Proof. Let us sketch the main idea of the argument. Start with a Morse function f : X → R. Connect each critical point in the interior of X 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 X small regular neighborhoods of those paths ("dig a system of dead-end tunnels") and restrict f to the remaining portion X ⊙ of X. Smoothen the entrances of the tunnels so that the boundary of X ⊙ will be a smooth manifold which is diffeomorphic to X. We got a nonsingular function f on X ⊙ . A slight perturbation of f on X ⊙ will not introduce critical points in the interior of X ⊙ and will deliver a Morse function on its boundary. Indeed, recall that the sets of Morse functions on X and ∂X are open and dense in the spaces C ∞ (X) and C ∞ (∂X) of all smooth functions, respectively (for example, see [GG]).
Of course v = 0 is an open condition imposed on a vector field on a compact manifold. On the other hand, if df (v) > 0, then any field v ′ , sufficiently close to v, will have the property df (v ′ ) > 0. 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 V(X) of all all vector fields on X.
Eliminating isolated critical points of a given function f : X → R on a manifold with boundary is not "a free lunch": the elimination introduces new critical points of the restricted function f : ∂X → R. This is a persistent theme throughout our program: Expelling critical points of gradient flows from a manifold X leaves crucial residual geometry on its boundary.
This boundary-confined geometry allows for a reconstruction of the topology of X.
Ideas like these will be developed in the future papers from this series. Meanwhile, the following lemma gives a taste of things to come. Proof. Let p ⋆ be a Morse singularity of f in the interior of X. Denote by S p⋆ a sphere which bounds a small disk D p⋆ centered on p ⋆ and such that f | Sp ⋆ is a Morse function. Without loss of generality, we can assume that, in the Morse coordinates {x i }, S p⋆ is given by Then f | Sp ⋆ has only Morse-type singularities at the points where the coordinate axes pierce the sphere S p⋆ . With respect to the pair (X \ D p⋆ , f ), these points come in two flavors: positive and negative. The two types are separated by the hypersurface of the cone In the vicinity of p ⋆ , the intersection C ∩ S p⋆ is exactly the locus ∂ 2 (X \ D p⋆ ) = ∂ + 2 (X \ D p⋆ ) , so that the f -gradient field v (tangent to S p⋆ along C ∩ S p⋆ ) is transversal to C ∩ S p⋆ , the product of two spheres. Therefore, in the vicinity of x ⋆ , ∂ 3 (X \ D p⋆ ) = ∅!
The function f | Sp ⋆ has exactly 2 · i(p ⋆ ) critical points of the positive type and exactly 2(n + 1 − i(p ⋆ )) critical points of the negative type. We shall denote these sets by Σ ± 1 (p ⋆ ) and the two domains in which C divides S p⋆ -by S ± p⋆ . Let x ∈ S + p⋆ be a local maximum of f | Sp ⋆ . Note that it is possible to connect x to a nonsingular (for f | ∂X ) point y ∈ ∂X by a smooth path γ along which f is increasing. Indeed, any non-extendable path γ such that df (γ) > 0 either approaches a critical point or reaches the boundary ∂X. By a small perturbation, we can insure that γ 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 γ can be extended until it reaches the boundary ∂X at a point y.
Drilling a narrow tunnel U diffeomorphic to the product γ × D n along γ does not change the topology of X; the function f | X\U retains almost the same list of singularities at the boundary as the function f | X\Dp ⋆ has: more accurately, the local maximum at x ∈ S + p⋆ disappears in ∂(X \ U ) and a negative critical point of index 1 of f | ∂(X\U ) appears near the y-end of the tunnel U . Thus we have modified f and have eliminated the critical point p ⋆ in the interior of X at the cost of introducing on the boundary 2(n + 1 − i(p ⋆ )) − 1 critical points of positive type and 2i(p ⋆ ) + 1 critical points of negative type.
Soon, motivated by Lemma 3.2, we will restrict our attention to nonsingular functions f : X → R and their gradient-like fields v-an open subset in the space of all gradient-like pairs (f, v); but for now, let us investigate a more general case.
Consider Morse data (f, v), where the field v is nonsingular along the boundary ∂ 1 X. Extend (f, v) toX := X ∪ C andv, where C is some external collar of ∂ 1 X so that the extension (f ,v) is nonsingular in C. At each point x ∈ ∂ 1 X, thev-flow defines a projection p x of the germ of ∂ 1 X into the germ of the hypersurfacef −1 (f (x)).
Let ∂ j X • and ∂ ± j X • denote the pure strata ∂ j X \ ∂ j+1 X and ∂ ± j X \ ∂ j+1 X, respectively. At the points x ∈ ∂ 1 X • , p x is a surjection; at the points of x ∈ ∂ 2 X • , it is a folding map; at the points x ∈ ∂ 3 X • , it is a cuspidal map. Often we will refer to points x ∈ ∂ 1 X by the smooth types of their p x -projections.
As the theorem and the corollary below testify, for a given function f : X → R, we enjoy a considerable freedom in changing the given Morse stratification {∂ + j X := ∂ + j X(v)} by deforming the f -gradient-like field v (cf. Section 3 in [K]).
Theorem 3.2. Let X be a compact smooth (n+1)-manifold with nonempty boundary. Take a smooth function f : X → R with no singularities along ∂X, and let v be its gradient-like field. Consider a stratification of X by compact smooth manifolds {Y j }, and let S j and S ∂ j denote the critical sets of the restrictions f | Y j and f | ∂Y j , respectively. Assume that the following properties are satisfied: • dim(Y j ) = n + 1 − j, • Y 1 ⊂ ∂X and {Y j ⊂ ∂Y j−1 } are regular embeddings for all j ∈ [2, n + 1], • for each j ≤ n + 1 the functions f | Y j and f | ∂Y j have Morse-type critical points at the loci S j and S ∂ j , respectively, • at the points of S j , df (ν) > 0 and, at the points of Then, within the space of f -gradient-like fields, there is a deformation of v into a new boundary generic gradient-like fieldṽ, such that the stratification {∂ + j X(ṽ)} 0≤j≤n+1 , defined byṽ, coincides with the given stratification {Y j } 0≤j≤n+1 . Proof. We pick a Riemannian metric g in a collar U of ∂X in X so that v becomes the gradient field of f . Consider auxiliary vector fields {v j }, where v j denotes the orthogonal projection of v on the tangent spaces of closed manifold ∂Y j−1 . The construction of the desired fieldṽ is inductive in nature, the induction being executed in increasing values of the index k. Fig. 4 illustrates a typical inductive step.
Assume that v :=ṽ has been already constructed so that ∂ + j X(v) = Y j and Σ + j (v) = S j for all j < k. This assumption implies that v is tangent to Y j exactly along its boundary ∂Y j for all j < k. Along ∂Y k−1 = ∂(∂ + k−1 X(v)) = ∂ k X(v) (and thus along Y k ⊂ ∂Y k−1 ), we decompose v as v k + k−1 j=0 n j , where n j := v j−1 − v j . The idea is to modify v in the direction normal to ∂ k X(v) in ∂ k−1 X(v), while keeping the rest of its components {n j } unchanged.
Denote by T x the tangent space of Y k−1 at x ∈ ∂Y k−1 . Let T + x be the open half-space of T x positively spanned by the vectors that point inside of Y k−1 . Let T + x (f ) be half of the tangent space T x , defined by df (u) > 0, where u ∈ T x . We introduce the complementary to T + x and T + Fig. 4). These cones are non-empty, except perhaps at the points of S ∂ k−1 , where ±v k−1 is anti-parallel to the inward normal ν k of ∂Y k−1 ⊂ Y k−1 . However, at x ∈ S k , C + x = ∅, and at x ∈ S ∂ k−1 \ S k , C − x = ∅ due to the last bullet in the hypotheses of the theorem. Thus, for each x ∈ Y • k , there is a number h so that the vector x . By the partition of unity argument, which employes convexity of the cones C ± x , there is a smooth function h : ∂Y k−1 → R which delivers the desired field u k along ∂Y k . In order to insure the continuity of h and u k across the boundary ∂Y k ⊂ ∂Y k−1 , we require h| ∂Y k = 0. Thus u k = v k = 0 on ∂Y k .
Put v ′ = u k + k−1 j=0 n j . Now, ∂ + j X(v ′ ) = ∂ + j X(v) = Y j for all j < k (these strata depend on the n j 's only), and ∂ + k X(v ′ ) = Y k by the construction of u k . Moreover, Σ + j (v ′ ) = Σ + j (v) = S j for all j ≤ k. In fact, v ′ is tangent to Y k−1 along ∂Y k−1 . Note that this inductive argument should be modified for k = n + 1 since Y n+1 = S n+1 is 0-dimensional.
We smoothly extend v ′ into a regular neighborhood V of ∂Y k−1 in X. Abusing notations, we denote this extension by v ′ as well. The neighborhood V is chosen so that there df (v ′ ) > 0.
To complete the proof of the inductive step k − 1 ⇒ k, we form the fieldṽ := ψ 0 v + ψ 1 v ′ , where the functions {ψ 0 , ψ 1 } deliver a smooth partition of unity subordinate to the cover {X \ V, V } of X. Since df (∼) > 0 defines a convex cone in the space of vector fields,ṽ is a f -gradient-like field with the desired Morse stratification.
Theorem 3.2 has an immediate implication: Corollary 3.2. Let f : X → R be a Morse function and v its boundary generic gradientlike field with the Morse stratification Then, within the space of f -gradient-like fields, it is possible to deform v into a new gradient-like boundary generic fieldṽ, such that the stratification {∂ + j X(ṽ)} 0≤j≤n+1 coincides with the given stratification In particular, if Σ + k (v) = ∅, the claim is valid for any stratification {Y j } 0≤j≤n+1 as above that terminates with Y k = ∅.
The next proposition (based on Corollary 3.2) shows that, for a given Morse function f : X → R, by an appropriate choice of gradient-like field v, the Morse stratification ∂ + j X can be made topologically very simple and regular: namely, each stratum ∂ + j X is a disjoint union of (n + 1 − j)-dimensional disks. Moreover, when the boundary ∂ 1 X is connected and j ∈ [1, n − 1], each stratum ∂ + j X is a just a single disk. Corollary 3.3. Let f : X → R be a Morse function on a compact (n + 1)-manifold X, f being nonsingular along the boundary ∂ 1 X. We divide the connected components {∂ 1 X α } α of the boundary into two types, A and B. By definition, for type A, the singularity set Σ + 1 (f ) ∩ ∂ 1 X α = ∅, and for type B, Σ + 1 (f ) ∩ ∂ 1 X α = ∅. Then any f -gradient-like field v can be deformed, within the space of f -gradient-like fields, into a boundary generic fieldṽ so that, for each component ∂ 1 X α of type A and all j < n, the stratum ∂ + j X(ṽ) ∩ ∂ 1 X α is diffeomorphic to a disk D n+1−j . At the same time, for the components of type B and all j ≥ 1, the stratum ∂ + j X(ṽ) ∩ ∂ 1 X α = ∅. For the components of type A, in contrast, the 1-dimensional stratum ∂ + n X(ṽ) ∩ ∂ 1 X α is a finite union of arcs residing in the circle ∂ n X(ṽ) ∩ ∂ 1 X α . Moreover, χ(∂ + n X(ṽ)), the number or arcs in ∂ + n X(ṽ), and the number of points in ∂ + n+1 X(ṽ) are linked via the formula , where Ind(v) = Ind(ṽ) is the index of the field v, and m is the number of boundary components of type A.
Since v andṽ both are the gradient-like fields for the same Morse function f , their indexes, Ind(v) and Ind(ṽ), are equal. Thus we get , where m 2 [(−1) n+1 − 1] is the contribution of all the disk-shaped strata {∂ + j X(ṽ)} 1≤j<n to the Morse formula 2.2.
Recall that, by Corollary 4.4 [K], for any 3-fold X and a boundary generic field v = 0 on it, we get |∂ + 3 X(v)| ≥ 2χ(X) − 2, provided ∂ + 1 X(v) ≈ D 2 . Thus, as a positive χ(X) increases, the boundary of the disk ∂ + 1 X(v) becomes more "wavily". If X is the Poincaré homology 3-sphere with a 3-ball being deleted, then by Corollary 4.4 [K], |∂ + 3 X(v)| > 0 for any gradient-like field v = 0 such that ∂ + 1 X(v) ≈ D 2 . These examples motivate Question 3.1 For boundary generic gradient-like fields v with a fixed value i of the index Ind(v) and a disk-shaped stratification {∂ + j X(v)} 1≤j<n as in Corollary 3.3, what is the minimum µ(X, i) of |∂ + n+1 X(v)|? Evidently, such number µ(X, i) is an invariant of the diffeomorphism type of X. It seems that µ(X, i) is semi-additive under the connected sum operation: that is,

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 v (see Definition 2.1), we say that v is boundary s-convex, if ∂ + s X = ∅. In particular, if ∂ + 2 X = ∅, we say that v is boundary 2-convex, or just boundary convex.
We say that v is boundary s-concave, if ∂ − s X = ∅. In particular, if ∂ − 2 X = ∅, we say that v 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 Y , given by an inequality {x : h(x) ≥ 0}, where h : Y → R is a smooth function with 0 being a regular value. Then the boundary convexity of a gradient field v := ∇f in X can be expressed in terms of the Hessian matrix Hess(h) by the inequality where v(x) is tangent to ∂X, then the field v 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 f -critical point, is boundary concave with respect to the gradient field v = ∇f . In fact, the field v is both boundary 3-concave and 3-convex! So, if f : Y → R is a Morse function on a closed manifold Y with a critical set Σ, then the complement X in Y to a small locally convex neighborhood of Σ admits a boundary concave f | X -gradient-like field (with ∂ 3 X = ∅)! 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" X (or even the space X itself) from the appropriate flow-generated structures ("observables") on its boundary ∂X. 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 f : ∂ 1 X → R of opposite polarities can serve as an indicator of the convexity/concavity of the gradient field ∇f in X (recall that the convexity/concavity properties of the v-flow do require knowing the field in the vicinity of ∂ 1 X in X!).
Theorem 4.1. Let f : X → R, f | : ∂ 1 X → R be Morse functions and v and v 1 their gradient fields with respect to a Riemannian metric g on X and its restriction to ∂ 1 X, respectively. Assume that v is boundary generic.
Proof. First consider the convex case, that is, the relation between the property ∂ + 2 X(v) = ∅ and the absence of an ascending v 1 -trajectory γ : R → ∂ 1 X which connects Σ − 1 to Σ + 1 . Consider the function h : ∂ 1 X → R, defined via the formula v = v 1 + h · n, where n denotes a unitary field inward normal to ∂ 1 X in X. Since v is boundary generic, zero is a regular value of h. Then Figure 5.
On the other hand, if no such v 1 -trajectory γ exists, then we claim the existence of a codimension one closed submanifold N ⊂ ∂ 1 X, which separates ∂ 1 X in two manifolds, A ⊃ Σ + 1 and B ⊃ Σ − 1 (∂A = N = ∂B), such that the vector field v 1 , or rather its perturbationṽ 1 , is transversal to N and points outward of A. Indeed, for each critical point x ∈ Σ + 1 , in the local Morse coordinates (y 1 , . . . , y n ) on ∂ 1 X, consider a small closed ǫ-disk D n ǫ (x) = { k y 2 k = ǫ 2 } centered on the critical point x. Denote by U ǫ (x) ⊂ ∂ 1 X the closure of the union of downward trajectories of the v 1 -flow passing through the points of D n ǫ (x) (see Fig. 6, the left diagram). Let A ǫ be the union ∪ x∈Σ + 1 U ǫ (x) (see Fig. 6, the right diagram).
Since we assume that no descending v 1 -trajectory γ links a point of Σ + 1 to a point of Σ − 1 , we can choose the disks {D n ǫ (x)} x∈Σ + 1 so small that the set Σ − 1 belongs to the complement ∂ 1 X \ A ǫ .
For each x ∈ Σ + 1 , the zero cone {Hess x (f | ∂ 1 X ) = 0} of the Morse function f | ∂ 1 X separates the sphere ∂D n ǫ (x) into two handles, H − ǫ (x) and H + ǫ (x) (each being a product of a sphere with a disk). We denote by H − (x) the handle in ∂D n ǫ (x) whose spherical core is formed by the intersection of the unstable disk through x with the sphere ∂D n ǫ (x). Then, by definition, the set U ǫ (x) is a collection of downward trajectories through the points of H − (x) union with D n ǫ (x). Note that the downward trajectories from a different set U ǫ (y) could enter the disk D n ǫ (x) only through the complementary handle H + (x) := ∂D n ǫ (x) \ H − (x) in its boundary. As a result, U ǫ (x) ∪ U ǫ (y) is a manifold whose piecewise smooth boundary could have corners (see Fig. 6, the right diagram) Similarly, A ǫ is a domain in ∂ 1 X whose boundary is piecewise smooth manifold with corners.
Since A ǫ consists of the downward trajectories of v 1 , if x ∈ A ǫ , then any point y ∈ γ x which can be reached from x following the field −v 1 (for short, "is below x") belongs to A ǫ as well. Therefore the boundary ∂A ǫ is assembled either from downward trajectories or from singletons; the singletons are contributed by some portions of x∈Σ + 1 ∂D n ǫ (x) where v 1 points outside of the relevant disk D n ǫ (x). Thus either v 1 is tangent to ∂A ǫ , or it points outside A ǫ .
Away from Σ + 1 ∪ Σ − 1 , v 1 = 0 is of the f -gradient type. Thus, in each tangent space T x (∂ 1 X), where x ∈ ∂ 1 X \ Σ 1 , there is an open cone C x (f ) comprised of f -gradient type vectors, and v 1 (x) ∈ C x (f ). Therefore, in the vicinity of ∂A ǫ , we can perturb v 1 to a new fieldṽ 1 of the f -gradient type, so thatṽ 1 points strictly outside A ǫ and stillṽ 1 (x) ∈ C x (f ) for all x ∈ ∂A ǫ . It is possible to smoothen the boundary ∂A ǫ so that, with respect to a new smooth boundary ∂Ã ǫ , the fieldṽ 1 still points outsideÃ ǫ ⊃ Σ + 1 , the new domain bounded by ∂Ã ǫ , andṽ 1 (x) ∈ C x (f ) for all x ∈ ∂Ã ǫ .
Note that if f (Σ + 1 ) < c < f (Σ − 1 ), then N := f −1 (c) can serve as a separator. LetÃ :=Ã ǫ andB := ∂ 1 X \Ã. With the separator N = ∂Ã in place, consider a smooth functionh : ∂ 1 X → R with the properties: (1) zero is a regular value ofh, andh −1 (0) = N , ṽ :=ṽ 1 +h · n ∈ C(f ), where n is the inward normal to ∂ 1 X in X. Note that the fieldṽ points inside of X alongÃ and outside of X alongB. It also points outside ofÃ along N =Ã∩B. As a result, we conclude that ∂ − 2 X(ṽ) = N and ∂ + 2 X(ṽ) = ∅; in other words,ṽ is boundary convex. Note thatṽ 1 can be chosen arbitrary close to v 1 . Ineeded, employing Theorem 3.2, we can perturbṽ 1 to insure its genericity with respect to the pair (∂ + 1 X(ṽ), ∂ 2 X(ṽ)), and thus the boundary genericity ofṽ itself. The argument in the concave case, which deals with the relation between the property ∂ − 2 X(v) = ∅ and the absence of an ascending v 1 -trajectory γ : R → ∂ 1 X, connecting Σ + 1 to Σ − 1 , 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 ω be a differential 1-form on a manifold Y .
We say that a path γ : [0, 1] → Y is ω-positive (ω-negative), if , ω(γ(t)) > 0 (< 0) for all values of the parameter t ∈ (0, 1). Definition 4.3. Let ω be a closed differential 1-form on a manifold Y , equipped with a Riemannian metric g. We say that a vector field v on Y is the gradient of ω (and denote it "∇ g ω"), if ω(w) = v, w g for any vector field w on Y .
Definition 4.4. Let ω be a differential 1-form on a manifold Y and let Σ ω be the set of points y ∈ Y , where ω : T y Y → R is the zero map. Assume that ω = df for some smooth function f in the vicinity of Σ ω .
We say that a vector field v is of ω-gradient type if ω(v) > 0 on Y \ Σ ω and v = ∇ g f in the vicinity of Σ ω . Here g is some Riemannian metric in the vicinity of Σ ω (cf. Definition 2.2).
We are in position to formulate a generalization of Theorem 4.1 for closed differential 1-forms-another instance of somewhat weaker "holographic phenomenon", this time for fields which may not be gradient-like globally.
Theorem 4.2. Let ω be a closed 1-form on a compact manifold X, equipped with a Riemannian metric g. Assume that ω and ω| ∂ 1 X have only Morse-type singularities. Let the gradient v := ∇ g ω be a boundary generic field, and let v 1 := ∇ g| ∂ 1 X (ω| ∂ 1 X ).
If ∂ ± 2 X(v) = ∅, then there is no ω-ascending v 1 -trajectory γ ⊂ ∂ 1 X, such that lim Assume that there exists a codimension one submanifold N ⊂ ∂ 1 X, which separates Σ + 1 and Σ − 1 and such that the field v 1 is transversal to N and points outwards/inwards of the domain in ∂ 1 X that is bounded by N and contains Σ + 1 . Then one can deform the ω-gradient vector fields (v, v 1 ) to a new boundary generic pair (ṽ,ṽ 1 ) of the ω-gradient type so that ∂ ± 2 X(ṽ) = ∅. Proof. The (ω| ∂ 1 X )-gradient fields v 1 on ∂ 1 X are characterized by the property ω(v 1 ) > 0, valid on the locus where ω| ∂ 1 X = 0. Usually, in this setting, we do not have a natural choice for the wall N ⊂ ∂ 1 X which would separate the singularities of opposite polarities Σ + 1 = Σ + 1 (ω) and Σ − 1 = Σ − 1 (ω) and such that the field v 1 would be transversal to N . It seems unlikely that the absence of an ascending v 1 -trajectory which links Σ − 1 with Σ + 1 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.
Given a metric g on a Riemannian (n + 1)-manifold X, let us recall a definition of the Hodge Star Operator * g : T * (X) → n T * (X).
Assume that, in the dual to α basis α * of T (X), the metric g is locally given by a matrix g = (g ik ). Then the matrix G of the * g -operator in the bases α, α ∨ is given by the formula G = det(g) · g −1 (4.1) , whence det(G) = (det(g)) n−1 2 .
Definition 4.5. A closed differential 1-form ω on a compact manifold Y is called intrinsically harmonic if there exists a Riemannian metric g on Y such that the form * g (ω) is closed.
Example 4.3. Let Y be a closed smooth manifold and H : Y → S 1 a smooth map with isolated Morse-type singularities. Consider the closed 1-form ω := H * (dθ), the pull-back of the canonic 1-form dθ on the circle S 1 . Assume that one of the H-fibers, F 0 := H −1 ( * ), is connected. Then ω is intrinsically harmonic [FKL].
Let Σ ω denote the singularity set of a closed 1-form ω on a compact manifold Y . We assume that Σ ω ⊂ int(Y ). By Calabi's Proposition 1 [Ca], ω is intrinsically harmonic if and only if through every point y ∈ Y \ Σ ω there is a ω-positive path γ which either is a loop, or which starts and terminates at the boundary ∂Y .
Theorem 4.3. Let ω be a closed 1-form on a Riemannian manifold X, such that Σ ω ⊂ int(X). Assume that ω| ∂ 1 X , the restriction of ω to T (∂ 1 X), is a harmonic form 7 .
Then the gradient field v := ∇ω is not boundary convex or boundary concave (that is, Proof. We abbreviate ∂ ± j X(v) to ∂ ± j X and * g| ∂ 1 X to * ∂ . Here * ∂ is the * -operator on the boundary of X with respect to the given Riemannian metric g on X.
Example 4.4. Let X be a compact smooth manifold and H : X → S 1 a smooth map with isolated Morse-type singularities. Consider the closed 1-form ω := H * (dθ), the pullback of the canonic 1-form dθ on the circle S 1 . Assume that one of the fibers of the map H : ∂ 1 X → S 1 is connected. Then there exists a metric g on X such that the form ω ∂ := H * (dθ)| ∂ 1 X is harmonic ( [Ca], [FKL]). Consider the gradient field v := ∇ g (ω). Then by Theorem 4.3, ∂ + 2 X(v) = ∅ and ∂ − 2 X(v) = ∅ for any metric g that "harmonizes" ω ∂ .
Definition 4.6. A non-vanishing vector field v on a compact manifold X is called traversing if each v-trajectory is either a closed segment or a singleton which belongs to ∂X.
Remark 4.2. The definition excludes fields with zeros in X (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 gradientlike fields of nonsingular closed 1-forms may not be traversing! Lemma 4.1. Any traversing vector field is of the gradient type.
Proof. Let v be a traversing field on X. We extend the pair (X, v) to a pair (X,v) so that X is properly contained inX andv = 0. Then every v-trajectory γ ⋆ ⊂ X has a local transversal compact section S γ ⋆ ⊂ int(X) of thev-flow. We can choose S γ ⋆ to be diffeomorphic to a n-dimensional ball with its center at the singleton γ ⋆ ∩ S γ ⋆ . We denote byŨ γ ⋆ the union ofv-trajectories through S γ ⋆ .
Then the collection U := {Û γ ⋆ ∩ X} γ ⋆ forms a cover of X. Since X ⊂ int(X) is compact, we can choose a finite subcover U ′ ⊂ U of X.
For eachÛ γ ⋆ ∩ X ∈ U ′ and the corresponding section S γ ⋆ , we produce a smooth function φ γ ⋆ :Û γ ⋆ → R by integrating the vector fieldv and using S γ ⋆ as the initial location for the integration. More accurately, let for all τ ∈ [−a γ ⋆ , b γ ⋆ ] and ψγ(0) =γ ∩ S γ ⋆ . This bijective parametrization introduces a smooth product structure Φ : . We define a smooth functionφ γ ⋆ :Û γ ⋆ → R by the formula x → ψ −1 γx (x) and denote it (quite appropriately) by the symbol x S γ ⋆v . Let χ γ ⋆ : S γ ⋆ → R + be a smooth non-negative function that vanishes only on the boundary ∂S γ ⋆ . Letχ γ ⋆ :Û γ ⋆ → R + denote the composition of thev-directed projection π γ ⋆ :Û γ ⋆ → S γ ⋆ with the function χ γ ⋆ . Sinceχ γ ⋆ vanishes on ∂Û γ ⋆ ∩ X, the function extends smoothly on X to produce a smooth functionχ γ ⋆ : X → R + with the support in U γ ⋆ ∩ X. Now consider the smooth function It is well-defined on X. Let us compute its v-directional derivative: Let us explain formula 4.3. By the very definition ofχ γ ⋆ , it is constant on eachvtrajectory, so that L v (χ γ ⋆ ) = 0. Also,χ γ ⋆ > 0 in int(Û γ ⋆ ). At the same time, L v x S γ ⋆v > 0, since d dt ψγ =v(ψγ) = 0 and ψγ increases in the direction of v. Finally, each x ∈ X belongs to the interior of some setÛ γ ⋆ . Therefore Corollary 4.1. Let X be a smooth compact manifold with boundary. Then V trav (X)-the space of traversing vector fields on X-is nonempty and coincides with the intersection V grad (X) ∩ V =0 (X), where V grad (X) denotes the space of gradient-like fields, and V =0 (X) the space of all non-vanishing fields on X.
Proof. By definition, any traversing field v on X does not vanish. By Lemma 4.1, v must be of the gradient type. Thus On the other hand, for a compact X with a gradient-like v = 0, each v-trajectory γ x through x ∈ int(X) must reach the boundary in both finite positive and negative times (since it is controlled by some Lyapunov function f ).
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 v imposes severe restrictions on the topology of the manifold X. Figure 7. The existence of a traversing boundary convex field v (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.
Lemma 4.2. A connected (n + 1)-manifold X admits a boundary convex traversing 8 field v, 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. By Lemma 4.2, X is diffeomorphic to a product of a fake n-disk Y with [0, 1], the corners of the product being rounded.
For n = 4, we do not know whether Y is a standard 4-disk. For n ≥ 5, the h-cobordism theorem [Sm] 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 H n (X; Z) = 0 is an obstruction to finding boundary convex traversing v on a (n + 1)-dimensional manifold X with a connected boundary.
Corollary 4.3. Let X be a smooth connected compact (n + 1)-manifold with boundary, which admits a boundary convex traversing field.
If H n (X; Z) = 0, then X is diffeomorphic to the product Y × [0, 1], where Y is a closed manifold.
In particular, no connected X with boundary ∂X = ∅, whose number of connected components differs from two, and with the property H n (X; Z) = 0 admits a boundary convex traversing field.
Proof. If such boundary convex traversing field v exists, ∂ + 1 X must be a deformation retract of X. Therefore, for a connected X, ∂ + 1 X must be connected as well. On the other hand, if ∂(∂ + 1 X) = ∂ 2 X = ∅, then the connected ∂ + 1 X must be of a homotopy type of a (n−1)-dimensional complex. In such a case, the groups H n (∂ + 1 X; Z) ≈ H n (X; Z) must vanish.
Thus when H n (X; Z) = 0 and v is boundary convex, the only remaining option is ∂ 2 X = ∅, which implies that ∂(∂ + 1 X) = ∅-the manifold ∂ + 1 X is closed. In such a case, X is a product of a connected closed n-manifold with an interval; so the boundary ∂X 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 D 2 does not admit a non-vanishing boundary concave field (see Example 4.4).
Lemma 4.3. If a boundary generic vector field v on an even-dimensional compact orientable manifold X is boundary concave, then its index If a boundary generic vector field v on an odd-dimensional compact orientable manifold X is boundary concave, then its index Ind(v) = χ(X) − χ(∂ + 1 X). Thus, for all boundary concave fields v with a fixed value of index Ind(v), the Euler number χ(∂ + 1 X) is a topological invariant. Proof. For a boundary concave field v, ∂(∂ + 1 X) := ∂ 2 X = ∂ + 2 X. Therefore, the Morse formula 2.2 reduces to the equation Recall that, for any orientable odd-dimensional manifold Y , χ(∂Y ) = 2·χ(Y ). Therefore, when dim(X) ≡ 0 mod 2, we get 2 · χ(∂ + 1 X) = χ(∂ 2 X). Thus formula 4.4 transforms into For an odd-dimensional X, the closed manifold ∂ 2 X is odd-dimensional, so χ(∂ 2 X) = 0. Therefore Corollary 4.4. Let X be a 4-dimensional oriented smooth and compact manifold with boundary.
Example 4.4. Let X = D 2 , the 2-dimensional ball. If v = 0 on X, then by the Morse formula, 1 − χ(∂ + 1 X) + χ(∂ + 2 X) = 0. If ∂ + 1 X consists of k arcs, then by this formula, #(∂ + 2 X) = k − 1. At the same time, #(∂ 2 X) = 2k. Therefore, #(∂ − 2 X) = k + 1 > 0. So we conclude that D 2 does not admit a non-vanishing field with ∂ − 2 X = ∅, that is, a boundary concave field. At the same time, if we delete any number of disjoint open disks from D 2 , 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 v on A, being restricted to X, is evidently of the gradient type and concave with respect to ∂X.
Note that, if a connected compact surface X admits a generic traversing concave field v, then X is homeomorphic either to a thickening of a finite graph Γ whose vertexes all have valency 3, or to an annulus.
In the previous example, we have seen that the disk D 2 does not admit a non-vanishing concave field. In contrast, D 3 does admit a boundary generic concave non-vanishing field: just consider the restriction of the Hopf field v on S 3 to the northern hemisphere D 3 ⊂ S 3 . For the unitary disk D 3 ⊂ R 3 centered at the origin, informally, we can describe v as the sum of the velocity field of the solid D 3 , spinning around the z-axis, with the solenoidal field of the loop L := {x 2 + y 2 = 4/9, z = 0}. However, this field v is not of the traversing type: it has closed trajectories (residing in the solid torus dist(∼, L) ≤ 1/3).
These observations encourage us to formulate Conjecture 4.1. The standard (n + 1)-disk D n+1 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 n-manifold Y . Let {Z i ⊂ Y } 1≤i≤s be compact submanifolds also of dimension n. Let W := Y × [0, 1]. We pick s disjointed close intervals {I i } i in the interval [0, 1]. Then we form the product U i := Z i × I i . By rounding the corners of U i , we get a (n + 1)-manifold V i ⊂ U i so that each segment z × I i , where z ∈ Int(Z i ), hits V i along a closed segment, and each segment z × I i , where z ∈ ∂(Z i ), hits V i along a singleton.
Form the manifold X := W \ i V i . Its boundary consists of two copies of Y together with the disjoint union of ∂V i (they are the doubles of Z i 's). The obvious vertical field v on W , being restricted to X, is boundary concave. In fact, 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 [BP], [K]).
We conjecture that an (n + 1)-dimensional X admits a traversing concave field v such that ∂ 2 X(v) = ∂ + 2 X(v) = ∅ if (perhaps, if and only if) X has a "special trivalent" simple n-dimensional spine K ⊂ T X , where T X denotes a smooth triangulation of X (see [Ma] for the definitions of simple spines and for the description of their local topology). Here "special trivalent" means that each (n − 1)-simplex from the singular set SK of K is adjacent to exactly three n-simplexes from K. Moreover, the vicinity of SK in K admits an oriented branching as in [BP].
When the (n + 1)-manifold in question is specially manufactured from a closed (n + 1)manifold by removing a number of (n + 1)-disks, another paper from this series will provide us with a wast gallery of manifolds which admit traversing concave fields.

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 [K] that, for every 3-fold X, any boundary generic v of the gradient type can be deformed into new such fieldṽ with ∂ 3 X(ṽ) = ∅. However, in conjunction with certain topological constraints on ∂ + 1 X (like being connected), the boundary 3-convexity has topological implications (see [K], 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 {∂ + j X(v) ⊂ ∂ j X(v)} j are realizable by boundary generic traversing fields v on X? 9 • Given two such fields, v 0 and v 1 , can we find a linking path {v t } t∈ [0,1] in the space V trav (X) that avoids certain types of singularities? 10 Specifically, if for some j > 0, ∂ j X(v 0 ) = ∅ = ∂ j X(v 1 ), is there a linking path so that ∂ j+1 X(v t ) = ∅ for all t ∈ [0, 1]?
Remark 5.1. The property of the field v 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 {∂ + j X(v) ⊂ ∂ j X(v)} j . Let us illustrate this remark for the fields v such that ∂ 3 X(v) = ∅. We divide the boundary ∂ 1 X into two complementary domains, Y + and Y − , which share a common boundary ∂Y + = ∂Y − -a closed manifold of dimension n − 1. It may have several connected components. Next, we divide the manifold ∂Y + into two complementary closed manifolds Z + and Z − . We claim that it is possible to find a boundary generic field v with the properties: ∂ ± 1 X(v) = Y ± , ∂ ± 2 X(v) = Z ± , and ∂ 3 X(v) = ∅. The construction of such v is quite familiar (see the arguments in Theorem 3.2).
We start with a field ν 1 which is normal to ∂Y + and points outside of Y + along Z − and inside of Y + along Z + . We extend ν 1 to a field v 1 tangent to the boundary ∂ 1 X so that v 1 has only isolated zeros. Let ν be the outward normal field of ∂ 1 X in X and h : ∂ 1 X → R a smooth function such that 0 is its regular value and Along ∂ 1 X, form the field v ′ = v 1 + h · ν and extend it to a field v on X with isolated singularities in int(X). By its construction, v has all the desired properties. Note that here we do not insist on the property v = 0.
In our inquiry, we are inspired by the Eliashberg surgery theory of folding maps [E1], [E2]. In many cases, Eliashberg's results give criteria for realizing given patterns of ∂ ± 2 X ⊂ ∂ ± 1 X, provided that ∂ 3 X = ∅, thus answering Question 5.1. Let us state one such result, Theorem 5.3 from [E2].
Theorem 5.1 (Eliashberg). Let X ⊂ R n+1 , n ≥ 2, be a compact connected smooth submanifold of dimension (n + 1). Consider two disjoint closed and nonempty (n − 1)submanifolds Z + and Z − of ∂X whose union separates ∂X into two complementary nmanifolds, Y + and Y − . Let ν be the outward normal field of ∂X in X, and denote by deg(ν) the degree of the Gauss map G ν : ∂X → S n . Let π : R n+1 → R n be a linear surjection.
Considering a traversing field v = 0 which is tangent to the fibers of the map π • h 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 v on X so that: Thus, at least for smooth domains X ⊂ R n+1 and for boundary generic traversing fields v, which are both 3-convex and 3-concave, the patterns for the strata are indeed very flexible. However, the requirement that both Z + = ∅ and Z − = ∅ puts breaks on any applcation of Corollary 5.1 to boundary concave and boundary convex traversing fields on X! Example 5.1. Let us illustrate how non-trivial the conclusions of Theorem 5.1 and Corollary 5.1 are.
We suspect that an important for our program generalization of Theorem 5.1 is valid and can be established by the methods as in [E1], [E2].
Conjecture 5.1. Let X be a compact connected smooth manifold of dimension n + 1 ≥ 3, equipped with a traversing vector field v. Let Z + and Z − be two disjoint closed and nonempty (n − 1)-submanifolds of ∂X whose union separates ∂X into two n-manifolds, Y + and Y − .
Then the topological constraints χ(Y + ) = χ(X), when n ≡ 0 mod 2 (5.1) χ(Z + ) − χ(Z − ) = 2 · χ(X), when n ≡ 1 mod 2 (5.2) are necessary and sufficient for the existence of an orientation-preserving diffeomorphism h : X → int(X) with the following properties: • the restriction of v to the image h(X) is boundary generic in the sense of Definition 2.1 11 , Moreover, in a given collar U of ∂X in X, there is a U -supported diffeomorphism h as above which is arbitrary close in the C 0 -topology to the identity map.
To prove the necessity of the topological constraints 5.1 and 5.2 is straightforward. By the Morse formula 5.2 (see also Corollary 5.1), a necessary condition for the existence of a diffeomorphism h with the desired properties, described in the bullets, is the constraint χ(h(X)) − χ(h(Y + )) + χ(h(Z + )) = i(v| h(X) ) = 0.
To prove the sufficiency of these conditions may require a clever application of the hprinciple in the spirit of [E1], [E2].
Corollary 5.2. Assuming the validity of Conjecture 5.1, any compact smooth manifold X with boundary admits a boundary generic traversing field v with the property ∂ 3 X(v) = ∅.
Proof. By Corollary 4.1, V trav (X) = ∅. So we can start with a traversing field v and apply Conjecture 5.1 to it to get the pull-back field h * (v) with the desired properties.