Causal Holography in Application to the Inverse Scattering Problems
For a given smooth compact manifold , we introduce an open class of Riemannian metrics, which we call metrics of the gradient type. For such metrics , the geodesic flow on the spherical tangent bundle admits a Lyapunov function (so the -flow is traversing). It turns out, that metrics of the gradient type are exactly the non-trapping metrics.
For every , the geodesic scattering along the boundary can be expressed in terms of the scattering map . It acts from a domain in the boundary to the complementary domain , both domains being diffeomorphic. We prove that, for a boundary generic metric the map allows for a reconstruction of and of the geodesic foliation on it, up to a homeomorphism (often a diffeomorphism).
Also, for such , the knowledge of the scattering map makes it possible to recover the homology of , the Gromov simplicial semi-norm on it, and the fundamental group of . Additionally, allows to reconstruct the naturally stratified topological type of the space of geodesics on .
We aim to understand the constraints on , under which the scattering data allow for a reconstruction of and the metric on it, up to a natural action of the diffeomorphism group . In particular, we consider a closed Riemannian -manifold which is locally symmetric and of negative sectional curvature. Let is obtained from by removing an -domain , such that the metric is boundary generic, of the gradient type, and the homomorphism of the fundamental groups is trivial. Then we prove that the scattering map makes it possible to recover and the metric on it.
Let be a compact connected and smooth Riemannian -manifold with boundary. In this paper, we apply the Holographic Causality Principle ([K4], Theorem 3.1) to the geodesic flow on the space of unit tangent vectors on .
Our main observation is that the holographic causality is intimately linked to the classical inverse scattering problems. So the geodesic scattering is the focus of our present investigation.
Let us briefly explain what we mean by the scattering data on a given compact connected Riemannian manifold with boundary . For each geodesic curve which “enters” through a point in the direction of an unitary tangent vector , we register the first along “exit point” and the exit direction, given by a unitary tangent vector at . Of course, not for any geodesic on , this construction makes sense: may belong to the interior of . In such case, the geodesic through never reaches the boundary again.
In any case, when available, we call the correspondence “the metric-induced scattering data”.
We strive to restore the metric on , up to the action of -diffeomorphisms that are the identity maps on , from the scattering data111This resembles the problem of reconstructing the mass distribution from the gravitational lensing.. This restoration seems harder when has closed geodesics or geodesics that originate at a boundary point, but never reach the boundary again.
In special cases, the restoration of is possible. This conclusion is very much inline with the results from [Cr], [Cr1], [We], as well as with [SU] - [SU4] and [SUV] - [SUV3]. The recent paper [SUV3], which reflects the modern state of the art, contains the strongest results.
Recall that there are examples of two analytic Riemannian manifolds with isometric boundaries and identical scattering (even lens) data, but with different -jets of the metric tensors at the boundaries (see [Zh], Theorem 4.3)! However, these examples have trapped geodesics; the metrics there are fundamentally different from the ones we study here.
Moving towards the goal of -reconstruction from the scattering data, we introduce a class of metrics which we call metrics of the gradient type (see Definition 2.1). By Lemma lem9.2, the the gradient type metrics are exactly the nontrapping metrics. In Theorem 2.1, we prove that, given any compact connected Riemannian -manifold with boundary that admits a -flat triangulation, where is a universal constant that depends only on (see Definition 2.3), it is possible to delete several smooth -balls from , so that is diffeomorphic to , and the restriction is of the gradient type. In particular, any connected with boundary admits a gradient type Riemannian metric , provided that admits a -flat triangulation for a sufficiently small and a different metric . The gradient type metrics form an open nonempty set in the space of all Riemannian metrics on (Corollary 2.2).
Then we introduce another class of Riemannian metrics on such that the boundary is “generically curved” in (see Definition 2.4). We call such geodesically boundary generic, or boundary generic for short. We denote by the space of geodesically boundary generic metrics of the gradient type. We speculate (see Conjecture 2.2) that, for any , the space is open and dense in and prove that it is indeed open (see Theorem 2.2).
We also consider a subspace , formed by metrics for which the geodesic vector field on is traversally generic in the sense of Definition 3.2 from [K2]. Again, is open in .
In Theorem 3.3, the main result of this paper, we prove that, for a metric , the geodesic flow on is topologically rigid for given scattering data. This means that when two scattering maps, and , are conjugated with the help of a smooth diffeomorphism , then the un-parametrized geodesic flows on and are conjugated with the help of an appropriate homeomorphism (often a diffeomorphism) which extends .
Let denote the space of geodesics in . In Theorem 3.4, we prove that, for any metric , the scattering data are sufficient for a reconstruction of the stratified topological type of . In general, is not a smooth manifold, but for , it is a compact -complex [K4]. For , the space carries some “surrogate smooth structure” [K4]. This structure is also captured by the scattering data.
In Theorem 3.5, we prove that, for any , the geodesic scattering map allows for a reconstruction of the homology spaces and , equipped with the Gromov simplicial semi-norms (see [G] for the definition). In particular, the simplicial volume of the relative fundamental cycle can be recovered from the scattering map.
If , the geodesic scattering map also allows for a reconstruction of the fundamental group , together with all the homotopy groups . Moreover, if the tangent bundle of is trivial, allows for a reconstruction of the stable topological type of the manifold .
Let be a closed smooth locally symmetric Riemannian -manifold, , of negative sectional curvature. In Theorem 3.7, we prove that, if and is obtained from by removing the interior of a smooth -ball so that is of the gradient type and boundary generic, then the knowledge of the scattering map
makes it possible to reconstruct and the metric , up to a positive scalar factor. However, this result does not imply the possibility of reconstructing from .
In Section 4, we study the inverse scattering problem in the presence of additional information about the lengths of geodesic curves that connect each point to the “scattered” point . This information is commonly called the “lens data”. Our main result here, Theorem 4.1, claims the strong topological rigidity (see Definition 3.1) of the geodesic flow for given lens data. The proof of the theorem requires additional hypotheses about the metric , which we call “ballanced” (see Definition 4.1). We apply Theorem 4.1 to the case of manifolds , obtained from closed Riemannian manifolds by removing special domains . Then the scattering problem on is intimately linked to the geodesic flow on (see the “Cut and Scatter” Theorem 4.2). By combining Theorem 4.2 with some classical results from [CK], [BCG], we are able to reconstruct from the scattering and lens data on , provided that either is locally symmetric and of negative curvature, or admits a non-vanishing parallel vector field (see Corollary 4.2 and Corollary 4.3).
In general, our approach to the inverse scattering problem relies more on the methods of Differential Topology and Singularity Theory, and less on the more analytical methods of Differential Geometry and Operator Theory. Of course, this topological approach has its limitations: by itself it allows only for a reconstruction of the geodesic flow from the scattering and lens data.
The assumption that the Riemannian manifolds in this study are smooth seems to be crucial for the effectiveness of our methods. Perhaps, similar results are valid under the weaker assumption that the -manifolds we investigate have a -differentiable structure.
2. Boundary Generic Metrics of the Gradient Type
Let be a compact -dimensional smooth Riemannian manifold with boundary, and a smooth Riemannian metric on . Let denote the tangent spherical bundle. With the help of , we may interpret the bundle is a subbundle of the tangent bundle .
The metric on induces a partially-defined one-parameter family of diffeomorphisms , the geodesic flow. Each unit tangent vector at a point determines a unique geodesic curve trough in the direction of . When , the geodesic curve is uniquely-defined for unit vectors that point inside of .
By definition, is the point such that the distance along from to is , and is the tangent vector to at . We stress that may not be well-defined for all and all : some geodesic curves may reach the boundary in finite time, and some tangent vectors may point outside of . However, such constraints are common to our enterprise ([K], [K1] - [K5]), which deals with such boundary-induced complications.
In the local coordinates on the tangent space , the equations of the geodesic flow are:
where is the metric tensor.
This system can be rewritten in terms of the Hamiltonian function
—the kinetic energy—in the familiar Hamiltonian form:
Let be the field on the manifold , tangent to the trajectories of the geodesic flow on . Note that -flow is tangent to , so .
In fact, the integral trajectories of are geodesic curves in the Sasaki metric on ([Be], Prop. 1.106).
Let be a compact smooth -manifold with boundary. Any smooth vector field on , which does not vanish along the boundary , gives rise to a partition of the boundary into two sets: the locus , where the field is directed inward of or is tangent to , and , where it is directed outwards or is tangent to . 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 (see [Mo]) that, for a generic vector field , the tangent locus inherits a similar structure in connection to , as has in connection to . That is, gives rise to a partition of into two sets: the locus , where the field is directed inward of or is tangent to , and , where it is directed outward of or is tangent to . Again, we assume that , viewed as a section of the quotient line bundle over , is transversal to its zero section.
For, so called, boundary generic vector fields (see [K1] for a formal definition), this structure replicates itself: the cuspidal locus is defined as the locus where is tangent to ; is divided into two manifolds, and . In , the field is directed inward of or is tangent to its boundary, in , outward of or is tangent to its boundary. We can repeat this construction until we reach the zero-dimensional stratum .
To achieve some uniformity in the notations, put and .
Thus a boundary generic vector field on gives rise to two stratifications:
the first one by closed smooth submanifolds, the second one—by compact ones. Here , and .
We will use often the notation “” instead of “” when the vector field is fixed or its choice is obvious.
As any non-vanishing vector field, the geodesic field divides the boundary into two portions: , where points inside of or is tangent to its boundary, and , where it points outside of or is tangent to its boundary. In fact, and do not depend on : the first locus is formed by the pairs , where and points inside of or is tangent to . Therefore, both and are homeomorphic to the tangent -disk bundle of the manifold . The locus is also -independent; it is the space of the sphere bundle, associated with the tangent -bundle .
Let be a compact connected smooth Riemannian manifold with boundary.
We say that a metric on is of the gradient type if the vector field that governs the geodesic flow is of the gradient type: that is, there exists a smooth function such that .
We denote by the space of the gradient-type Riemannian metrics on .
Example 2.1. Consider a flat metric on the torus and form a punctured torus by removing an open disk from . If is convex in the fundamental square domain , then there exist closed geodesics (with a rational slope with respect to the lattice ) that miss . For such , the flat metric is not of the gradient type.
However, it is possible to position so that its lift to will have intersections with any line that passes through (see Figure 1). We restrict the flat metric to . For such a choice of , thanks to Lemma 2.2 below, the metric is of the gradient type. Moreover, by Theorem 2.2, any metric on , sufficiently close to this flat metric , is also of the gradient type.
The set of Riemannian metrics of the gradient type is open in the space of all Riemannian metrics on , considered in the -topology.
Let denote the zero section. If on , then extends in a compact neighborhood of in to a smooth function so that in . In this neighborhood, for all metrics , sufficiently close to . For such metrics , the space of unit spheres is fiberwise close to ; in particular, we may assume that . Recall that the geodesic field is tangent to , thus on . ∎
A Riemannian metric on a connected compact manifold with boundary is called non-trapping if has no closed geodesics and no geodesics of an infinite length (the later are homeomorphic to an open or a semi-open interval).
Let be a compact connected smooth manifold with boundary. A metric on is of the gradient type, if and only if, any trajectory of the geodesic flow is homeomorphic to a closed interval or to a singleton.
In other words, the non-trapping metrics and the metrics of the gradient type are the same.
If, for a smooth function , , then each -trajectory is singleton residing in or a closed segment with its both ends residing in (we call such vector fields traversing). Evidently, prevents from having a closed trajectory. Let be a trajectory that starts at a point and is homeomorphic to a semi-open interval. So extends beyond any point on that can be reached from ( cannot “exit” in a finite time). Consider the closure of . It is a compact and -invariant set. So attends its maximum at a point . However, at and a germ of a -trajectory through belongs to , a contradiction to the assumption that attends its maximum at .
A similar argument rules out the -trajectories that are homeomorphic to a semi-open interval.
Conversely, by Lemma 5.6 from [K1], any traversing field is of the gradient type. So admits a Lyapunov function .
Thus is of the gradient type if and only if the image of any -trajectory under the map is either a singleton in or a compact geodesic curve whose ends reside in (by our convention, does not extends beyond its two ends). In particular, any has no closed geodesics in and no geodesics that originate at the boundary and are trapped in for all positive times. ∎
Let be an open Riemannian manifold such that no geodesic curve in is closed or has an end that is contained in a compact set. Let be a smooth compact codimension zero submanifold. Then the restriction is a metric of the gradient type, and so are all the metrics on that are sufficiently close to .
In particular, for any compact domain with a smooth boundary in the Euclidean space or in the hyperbolic space , the Euclidean metric or the hyperbolic metric on are of the gradient type, and so are all the metrics that are sufficiently close to or , respectively.
Using the hypotheses, no positive time geodesic in the metric is an image of a semi-open interval or a closed loop. By Lemma 2.2, the pair is of the gradient type.
By Lemma 2.1, any metric on , which is sufficiently close to , is of the gradient type as well. ∎
In order to prove Theorem 2.1 below, we will need few lemmas, dealing with smooth triangulations of compact smooth Riemannian manifolds, triangulations that are specially adjusted to the given metric.
Let be a smooth compact -manifold. A smooth triangulation is a homeomorphism from a finite simplicial complex to . The triangulation is assembled out of several homeomorphisms , where denotes of the standard -simplex , the restriction of to the interior of each subsimplex being a smooth diffeomorphism. The homeomorphisms commute with affine maps of subsimplicies , the maps that assemble out of several copies of .
Consider a smooth triangulation of a smooth compact -manifold with a Riemannian metric .
Let be a geodesic arc, and its preimage in the standard -simplex . We denote by its length in the Euclidean metric on . Let be a line segment that shares its ends with . We denote by its length.
Pick a number . We say that a smooth triangulation is -flat with respect to if, for each index and any geodesic arc , the inequality is valid:
Note that the inequality in this definition remains valid under the conformal scaling of the simplex .
Let be a compact smooth -manifold, equipped with a Riemannian metric , and a convex domain in , equipped with the Euclidean metric . Consider a diffeomorphism . Let be a geodesic arc and its -preimage. We denote by the segment in that shares its ends with the arc .
Assume that, for some and any geodesic arc , the Euclidean lengths of and satisfy the inequality
Then the arc is contained in the -neighborhood of the segment .
Put . Consider the hyperplane , perpendicular to the segment at a typical point . That hyperplane must hit the curve at least at one point . Consider the maximum
Let be the point where this maximum is achieved. Put . We form two right triangles, and , with the verticies and , respectively. Consider the Euclidean distances:
and put , . Then and .
Since is the distance between and , and is the distance between and , we get
By the hypotheses, the LHS of this inequality does not exceed . Hence
Let . Then . This substitution transforms the previous inequalities into the following one:
So we get the desired inequality ∎
Let be the standard -simplex, residing in and equipped with the Euclidean metric. We denote by and the first and the second barycentric subdivisions of . For any vertex , we form its star in . For any , we denote by the -homothetic image of , the center of homothety being at . Consider the set
By a line in we mean an intersection of an affine line in with .
There exists a number such that every line has a point that belongs to the interior of .222It is desirable to find an elementary argument that explicitly computes as a function of .
Let be a typical simplex of . It will suffice to show that there exists an universal , such that any line has a point that belongs to the interior of some set , for a vertex .
For each , consider the polyhedron . For each simplex of , put .
The space of lines in is compact; in fact, it is a continuous image of a compact subset of the Grassmanian . consists of the -planes through the point that have a nonempty intersections with . Here the hyperplane is defined by equating the first coordinate with zero. This construction gives rise to a continuous map . Any line whose intersection with is not a singleton (equivalently whose intersection with is not a vertex of ), defines a unique point in .
Consider an increasing sequence that converges to 1. Contrarily to the claim of the lemma, assume that for each , there exists a line that is contained in the polyhedron .
Using compactness of , there exists a subsequence that converges to a limiting line . Then . Note that is missing the verticies of . On the other hand, by the construction of , if a line has a pair of distinct points such that the segment , then .
This contradiction proves that exists for which no line in is contained in . The definition of the polyhedron is given by an affine (metric-indepentent) construction. So the polyhedra for different match automatically.
Thus there is a number such that, for each , every line hits the set ∎
Let be a compact smooth Riemannian manifold. For any sufficiently small , admits a smooth -flat triangulation .
Let be as in Lemma 2.4. Put , and let denote the Euclidean distance between the sets and in the standard simplex .
Let be a closed connected smooth Riemannian -manifold that admits a -flat smooth triangulation333By Conjecture 2.2, any will do.. Then there exists a smooth -ball such that the restriction of the metric to is of the gradient type.
If is a compact connected Riemannian -manifold with boundary that admits a -flat smooth triangulation. Then, for each connected component of the boundary , there exists a relative -ball , where is the -ball, so that all the balls are disjointed and the restriction of the metric to the manifold is of the gradient type. The manifolds and are diffeomorphic.
The idea is first to construct a number of disjointed balls in so that each geodesic curve will hit some ball. Thus deleting such balls from will produce a “geodesically traversing swiss cheese”. When is closed, we will incorporate all the balls into a single smooth one. When has a boundary, then the balls will be incorporated in a domain, whose removal from does not change the smooth topology of .
Let and denote the first and second barycentric subdivisions of a given smooth triangulation . As before, is assembled from a collection of singular simplicies .
Put , . By the hypotheses, there exists a smooth -flat smooth triangulation .
Then, for any geodesic curve , its -preimage in the simplex is contained in the -neighborhood of a line . By Lemma 2.4, that line contains a point whose -neighborhood is contained in the set . So, by Definition 2.3, the curve must intersect the neighborhood . As a result, has a non-empty intersection with the set , a finite disjoint union of -dimensional -balls , centered on the vertices of in .
We can smoothen their boundaries by encapsulating each ball into a smooth ball so that for all distinct .
When is closed, we place the disjoint union inside of a single smooth ball . This may be accomplished by attaching -handles to so that the cores of the handles form a tree. Any geodesic in hits since it hits .
In the case of a non-empty boundary , by attaching first some relative -handles , whose cores reside in , transforms into a disjoint union of several -balls, each ball residing in its connected component of . Again, in each component of , the attaching the handles is guided by a tree.
In the process, we incapsulate into a disjoint union of several smooth balls that reside in the interior of and several relative balls, each of which is touching the corresponding boundary component . These relative balls are in 1-to-1 correspondence with the boundary components. Then connecting the balls in the interior on to the balls that touch the boundary by -handles (which reside in ) produces the desired relative pairs , one pair per component of . Again, any geodesic in hits the disjoint union of these pairs. The removal of the union from results in a smooth manifold , which is diffeomorphic to . ∎
If a compact connected smooth Riemannian -manifold with boundary admits a -flat triangulation, then admits a Riemannian metric of the gradient type.
In fact, the subspace of such metrics is nonempty and open in the space of all Riemannian metrics on .
Note that the smooth balls, whose removal from (the “geodesic Swiss cheese”) delivers, by Theorem 2.1, a metric of the gradient type on , are not necessarily convex in the original metric .
For any compact smooth Riemannian manifold , there exists a finite disjoint union of smooth convex balls whose removal from delivers the metric of the gradient type on their complement .
Let be a compact connected Riemannian manifold with boundary.
We say that a metric on is geodesically traversally generic if the vector field is of the gradient type and is traversally generic555See Definition 3.2 from [K2]. with respect to .
We denote the space of all gradient type metrics on by the symbol , the space of all geodesically boundary generic metrics of the gradient type on by the symbol , and the space of all geodesically traversally generic metrics on by the symbol . So we get .
Remark 2.2. If is such that there exists a geodesic curve whose arc is contained in , then the metric is not geodesically boundary generic in the sense of Definition 2.4. For example, the Euclidean metric on is not geodesicly boundary generic with respect to the ruled surface : indeed, is comprised of lines (geodesics).
Example 2.2. Let be a domain in the Euclidean plane , bounded by simple smooth closed curves. Then the flat metric is boundary generic on if and only if is comprised of strictly concave and convex loops or arcs that are separated by the cubic inflection points. In particular, no line, tangent to the boundary, has order of tangency that exceed 3 (see Example 3.1 for the details).
How to formulate the property of the geodesic vector field being boundary generic/traversally generic with respect to in terms of the geodesic curves and Jacobi fields in and their interactions with ?
Remark 2.3. Of course, not any metric on is of the gradient type. At the same time, thanks to Theorem 2.1, the gradient-type metrics form a massive set.
Examples of geodesically boundary generic metrics are also not so hard to exhibit. They require only a localized control of the geometry of in terms of (see Lemmas 3.2 and 3.3). For instance, if all the components of are either strictly convex or strictly concave in , then is geodesically boundary generic.
In contrast, to manufacture a geodesicly traversally generic metric is a more delicate task. In fact, we know only few examples, where gradient type metrics are proven to be of the traversally generic type: these examples have gradient-type metrics in which the boundary is strictly convex (see Corollary 3.7). However, we suspect that traversally generic metrics are abundant (see Conjecture 2.3). In any case, by Theorem 2.2 below, the property of a metric to be traversally generic is stable under small smooth perturbations of .
So we have only a weak evidence for the validity of following conjecture; however, the world in which it is valid seems to be a pleasing place…
The sets and are open and dense in the space .
The openness of and in follows from the theorem below.
Let be a compact smooth connected manifold with boundary.
In the space of all Riemannian metrics on , equipped with the -topology, the spaces and are open. Each of these spaces is invariant under the natural action of the smooth diffeomorphism group on .
If a metric on is of the gradient type, then the geodesic field on can be approximated arbitrary well in the -topology by a traversally generic field .
The construction of the geodesic flow defines a continuous map
where denotes the space of all vector fields on . By Theorem 6.7 and Corollary 6.4 from [K2], the subspace , formed by traversally generic (and thus gradient-like) vector fields, is open in . Similarly, the boundary generic and traversing fields form an open set in .
Since the germ of the geodesic through a point in the direction of a given unit tangent vector depends smoothly on metric , we conclude that is a continuous map. Therefore,
are open sets in .
By definition, for any , the geodesic field on is of the gradient type (and thus traversing). Again, by Theorem 6.7 from [K2], can be approximated by a traversally generic field . Note however that the projections of -trajectories under the map may not stay -close to the geodesic lines in the original metric due to the concave boundary effects.
By Theorem 2.1, . Nevertheless, the question whether for a given remains open!
Evidently, by the “naturality” of the geodesic flow, the spaces , are invariant under the natural action of the smooth diffeomorphism group on . ∎
Remark 2.4. Let be a codimension compact submanifold of a compact Riemannian manifold such that . If a metric on the ambient is of the gradient type, then, by Corollary 2.2, its restriction is of the gradient type on .
Of course, if is geodesicly traversally generic on a compact manifold , it may not be geodesically traversally generic on .
Example 2.3. Consider the hyperbolic space with its virtual spherical boundary and hyperbolic metric . is modeled by the open unit ball in the Euclidean space .
Each geodesic line hits at a pair of points, where it is orthogonal (in the Eucledian metric) to . For each oriented geodesic line trough a given point in the direction of a given vector , consider the distance between and the unique point that can be reached from by moving along in the direction of , that is, is the length of the circular arc in . Evidently, is strictly increasing, as one moves along the oriented .
Let be a compact codimension smooth submanifold, equipped with the induced hyperbolic metric. Then the geodesic field on the space is of the gradient type, since is strictly increasing along the oriented trajectories of .
Again, by Theorem 2.2, any metric on , sufficiently close to the hyperbolic metric , is also of the gradient type.
3. The Geodesic Scattering and Holography
In this section, we will apply the Holographic Causality Principle [K4], to geodesic flows on the spaces of unit tangent vectors on compact smooth Riemannian manifolds with boundary.
We will be guided by a single important observation: if a metric on is of the gradient type, then the causality map
introduced in [K4] (for generic smooth traversing vector fields on compact manifolds with boundary), is available! To get a feel for the nature of the causality map from [K4], the reader may glance at Figure 3. It depicts the causality map for a traversing field on a surface with boundary.
The map represents the -induced geodesic scattering: indeed, with the help of , each unit tangent vector