Boundary-Rigidity of Projective Metrics and the Geodesic X-Ray Transform

We prove that given a compact convex non-empty domain M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$\end{document} in the plane, a function δ:∂M×∂M→R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta :\partial \mathcal M\times \partial {\mathcal {M}}\rightarrow {\mathbb {R}}_+$$\end{document} can be extended to a projective metric d on M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$\end{document} if and only if δ(P,R)+δ(Q,S)-δ(P,S)-δ(Q,R)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \delta (P,R)+\delta (Q,S)-\delta (P,S)-\delta (Q,R)>0 $$\end{document} for any convex quadrangle □(PQRS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Box (PQRS)$$\end{document} inscribed in ∂M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial {\mathcal {M}}$$\end{document}. Moreover, this extension is unique.


Introduction
The problem of boundary-rigidity of Riemannian manifolds (see for example [12][13][14][15]22]) has recently reached a general solution in [20] (see [7] for how important consequences this may have in seismology). Obtaining similar results without using differentiability properties is what motivates investigating the rigidity of projective metrics.

Notations and Preliminaries
Points of R n are denoted as A, B, . . . , vectors as − → AB or a, b, . . . . The latter notations are also used for points if the origin is fixed. The affine line through A and B is AB, and the closed segment with endpoints A and B is AB. The convex hull of a point set P is denoted by ConvP, the interior of P is denoted by P • , and χ P denotes the indicator function of P.
The n-dimensional projective space is denoted by P n . We use P * to denote the set of hyperplanes through a point P ∈ P n . Accordingly, the set of all hyperplanes intersecting M ⊆ P n is M * , and P n * is the set of all hyperplanes of P n .
Let d e be a Euclidean metric on R n+1 , and ·, · e be the corresponding Euclidean product on the space of vectors of R n+1 . We denote the unit sphere and ball by S n−1 e and by B n e , respectively. The Euclidean metric d e induces a metric d S e : S n × S n → [0, π] by cos(d S e (u, v)) = u, v e . Modeling P n as the set of straight lines through the origin O in R n+1 shows that the map ψ : S n → P n given by ψ(X ) = O X is a double covering of P n , because   Y )). Let S P e (ψ(X )) denote the sphere with respect to d P e in P n that is centered at ψ(X ) and has the greatest radius. This corresponds in S n to the great circle S n−1 e (X ) = S n−1 e (−X ), the "equator", that is, the set of points having equal distances from X and −X . Let R n e (X ) denote the n-dimensional subspace of R n+1 that contains S n−1 e (X ), and let P n−1 e (ψ(X )) be the corresponding (n − 1)-dimensional hyperplane in P n . Metric d P e induces a bijective pairing : P n ↔ P n * , the elliptic polarity, by (ψ(X )) = R n e (X ) and (R n e (X )) = ψ(X ) 1 . Further, (P * ) = { (H) : P ∈ H ∈ P n * } = (P) is a hyperplane in P n * , and (H) = { (P) : P ∈ H} = (H) * for any H ∈ P n * .
Fix a point S ∈ S n−1 . The tangent space T S S n is naturally identified with R n by, say, ı. LetR n be R n supplemented with the ideal hyperplane. Then the mapping τ S : O X → O X ∩ T S S n naturally extends to a bijective pairing τ S : P n ↔R n . The pairing τ S maps hyperplanes into hyperplanes, so τ S ( (P)) is a hyperplane inR n .
We call a subset M ⊆ P n convex, if it is either P n , or, for a well-chosen S ∈ S n , τ S (M) is convex in T S S n ∼ = R n . With a slight abuse of notions, we call a subset M ⊆ P n compact, if for a well-chosen S ∈ S n , τ S (M) is compact in T S S n ∼ = R n . If M is a compact convex set, then the set M # := P n \ (M * ) is either empty, or a compact convex set. If M # is not empty, then the continuity of implies that Observe that { (X * ) : X ∈ (P * )} = P * , hence a segment P Q given by points P, Q ∈ P n corresponds to the two-edge that is bounded by the union of the hyperplanes (P) and (Q). We say that a two-edge supports a convex set M if it does not contain any inner point of M, and both of its bounding hyperplanes support M. We call a measure μ : 2 M * → R + p-admissible if for every non-collinear triple P, Q, R ∈ M of points we have 1 Given an elliptic polarity : P 2 ↔ P 2 * and a convex (maybe empty) M ⊆ P 2 , a measure μ : 2 M * → R + is p-admissible if and only if μ • is a strictly positive measure on P 2 such that μ • vanishes on straight lines.
The easy proof is left to the interested reader. Note however, that in higher dimensions no such easy equivalence exists.
The following observation was originally made by Busemann [6].

Lemma 2.2
Let M be a convex non-empty domain in P n , and let μ be a p-admissible measure μ : 2 M * → R + . Then the function d : M×M → R + defined by d(P, Q) = μ( X ∈P Q X * )/2 for every pair of points P, Q ∈ M is a projective metric on M.
Proof The function d is clearly non-negative and symmetric, it vanishes if P = Q, and it is positive if P = Q, because μ is definite and positive. It also satisfies the strict triangle inequality, because and μ is strict.
Given a set , we call a class of its subsets R ⊂ 2 a semiring if The smallest σ -algebra containing R is said to be generated by R, and is denoted by σ (R).
We say that a set function μ : A major tool in this paper is the following variation of Carathéodory's extension theorem [23]. It uses slightly weaker assumptions.

Extendibility of Boundary-Metrics in Dimension Two
Let M be a compact convex non-empty domain in P 2 . By the triangle inequality, because the diagonal point X = P R ∩ QS of any convex non-degenerate quadrangle (P Q RS) inscribed in ∂M falls in M, every projective metric d :  Proof By Lemmas 2.1 and 2.2, it is enough to construct a measure μ : (R 2 \ M # ) → R + such that δ(P, Q) = μ • ( X ∈P Q X * )/2 for pairs of points P, Q ∈ ∂M. Since the set X ∈P Q X * is the two-edge E( (P), (Q)), we need to construct μ from knowing it only on two-edge sets supporting M # .
First we define a positive set function ν on a semiring R in several steps. We start with two-edges, triangles and convex quadrangles all of whose side-lines support M # .
Observe that exactly those two-edges that support M # are given in the form E( (P), (Q)). So, by (2.1), if the requisite measure μ existed, then it should give δ(P, Q) for the two-edge of form E( (P), (Q)). Figure 2 shows what we have. Thus, we define Figure 3 shows what we have.
Observe that, E( p, q), E(q, r ) and R 2 \ E(r , p) cover the interior of (ABC) three times, while they cover lines p, q, r twice and every other point of the plane only once. So, outside lines p, q, r , we have 2χ (ABC) = χ E( p,q) +χ E(q,r) −χ E(r, p) . Therefore, using (3.2), we define ν( (ABC)) := (δ(P, Q) + δ(Q, R) − δ(P, R))/2. The side-lines of any half-closed connected supporting polygon P cut P into finitely many mutually disjoint half-closed supporting quadrangles Q i ∈ Q (i = 1, . . . , n), so we define (3.5) As every set R ∈ R is the union of such mutually disjoint closed polygons P j ∈ R ( j = 1, . . . , ), we can finish defining ν by Choose a point C outside M # and let r and q be the supporting lines of M # through C. These two lines determine two two-edges one of which, say E C contains M # . Let  Figure 6 shows what we have.
Let ν C;s denote the restriction of ν onto R C;s = {R : R R ⊂ L C;s }. We claim that ν C;s is extendable. The set function ν C;s is clearly additive and σ -subadditive, so we only need to prove that it is also σ -finite.
Let Thus, by Theorem 2.3, set function ν C;s extends to a σ -finite measure μ C;s on σ (R C;s ), the set of the Borel sets in L C;s .
Observe that μ C;s ( (ABC D)) = ν( (ABC D)) for every supporting half-closed quadrangle (ABC D) in R C;s , hence every measure μ C;s takes the same value on every quadrangle (ABC D) which is in the pointed lane L C;s , so all such measures are equal on every Borel set in their common domains. Now we can define the measure μ requested in the theorem as follows: Given a Borel set, divide it to disjoint parts so that every part falls in a pointed lane L C;s , then measure every such part by the appropriate μ C;s , and sum up the values (this summation is finite if the Borel set is covered by finitely many lanes, for example if the Borel set is bounded).
To finish the proof we only have to check that δ is a restriction of the projective metric d defined from μ • by Lemma 2.2.
Let (ABC D) be a convex quadrangle with side-lines p = AB = (P), q = BC = (Q), r = C D = (R), and s = D A = (S), such that E( p, q), E(q, r ), E(r , s), and E(s, p) support ∂M # . Observe that where the first and last equation are proved by the derivation of (3.4). Letting Q → R and S → P in this equality, the continuity of δ and d implies 2δ(P, R) = 2d(P, R) that completes the proof.

Rigidity by Boundary-Metrics in Dimension Two
In what follows we have a projective metric d : M × M → R + on a compact convex domain M. The restriction d ∂M : ∂M×∂M → R + of d defined by d ∂M (P, Q) := d(P, Q) is called the boundary-metric. We consider how much of a projective metric is determined by some restrictions of its boundary-metric.

Theorem 4.1 A projective metric is determined by its boundary-metric.
Proof By Theorem 3.1, we have a measure μ • : 2 M * → R + , such that d(P, Q) = μ(E( (P), (Q))). So we only need to show that μ is determined by the boundarymetric. Following where P, Q, R, S ∈ ∂M. Let Q be the set of the half-closed quadrangles all of whose side-lines support M # . Let R be the smallest semiring containing Q. Then the sets in R are the union of mutually disjoint half-closed connected polygons all of whose side-lines support M # . The side-lines of any half-closed connected supporting polygon P cut P into finitely many mutually disjoint half-closed supporting quadrangles Q i ∈ Q (i = 1, . . . , n). So we have μ(P) := n i=1 μ(Q i ). As every set R ∈ R is the union of such mutually disjoint half-closed connected polygons P j ∈ R ( j = 1, . . . , ), we also have μ(R) := j=1 μ(P i ). Since μ is a measure, it is σ -finite in every pointed lane L C;s (see Fig. 6 and the text above it), so μ is uniquely determined by its values on R by the unicity part of Carathéodory's Theorem 2.3.
This proves the theorem.
We can sharpen Theorem 4.1 by digging a hole in M. The result resembles the "peeling argument" of [20] in a way. Proof By Theorem 3.1, we have a measure μ • : 2 M * → R + , such that d(P, Q) = μ(E( (P), (Q))). As d is given on ∂M × ∂M, we have (4.1) for every two-edge E( (P), (Q)) that supports M # and has vertex in N # (See Fig. 7).
Just as in the proof of Theorem 4.1, we can calculate from this the μ-measure of those triangles and quadrangles in N # \ M # that support M # . Denoting the set of such half-closed quadrangles by Q and considering the semiring R generated by Q one can finish the proof in the same way as the proof of Theorem 4.1 was finished.
Just as in the proof of Theorem 4.1 we can calculate from this the μ-measure of those triangles and quadrangles outside of N # that support M # . Denoting the set of We can restrict the knowledge on the boundary-metric even more while being able to deduce some partial information on the projective metric. This resembles the Limited Data X-ray Tomography [19].
The following theorem is a clear consequence of Theorem 4.3. We have to make it clear here that the projective metric d is not determined on M \ ConvA. Let the endpoints of A be A ± , and the dual lines for these points be a ± = (A ± ), as shown on Fig. 9.

Theorem 4.4 Let
The two-edge E † , the complement of E(a − , a + ), is separated by M # into two connected domains E † ± . Let E † + be the one that is bounded by (A * ), and let E † − be the other one, that is bounded by ∂M # \ (A * ). Domain E † − is shown on Fig. 9 with red color. It is clear, that the measure μ that corresponds to d is given exactly for those two-edges that support M # in (A * ). Therefore, the only information about how μ behaves on E † − is the function ∂M \ A S → μ( (AS + S − )), where A is the vertex of E † − , and points S ± are the intersections of tangents a ± with tangent s = (S).

Generalized X-Ray Transform in Dimension Two
Let N ⊆ R 2 be a compact connected domain such that a set C of continuous curves in N exists, such that two curves intersect each other in at most one point, and there is a unique curve C P,Q ∈ C for any two different points P, Q ∈ N that contains both points P, Q. The generalized X-ray transform X C,μ maps the functions f : R 2 → R integrable on each curve C P,Q into where μ P,Q is a distribution on C P,Q . Following [11], the curves C P,Q are called petals, the set C is called flower, and μ P,Q is called the weight on the petal C P,Q . Identification of such Radon transforms by some partial data is a widely studied subject (see for example [8,10,11,17,18] etc.). Among the many known results [12][13][14][15]20] about the boundary-metric rigidity of Riemannian metrics, we find the following to be of particular interest: Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.