X-ray Transforms in Pseudo-Riemannian Geometry

Abstract

We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces, and tori. We give proofs of uniqueness and characterize non-uniqueness in different settings. Reconstruction is sometimes possible if the signature $(n_1,n_2)$ satisfies $n_1\ge 1$ and $n_2\ge 2$ or vice versa and always when $n_1,n_2\ge 2$. The proofs are based on a Pestov identity adapted to null geodesics (product manifolds) and Fourier analysis (other geometries). The problem in a Minkowski space of any signature is a special case of recovering a function in a Euclidean space from its integrals over all lines with any given set of admissible directions, and we describe sets of lines for which this is possible. Characterizing the kernel of the null geodesic ray transform on tori reduces to solvability of certain Diophantine systems.

Appendix 1: Bundles and Differential Operators

We give an overview of the various bundles, differential operators, and vector fields used throughout this paper, especially Sect. 2. We start with Riemannian manifolds and then proceed to pseudo-Riemannian ones.

1.1 Riemannian Manifolds

Let M be a compact, smooth Riemannian manifold of dimension n, and denote its unit sphere bundle by SM. The geodesic vector field X is the generator of the geodesic flow on SM, and it acts as a differential operator $C^\infty (SM)\rightarrow C^\infty (SM)$.

For $\theta =(x,v)\in SM$, denote by $N_\theta$ the hyperplane of $T_xM$ orthogonal to v. Let N denote the bundle over SM whose fiber at $\theta$ is $N_\theta$. This is a subbundle of TSM under the following identifications.

Vectors in TTM can be canonically split into horizontal and vertical components. Give any $\theta =(x,v)\in TM$, the horizontal and vertical fibers $H(\theta )$ and $V(\theta )$ of $T_\theta TM$ can both be identified with $T_xM$. Declaring these identifications as isometric and requiring horizontal and vertical vectors to be orthogonal to each other produces the Sasaki metric on TM.

A similar splitting is naturally induced on TSM. The gradient of a function $u\in C^\infty (SM)$ can be decomposed into geodesic, horizontal, and vertical components as

$$\begin{aligned} \nabla _{SM}u = ((Xu)X,{{\mathrm{\overset{\scriptstyle h}{\nabla }}}}u,{{\mathrm{\overset{\scriptstyle v}{\nabla }}}}u). \end{aligned}$$

(47)

The horizontal and vertical gradients ${{\mathrm{\overset{\scriptstyle h}{\nabla }}}}u$ and ${{\mathrm{\overset{\scriptstyle v}{\nabla }}}}u$ are originally vector fields on SM. Via the identification made above they can both be considered sections of the bundle N.

The horizontal divergence ${{\mathrm{\overset{\scriptstyle h}{{\text {div}}}}}}$ is the formal adjoint of $-{{\mathrm{\overset{\scriptstyle h}{\nabla }}}}$ with respect to the measure on SM induced by the Sasaki metric. The vertical divergence ${{\mathrm{\overset{\scriptstyle v}{{\text {div}}}}}}$ is defined similarly as an adjoint of $-{{\mathrm{\overset{\scriptstyle v}{\nabla }}}}$. Both divergences map sections of N to smooth functions on SM.

The geodesic vector field X acts also on the sections of N by covariant differentiation along the geodesic flow. The actions of the geodesic vector field on the scalars on SM and the sections of N both obey $X^*=-X$.

The curvature operator R maps the sections of N to sections of N.

Let x be some local coordinates on M and (x, y) the associated coordinates on TM. We define the vector fields $\delta _{x^i}=\partial _{x^i}-\Gamma ^j_{\phantom {j}ik}y^k\partial _{y^j}$ on TM. These vectors $\delta _{x^i}$ span the horizontal fiber H(x, y) and the vectors $\partial _{y^i}$ span V(x, y). These vector fields act on TM, and we can turn them into vector fields on SM as follows.

Let $p:TM{\setminus }0\rightarrow SM$ be the radial projection $p(x,y)=(x,y/\left| y \right| )$. The local vector fields $\delta _i$ and $\partial _i$ on SM are defined so that

$$\begin{aligned} \begin{aligned} \delta _i u&= \delta _{x^i}(u\circ p)|_{SM} \quad \text {and} \\ \partial _i u&= \partial _{y^i}(u\circ p)|_{SM} \end{aligned} \end{aligned}$$

(48)

for any $u\in C^\infty (SM)$. The geodesic vector field and the two gradients of the decomposition (47) are then $Xu=y^i\delta _iu$, ${{\mathrm{\overset{\scriptstyle h}{\nabla }}}}u =(\delta ^i-y^iy^j\delta _ju)\partial _{x^i}$, and ${{\mathrm{\overset{\scriptstyle v}{\nabla }}}}u=(\partial ^iu)\partial _{x^i}$. For details on the coordinate representations of these operators, we refer to [17, Appendix A]. The commutator formulas listed in Sect. 2.2 were also proved there.

1.2 Pseudo-Riemannian Product Manifolds

As was argued in the proof of Theorem 1, the conformal factor is irrelevant for the ray transform problem studied here. Therefore we restrict our attention to pseudo-Riemannian products of Riemannian manifolds with no conformal factor.

Let $M_1$ and $M_2$ be two Riemannian manifolds and $M=M_1\times M_2$ their product manifold. We add subindices to all Riemannian operators introduced in Sect. 1 to indicate which manifold they act on. We equip it with the pseudo-Riemannian product metric $g_1\ominus g_2$, but this choice is irrelevant for the present discussion of operators and bundles. The light cone bundle $LM=SM_1\times SM_2$ comes with two natural projections, $\pi _i:LM\rightarrow SM_i$, $i=1,2$.

Let u be a smooth function on LM. Keeping $(x_2,v_2)\in SM_2$ fixed, one can regard it as a function on $SM_1$ and apply the Riemannian operators $X_1$, ${{\mathrm{\overset{\scriptstyle h}{\nabla }}}}_1$, and ${{\mathrm{\overset{\scriptstyle v}{\nabla }}}}_1$ introduced in Sect. 1. Now $X_1u$ is simply a scalar function on LM. If the point on $SM_2$ is fixed, the gradients ${{\mathrm{\overset{\scriptstyle h}{\nabla }}}}_1u$ and ${{\mathrm{\overset{\scriptstyle v}{\nabla }}}}_1u$ are sections of the bundle $N_1$. It is therefore natural to consider the two gradients to be sections of the pullback bundle $\pi _1^*N_1$. We extend similarly operators from $SM_2$ to LM.

Given any point in LM, the gradients ${{\mathrm{\overset{\scriptstyle h}{\nabla }}}}_1u$ and ${{\mathrm{\overset{\scriptstyle v}{\nabla }}}}_1u$ are in the same space via the identifications we use freely. However, the gradients ${{\mathrm{\overset{\scriptstyle h}{\nabla }}}}_2u$ and ${{\mathrm{\overset{\scriptstyle v}{\nabla }}}}_2u$ are not in the same space with them.

The product structure ensures that operators on $SM_1$ commute with those acting on $SM_2$. The coordinate representations of these operators are readily found using coordinates on the two base manifolds $M_1$ and $M_2$ to give coordinates on M, and by applying the Riemannian coordinate descriptions mentioned at the end of Sect. 1.