# Asian Journal of Mathematics

## Volume 23 (2019)

### Determination of a Riemannian manifold from the distance difference functions

Pages: 173 – 200

DOI: https://dx.doi.org/10.4310/AJM.2019.v23.n2.a1

#### Authors

Matti Lassas (Department of Mathematics and Statistics, University of Helsinki, Finland)

Teemu Saksala (Department of Mathematics and Statistics, University of Helsinki, Finland)

#### Abstract

Let $(N, g)$ be a Riemannian manifold with the distance function $d(x, y)$ and an open subset $M \subset N$. For $x \in M$ we denote by $D_x$ the distance difference function $D_x : F \times F \to \mathbb{R}$, given by $D_x(z_1, z_2) = d(x, z_1) - d(x, z_2), z_1, z_2 \in F = N \setminus M$. We consider the inverse problem of determining the topological and the differentiable structure of the manifold $M$ and the metric ${g \vert}_M$ on it when we are given the distance difference data, that is, the set $F$, the metric ${g \vert}_F$, and the collection $\mathcal{D}(M) = \lbrace D_x ; x \in M \rbrace$. Moreover, we consider the embedded image $\mathcal{D}(M)$ of the manifold $M$, in the vector space $C(F \times F)$, as a representation of manifold $M$. The inverse problem of determining $(M, g)$ from $\mathcal{D}(M)$ arises e.g. in the study of the wave equation on $\mathbb{R}\times N$ when we observe in $F$ the waves produced by spontaneous point sources at unknown points $(t, x) \in \mathbb{R}\times M$. Then $D_x (z_1, z_2)$ is the difference of the times when one observes at points $z_1$ and $z_2$ the wave produced by a point source at $x$ that goes off at an unknown time. The problem has applications in hybrid inverse problems and in geophysical imaging.

#### Keywords

inverse problems, distance functions, embeddings of manifolds, wave equation

35R30, 53C22