3 corrected a typo

Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for $H$ running over the set of affine hyperplanes. Radon's inversion theorem gives a formula to recover $f$ from its transform.

I needed a similar result in a slightly different context in one of my project. We are on a simply connected, non-positively curved Riemannian manifold $X$ (or more generally a Hadamard space) and we have compactly supported smooth (positive if needed) functions $f$ and $g$. For all geodesic $\gamma\subset X$ (which is globally minimizing thanks to the curvature assumption, in particular it is convex), the metric projection $p_\gamma:X\to\gamma$ is well defined and $1$-Lipschitz. Let us call a perpendicular to $\gamma$ any level of $p_\gamma$. If we know that $\int_P f = \int_P g$ for all perpendicular $P$ of all geodesics, can we deduce that $f=g$?

In fact, I managed to get a result along these lines which is sufficient for my needs; my question is: is this well-known? Does it have a name? Do you know a reference?

Note that in the case of $\mathbb{R}^n$ or of the real hyperbolic space, then perpendiculars are totally geodesic hyperplanes so that we are reduced to the usual Radon transform, and the book by Helgason contains more than I really need. But even when $X$ is the complex hyperbolic space, then the perpendiculars are not the subsets used in the usual Radon transform on symmetric spaces (note that they are not convex).

Note also that the case of trees is an easy but nice combinatorial exercise; I could not find a reference either (I only found a paepr paper studying the Radon transform defined using horospheres).

2 edited tags
1

Reference for a particular Radon transform on non-positively curved spaces

Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for $H$ running over the set of affine hyperplanes. Radon's inversion theorem gives a formula to recover $f$ from its transform.

I needed a similar result in a slightly different context in one of my project. We are on a simply connected, non-positively curved Riemannian manifold $X$ (or more generally a Hadamard space) and we have compactly supported smooth (positive if needed) functions $f$ and $g$. For all geodesic $\gamma\subset X$ (which is globally minimizing thanks to the curvature assumption, in particular it is convex), the metric projection $p_\gamma:X\to\gamma$ is well defined and $1$-Lipschitz. Let us call a perpendicular to $\gamma$ any level of $p_\gamma$. If we know that $\int_P f = \int_P g$ for all perpendicular $P$ of all geodesics, can we deduce that $f=g$?

In fact, I managed to get a result along these lines which is sufficient for my needs; my question is: is this well-known? Does it have a name? Do you know a reference?

Note that in the case of $\mathbb{R}^n$ or of the real hyperbolic space, then perpendiculars are totally geodesic hyperplanes so that we are reduced to the usual Radon transform, and the book by Helgason contains more than I really need. But even when $X$ is the complex hyperbolic space, then the perpendiculars are not the subsets used in the usual Radon transform on symmetric spaces (note that they are not convex).

Note also that the case of trees is an easy but nice combinatorial exercise; I could not find a reference either (I only found a paepr studying the Radon transform defined using horospheres).