ACTA issues

The range of the Radon transform on the real hyperbolic Grassmann manifold

Satoshi Ishikawa

Acta Sci. Math. (Szeged) 86:1-2(2020), 225-264

Abstract. Let $\Gamma ^n_k$ be the space of all the $k$-dimensional totally geodesic submanifolds of the $n$-dimensional real hyperbolic space where $1\leq k\leq n-1$. We prove that the Radon transform $R$ for double fibrations of the real hyperbolic Grassmann manifolds $\Gamma ^n_p$ and $\Gamma ^n_q$ with respect to the inclusion incidence relations maps $C^\infty _0(\Gamma ^n_p)$ bijectively onto the space of all the functions in $C^\infty _0(\Gamma ^n_q)$ which satisfy a certain system of linear partial differential equations explicitly constructed from the left infinitesimal action of the transformation group when $0\leq p<q\leq n-1$ and $\dim \Gamma ^n_p< \dim \Gamma ^n_q$. Our approach is based on the generalized method of gnomonic projections. We also treat the dual Radon transform $R^*$.

DOI: 10.14232/actasm-019-773-1

AMS Subject Classification (1991): 44A12; 43A85

Keyword(s): Radon transform, real hyperbolic space, Grassmann manifold

received 26.3.2019, revised 2.9.2019, revised 17.3.2020, accepted 18.3.2020. (Registered under 23/2019.)