Sankhya: The Indian Journal of Statistics

2000, Volume 62, Series A, Pt. 3, pp. 419--424

SOME QUESTIONS ON INTEGRAL GEOMETRY ON RIEMANNIAN MANIFOLDS

By

VISHWAMBHAR PATI and ALLADI SITARAM, * Indian Statistical Institute, Bangalore*

*SUMMARY. *It is a surprising but known fact that an *L*^{1} function on
**R**^{n} is
determined once *all* its spherical averages with centres coming
from certain ``small'' sets are known. We generalise this result to
complete connected real-analytic Riemannian manifolds with a real-analytic metric
(with some curvature restrictions), where superisothermal sets play
the same role as balls do in the case of Euclidean space.

*AMS (1991) subject classification. * 53C65.

*Key words and phrases. * Integral geometry, heat kernel, Riemannian
manifold.