  Necessary and sufficient conditions are obtained for injectivity of the
shifted Funk-Radon transform associated with $k$-dimensional totally geodesic
submanifolds of the unit sphere $S^n$ in $\mathbb{R}^{n+1}$. This result
generalizes the well known statement for the spherical means on $S^n$ and is
formulated in terms of zeros of Jacobi polynomials. The relevant harmonic
analysis is developed, including a new concept of induced Stiefel (or
Grassmannian) harmonics, the Funk-Hecke type theorems, addition formula, and
multipliers. Some perspectives and conjectures are discussed.
