  We discuss problems that relate curvature and concentration properties of
eigenfunctions and quasimodes on compact boundaryless Riemannian manifolds.
These include new sharp $L^q$-estimates, $q\in (2,q_c]$, $q_c=2(n+1)/(n-1)$, of
log-quasimodes that characterize compact connected space forms in terms of the
growth rate of $L^q$-norms of such quasimode for these relatively small
Lebesgue exponents $q$.
  No such characterization is possible for any exponent $q> q_c$.
