  Given a time T>0 and a region Omega on a compact Riemannian manifold M, we
consider the best constant, denoted C_{T,Omega}, in the observation inequality
for the Schroedinger evolution group of the Laplacian Delta with Dirichlet
boundary condition: for all f in L^2(M), ||f||_{L^2(M)} \leq C_{T,Omega}
||exp(itDelta)f||_{L^2((0,T)xOmega)}. We investigate the influence of the
geometry of Omega on the growth of C_{T,Omega} as T tends to 0. By duality,
C_{T,Omega} is also the controllability cost of the free Schroedinger equation
on M with Dirichlet boundary condition in time T by interior controls on Omega.
It relates to hinged vibrating plates as well. We emphasize a tool of wider
scope: the control transmutation method. We prove that C_{T,Omega} grows at
least like exp(d^2/4T), where d is the largest distance of a point in M from
Omega, and at most like exp(alpha L^2/T), where L is the length of the longest
generalized geodesic in M which does not intersect Omega, and alpha is a
constant in ]0,4[ (it is the growth rate of the controllability cost in a
similar one dimensional problem). We also deduce such upper bounds on product
manifolds for some control regions which are not intersected by all geodesics.
