  The Grassmann manifold G_{n,p}(L) is the set of all p-dimensional planes
(through the origin) in the n-dimensional Euclidean space L^{n}, where L is
either R or C. This paper considers an unequal dimensional quantization in
which a source in G_{n,p}(L) is quantized through a code in G_{n,q}(L), where p
and q are not necessarily the same. It is different from most works in
literature where p\equiv q. The analysis for unequal dimensional quantization
is based on the volume of a metric ball in G_{n,p}(L) whose center is in
G_{n,q}(L). Our chief result is a closed-form formula for the volume of a
metric ball when the radius is sufficiently small. This volume formula holds
for Grassmann manifolds with arbitrary n, p, q and L, while previous results
pertained only to some special cases. Based on this volume formula, several
bounds are derived for the rate distortion tradeoff assuming the quantization
rate is sufficiently high. The lower and upper bounds on the distortion rate
function are asymptotically identical, and so precisely quantify the asymptotic
rate distortion tradeoff. We also show that random codes are asymptotically
optimal in the sense that they achieve the minimum achievable distortion with
probability one as n and the code rate approach infinity linearly. Finally, we
discuss some applications of the derived results to communication theory. A
geometric interpretation in the Grassmann manifold is developed for capacity
calculation of additive white Gaussian noise channel. Further, the derived
distortion rate function is beneficial to characterizing the effect of
beamforming matrix selection in multi-antenna communications.
