  This paper considers the quantization problem on the Grassmann manifold
\mathcal{G}_{n,p}, the set of all p-dimensional planes (through the origin) in
the n-dimensional Euclidean space. The chief result is a closed-form formula
for the volume of a metric ball in the Grassmann manifold when the radius is
sufficiently small. This volume formula holds for Grassmann manifolds with
arbitrary dimension n and p, while previous results pertained only to p=1, or a
fixed p with asymptotically large n. Based on this result, several quantization
bounds are derived for sphere packing and rate distortion tradeoff. We
establish asymptotically equivalent lower and upper bounds for the rate
distortion tradeoff. Since the upper bound is derived by constructing random
codes, this result implies that the random codes are asymptotically optimal.
The above results are also extended to the more general case, in which
\mathcal{G}_{n,q} is quantized through a code in \mathcal{G}_{n,p}, where p and
q are not necessarily the same. Finally, we discuss some applications of the
derived results to multi-antenna communication systems.
