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We study the performance of multiple-antenna systems under finite-rate feedback of some function of the current channel realization from a channel-aware receiver to the transmitter. Our analysis is based on a novel geometric paradigm whereby the feedback information is modeled as a source distributed over a Riemannian manifold. While the right singular vectors of the channel matrix and the subspace spanned by them are located on the traditional Stiefel and Grassmann surfaces, the optimal input covariance matrix is located on a new manifold of positive semi-definite matrices - specified by rank and trace constraints - called the Pn manifold. The geometric framework developed enables the results to hold for arbitrary distributions of the channel matrix and extends to all covariance computation strategies including, waterfilling in the short-term/long-term power constraint case, antenna selection and other rank-limited scenarios that could not be analyzed using previous probabilistic approaches. This book will be useful to both graduate students and practicing industry professionals.