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Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra.

In this paper, under certain conditions on the mean and scalar curvatures, we prove that there are no strongly stable linear Weingarten closed two-sided hypersurfaces immersed in a certain region determined by a geodesic sphere of the ( n + 1 ) -dimensional real projective space R P n + 1 . We also provide a rigidity result for these hypersurfaces.

This book provides the first unified examination of the relationship between Radon transforms on symmetric spaces of compact type and the infinitesimal versions of two fundamental rigidity problems in Riemannian geometry. Its primary focus is the spectral rigidity problem: Can the metric of a given Riemannian symmetric space of compact type be characterized by means of the spectrum of its Laplacian? It also addresses a question rooted in the Blaschke problem: Is a Riemannian metric on a projective space whose geodesics are all closed and of the same length isometric to the canonical metric? The authors comprehensively treat the results concerning Radon transforms and the infinitesimal versions of these two problems. Their main result implies that most Grassmannians are spectrally rigid to the first order. This is particularly important, for there are still few isospectrality results for positively curved spaces and these are the first such results for symmetric spaces of compact type of rank >1. The authors exploit the theory of overdetermined partial differential equations and harmonic analysis on symmetric spaces to provide criteria for infinitesimal rigidity that apply to a large class of spaces. A substantial amount of basic material about Riemannian geometry, symmetric spaces, and Radon transforms is included in a clear and elegant presentation that will be useful to researchers and advanced students in differential geometry.

Linear combinations of translates of a given basis function have long been successfully used to solve scattered data interpolation and approximation problems. We demonstrate how the classical basis function approach can be transferred to the projective space ℙ d −1 . To be precise, we use concepts from harmonic analysis to identify positive definite and strictly positive definite zonal functions on ℙ d −1 . These can then be applied to solve problems arising in tomography since the data given there consists of integrals over lines. Here, enhancing known reconstruction techniques with the use of a scattered data interpolant in the “space of lines”, naturally leads to reconstruction algorithms well suited to limited angle and limited range tomography. In the medical setting algorithms for such incomplete data problems are desirable as using them can limit radiation dosage.

The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.

We establish the formulas on the power τk\\tau ^k of the tangent bundle τ=τ(RPn)\\tau =\\tau (RP^n) of the real projective nn-space RPnRP^n and its complexification cτkc\\tau ^k, and apply the formulas to the problem of extendibility and stable extendiblity of τk\\tau ^k and cτkc\\tau ^k.

This book proves an analogue of William Thurston's celebrated hyperbolic Dehn surgery theorem in the context of complex hyperbolic discrete groups, and then derives two main geometric consequences from it. The first is the construction of large numbers of closed real hyperbolic 3-manifolds which bound complex hyperbolic orbifolds--the only known examples of closed manifolds that simultaneously have these two kinds of geometric structures. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. In an accessible and straightforward manner, Richard Evan Schwartz also presents a large amount of useful information on complex hyperbolic geometry and discrete groups. Schwartz relies on elementary proofs and avoids quotations of preexisting technical material as much as possible. For this reason, this book will benefit graduate students seeking entry into this emerging area of research, as well as researchers in allied fields such as Kleinian groups and CR geometry.

If VV is an elementary abelian 22-group, Ossa proved that the connective KK-theory of BVBV splits into copies of Z/2\\mathbf { Z}/2 and of the connective KK-theory of the infinite real projective space. We give a brief proof of Ossa’s theorem.

In this paper we utilize BP∗()B{P^*}(\\;), a generalized cohomology theory associated with the Brown-Peterson spectrum to prove a nonimmersion theorem for products of real projective spaces.

An elementary argument shows that the geometric dimension of any vector bundle of order 2e{2^e} over RPnR{P^n} depends only on ee and the residue of nmod8n\\,\\bmod \\,8 for nn sufficiently large. In this paper we calculate this geometric dimension, which is approximately 2e2e. The nonlifting results are easily obtained using the spectrum bJbJ. The lifting results require bobo-resolutions. Half of the paper is devoted to proving Mahowald’s theorem that beginning with the second stage bobo-resolutions act almost like K(Z2)K({Z_2})-resolutions.