In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown

These notes are about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using conserved quantities, which can be converted into a discrete theory in a natural way. About referencing: We do not attempt to give a complete reference list, and omit what is already referenced in [59]. We list only articles referenced in the body of the text, or that were written after [59] was published, or were otherwise not included in the reference list in [59], or that were referenced in [59] but need to be updated. About using quaternions: In following with the historical development of the field, we use a model that involves quaternions. However, the use of a more standard model has some advantages, as it can be applied in more general dimensions and settings (see Chapter 10 here, for example), and sometimes gives less cluttered computations. It would be a good exercise to convert this text into one involving a more standard quaternion-free model, but we do not do that here (see [27]), and instead only make occasional comments about this. Acknowledgements: Primary thanks must go to Udo Hertrich-Jeromin, who carefully and patiently taught the author more than half of the material in this text. The author also owes thanks to many others for numerous mathematical tips: Fran

Abstract. Let M and ¯ M be n-dimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M × ¯ M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry. Using the pseudo-Riemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principal characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and ¯ M of Rn. This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered. 1.

ABSTRACT. We prove an extension of the Index Theorem for Morse–Sturm systems of the form −V ′ ′ + RV = 0, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M, g), we consider a geodesic γ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M, g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at γ and the sum of the Maslov index of γ with the index of the metric g on P. Under generic circumstances, the Maslov index of γ is given by an algebraic count of the P-focal points along γ. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold. 1.

manifolds with large isometry groups

Abstract. In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry. Namely, the complete classification of strongly convex Randers metrics of constant flag curvature. 1.

ABSTRACT. We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the Maslov index of a semi-Riemannian geodesic, which is a homological invariant and it substitutes the notion of geometric index in Riemannian geometry. Under generic circumstances, the Maslov index of a geodesic is computed as a sort of algebraic count of the conjugate points along the geodesic. For non positive definite metrics the index of the index form is always infinite; in this paper we prove that the space of all variations of a given geodesic has a natural splitting into two infinite dimensional subspaces, and the Maslov index is given by the difference of the index and the coindex of the restriction of the index form to these subspaces. In the case of variable endpoints, two suitable correction terms, defined in terms of the endmanifolds, are added to the equality. Using appropriate change of variables, the theory is entirely extended to the more general case of symplectic differential systems, that can be obtained as linearizations of the Hamilton equations. The main results proven in this paper were announced in [23].

ABSTRACT. Helfer in [6] was the first to produce an example of a spacelike Lorentzian geodesic with a continuum of conjugate points. In this paper we show the following result: given an interval [a, b] of IR and any closed subset F of IR contained in]a, b], then there exists a Lorentzian manifold (M, g) and a spacelike geodesic γ: [a, b] → M such that γ(t) is conjugate to γ(a) along γ iff t ∈ F. 1.

The concept of generalized functions taking values in a differentiable manifold ([15, 19]) is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of manifold-valued generalized functions and of generalized vector bundle homomorphisms. As a consequence, a characterization of equivalence that does not resort to derivatives (as provided in [11] for the scalar-valued cases of Colombeau’s construction) is achieved. These results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly. Finally, a new concept of association adapted to the present setting is introduced.