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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.

This book provides the first-ever systematic introduction to the theory of Riemannian submersions, which was initiated by Barrett O'Neill and Alfred Gray less than four decades ago. The authors focus their attention on classification theorems when the total space and the fibres have nice geometric properties. Particular emphasis is placed on the interrelation with almost Hermitian, almost contact and quaternionic geometry. Examples clarifying and motivating the theory are included in every chapter. Recent results on semi-Riemannian submersions are also explained. Finally, the authors point out the close connection of the subject with some areas of physics.

We study Lagrangian submersions whose total manifolds are locally conformal Kähler manifolds. We first investigate the necessary and sufficient conditions for the horizontal and vertical distributions of a Lagrangian submersion from a locally conformal Kähler manifold to be totally geodesic. Then, we examine the harmonicity of these submersions. We prove that the Lee vector field of the total manifold of such a submersion cannot be vertical. In the case of the Lee vector field is horizontal, we show that the horizontal distribution is always integrable and totally geodesic while its fibers cannot be totally geodesic. We obtain fundamental equations for a curve on the total manifold of such submersions to be geodesic. Consequently, we give a necessary and sufficient condition for a Lagrangian submersion to be Clairaut. Finally, we prove that if a Lagrangian submersion from a locally conformal Kähler manifold is a Clairaut submersion, then either its mean curvature vector field is proportional to the horizontal part of its Lee vector field or the vertical distribution of the submersion is one dimensional.

Our purpose in this article is to study anti-invariant statistical submersions from holomorphic statistical manifolds. Firstly we introduce holomorphic statistical submersions satisfying the certain condition, after we give anti-invariant statistical submersions satisfying the certain condition. And we supported our results with examples.

In this paper, we study conformal submersions from Ricci solitons to Riemannian manifolds with non-trivial examples. First, we study some properties of the O’Neill tensor A in the case of conformal submersion. We also find a necessary and sufficient condition for conformal submersion to be totally geodesic and calculate the Ricci tensor for the total manifold of such a map with different assumptions. Further, we consider a conformal submersion F : M → N from a Ricci soliton to a Riemannian manifold and obtain necessary conditions for the fibers of F and the base manifold N to be Ricci soliton, almost Ricci soliton and Einstein. Moreover, we find necessary conditions for a vector field and its horizontal lift to be conformal on N and ( Ker F ∗ ) ⊥ , respectively. Also, we calculate the scalar curvature of Ricci soliton M . Finally, we obtain a necessary and sufficient condition for F to be harmonic.

In this paper, we describe anti-invariant and Clairaut anti-invariant pseudo-Riemannian submersions (AIPR and CAIPR submersions, respectively, briefly) from para-Kenmotsu manifolds onto Riemannian manifolds. We introduce new Clairaut circumstances for anti-invariant submersions whose total space is para-Kenmotsu manifold. Also, we offer a obvious example of CAIPR submersion.

In the present paper, we introduce a new kind of Riemannian submersion such that the fibers of such submersion are generic submanifolds in the sense of Ronsse that we call generic submersion. Some examples are given for generic submersion. Necessary and sufficient conditions are found for the integrability and totally geodesicness of the distributions which are mentioned in the definition. The geometry of the fibers is investigated. New results are obtained by considering the parallelism condition of canonical structures.

The aim of the present paper is to study statistical submersions with parallel almost complex structures. First, we define the notion of the generalized Kähler-like statistical submersion and give examples of the Kähler-like statistical submersions. In addition, we investigate total space and fibers under certain conditions. After, we introduce some results on J -invariant, J ∗ -invariant and anti-invariant generalized Kähler-like statistical submersions.

In this paper, when a given curve on the total manifold of a Riemannian submersion is transferred to the base manifold, the character of the corresponding curve is examined. First, the case of a Frenet curve on the total manifold being a Frenet curve on the base manifold along a Riemannian submersion is investigated. Then, the condition that a circle on the total manifold (respectively a helix) is a circle (respectively, a helix) or a geodesic on the base manifold along a Riemannian submersion is obtained. We also investigate the curvatures of the original curve on the total manifold and the corresponding curve on the base manifold in terms of Riemannian submersions.

We introduce Chen inequalities for slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We provide trivial and non-trivial examples for slant Riemannian submersions, investigate some curvature relations between the total space, the base space and fibres. Moreover, we obtain Chen-Ricci inequalities on the vertical and the horizontal distributions for slant Riemannian submersions from cosymplectic space forms.