**Meet the editor**

Paul Bracken is a professor in the Department of Mathematics at the University of Texas in Edinburg, TX. His BSc degree is from the University of Toronto and he has a PhD degree from the University of Waterloo, Canada. His research interests are quite broad and include studying problems in mathematical physics such as quantum mechanics and quantum field theory. He has

also worked in the areas of partial differential equations and differential geometry, as well as gravity. He has written some 160 chapters and several more that have appeared in books. He has also published two short books and edited several books for IntechOpen publisher, including this one, which is the fourth volume that he has contributed to with IntechOpen. He is on the editorial boards of several journals, has presented many talks at various meetings and conferences, and has taught courses in mathematics and physics at all levels over the years.

Contents

**Preface VII**

**Euclidean Space 3** Paul Bracken

Mehmet Akif Akyol

Samuel Kadoury

Zhengsheng Chen

Yuichi Kobayashi

Xueqi Ma and Weifeng Liu

Chapter 4 **Manifold Learning in Medical Imaging 81**

Chapter 1 **The Generalized Weierstrass System in Three-Dimensional**

Chapter 2 **On Conformal Anti-Invariant Submersions Whose Total Manifolds Are Locally Product Riemannian 25**

Chapter 5 **Trajectory Tracking Control of Parallel Manipulator with Integral Manifold and Observer 101**

Chapter 6 **Manifold-Based Robot Motion Generation 121**

Chapter 3 **Recent Advances of Manifold Regularization 47**

**Section 1 Theoretical 1**

**Section 2 Applications 79**

### Contents

**Preface VII** 


Preface

 

This volume is a collection of chapters dedicated to the investigation of manifolds. Research in this area, in particular differentiable manifolds, which often have physical applications, forms an integral part of mathematics research. In addition to the significant interest mani‐ folds hold for pure mathematicians, manifolds also have very important applications to many areas of modern applied mathematics and the physical sciences. As the book will show, there are numerous applications to such diverse areas as partial differential equa‐ tions, dynamical systems, and even constructing computer images. *Manifolds II: Theory and Applications* is basically divided into two groups. The first part is broken down into a group of three chapters underlying theoretical aspects of manifolds and a group of three chapters

The first group presents chapters of a theoretical nature on the ideas behind manifold regu‐ larization and conformal anti-invariant submersions whose total manifolds are locally prod‐ uct Riemannian. There is also a chapter on the generalized Weierstrass system for inducing mean curvature surfaces in Euclidean three-space. This area has seen a lot of activity recent‐

The last three chapters form a collection of chapters that touch on manifolds in a very ap‐

The book has been put together by an international group of invited authors. It is a pleasure to thank them for their hard work and significant contributions. I gratefully acknowledge the assistance provided by Mr. Nino Popović, who was the author service manager through‐ out the publishing process, as well as the IntechOpen for the opportunity to edit this volume

> **Professor Paul Bracken** Department of Mathematics University of Texas, USA

directed toward applications of manifolds to applied areas of science.

plied manner, such as manifold-based robot motion generation.

that examines the subject of manifolds.

ly and the chapter is written with both mathematicians and physicists in mind.

### Preface

 This volume is a collection of chapters dedicated to the investigation of manifolds. Research in this area, in particular differentiable manifolds, which often have physical applications, forms an integral part of mathematics research. In addition to the significant interest mani‐ folds hold for pure mathematicians, manifolds also have very important applications to many areas of modern applied mathematics and the physical sciences. As the book will show, there are numerous applications to such diverse areas as partial differential equa‐ tions, dynamical systems, and even constructing computer images. *Manifolds II: Theory and Applications* is basically divided into two groups. The first part is broken down into a group of three chapters underlying theoretical aspects of manifolds and a group of three chapters directed toward applications of manifolds to applied areas of science.

 The first group presents chapters of a theoretical nature on the ideas behind manifold regu‐ larization and conformalanti-invariant submersionswhose total manifoldsare locally prod‐ uct Riemannian. There is also a chapter on the generalized Weierstrass system for inducing mean curvature surfacesin Euclideanthree-space. Thisarea has seen a lot of activity recent‐ ly and the chapter is written with both mathematicians and physicists in mind.

 The last three chapters form a collection of chapters that touch on manifolds in a very ap‐ plied manner, such as manifold-based robot motion generation.

 The book has been put together by an international group of invited authors. It is a pleasure to thank them for their hard work and significant contributions. I gratefully acknowledge theassistance provided by Mr. Nino Popović, who was theauthor service manager through‐ out the publishing process, as well as the IntechOpen for the opportunity to edit this volume that examines the subject of manifolds.

> **Professor Paul Bracken**  Department of Mathematics University of Texas, USA

**Section 1**

**Theoretical**

**Section 1** 
