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Global differential geometry. --- Ricci flow. --- Riemannian manifolds.

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Poincaré conjecture --- Ricci flow --- Differential equations, Partial --- Differential equations, Partial. --- Poincaré conjecture. --- Ricci flow.

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This book contains a self-consistent treatment of a geometric averaging technique, induced by the Ricci flow, that allows comparing a given (generalized) Einstein initial data set with another distinct Einstein initial data set, both supported on a given closed n-dimensional manifold. This is a case study where two vibrant areas of research in geometric analysis, Ricci flow and Einstein constraints theory, interact in a quite remarkable way. The interaction is of great relevance for applications in relativistic cosmology, allowing a mathematically rigorous approach to the initial data set averaging problem, at least when data sets are given on a closed space-like hypersurface. The book does not assume an a priori knowledge of Ricci flow theory, and considerable space is left for introducing the necessary techniques. These introductory parts gently evolve to a detailed discussion of the more advanced results concerning a Fourier-mode expansion and a sophisticated heat kernel representation of the Ricci flow, both of which are of independent interest in Ricci flow theory. This work is intended for advanced students in mathematical physics and researchers alike. .

Ricci flow. --- Flow, Ricci --- Evolution equations --- Global differential geometry

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This book provides an up-to-date introduction to the theory of manifolds, submanifolds, semi-Riemannian geometry and warped product geometry, and their applications in geometry and physics. It then explores the properties of conformal vector fields and conformal transformations, including their fixed points, essentiality and the Lichnerowicz conjecture. Later chapters focus on the study of conformal vector fields on special Riemannian and Lorentzian manifolds, with a special emphasis on general relativistic spacetimes and the evolution of conformal vector fields in terms of initial data. The book also delves into the realm of Ricci flow and Ricci solitons, starting with motivations and basic results and moving on to more advanced topics within the framework of Riemannian geometry. The main emphasis of the book is on the interplay between conformal vector fields and Ricci solitons, and their applications in contact geometry. The book highlights the fact that Nil-solitons and Sol-solitons naturally arise in the study of Ricci solitons in contact geometry. Finally, the book gives a comprehensive overview of generalized quasi-Einstein structures and Yamabe solitons and their roles in contact geometry. It would serve as a valuable resource for graduate students and researchers in mathematics and physics as well as those interested in the intersection of geometry and physics.

Global analysis (Mathematics). --- Manifolds (Mathematics). --- Geometry, Differential. --- General relativity (Physics). --- Global Analysis and Analysis on Manifolds. --- Differential Geometry. --- General Relativity. --- Conformal geometry. --- Ricci flow.

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Ricci Flow for Shape Analysis and Surface Registration introduces the beautiful and profound Ricci flow theory in a discrete setting. By using basic tools in linear algebra and multivariate calculus, readers can deduce all the major theorems in surface Ricci flow by themselves. The authors adapt the Ricci flow theory to practical computational algorithms, apply Ricci flow for shape analysis and surface registration, and demonstrate the power of Ricci flow in many applications in medical imaging, computer graphics, computer vision and wireless sensor network. Due to minimal pre-requisites, this book is accessible to engineers and medical experts, including educators, researchers, students and industry engineers who have an interest in solving real problems related to shape analysis and surface registration. .

Algorithms. --- Mathematical physics. --- Riemannian manifolds. --- Ricci flow --- Computer vision --- Discrete groups --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Computer science --- Computer vision. --- Discrete groups. --- Geometry. --- Mathematics. --- Math --- Groups, Discrete --- Machine vision --- Vision, Computer --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Computer graphics. --- Computer mathematics. --- Convex geometry. --- Discrete geometry. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Computational Mathematics and Numerical Analysis. --- Convex and Discrete Geometry. --- Science --- Euclid's Elements --- Infinite groups --- Artificial intelligence --- Image processing --- Pattern recognition systems --- Optical data processing. --- Convex geometry . --- Combinatorial geometry --- Optical computing --- Visual data processing --- Bionics --- Integrated optics --- Photonics --- Computers --- Optical equipment --- Ricci flow. --- Evolution equations.