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Gilkey, John V. Michor: Natural operations in differential geometry.

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Kutateladze: Fundamentals of Functional Analysis Russian. Kutateladze: Russian - English in writing. Advice to the occasional translator Russian. Marco Squassina: Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems. Electronic reprint of the original Originally published on paper as Volume 1 of the series Monographs and Studies in Mathematics by Pitman Publisher Ltd.

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Like mod-ern analysis itself, differential geometry originates in classical mechanics. For instance,geodesics and minimal surfaces are defined via variational principles and the curva-ture of a curve is easily interpreted as the acceleration with respect to the path lengthparameter. Modern differential geometry in its turn strongly contributed to modern physicswhen, for instance, at the beginning of the 20th century it was discovered by Einsteinthat a gravitational field is just a pseudo-Riemannian metric on space time. The basicequations of gravity theory were written in terms of the curvature of a metric, whichis a geometric quantity.

More recently the modern theory of elementary particles wasbased on gauge fields, which mathematically are connections on fiber bundles.

In this book we attempt to give an introduction to the basics of differential geometry,keeping in mind the natural origin of many geometrical quantities, as well as theapplications of differential geometry and its methods to other sciences. The book is divided into three parts. The first part covers the basics of curves andsurfaces, while the second part is designed as an introduction to smooth manifoldsand Riemannian geometry. In particular, in Chapter 5 we give short introductions tohyperbolic geometry and geometrical principles of special relativity theory.

Here, onlya basic knowledge of algebra, calculus and ordinary differential equations is required. The third part is more advanced and assumes that the reader is familiar with thefirst two parts of the book. It introduces the reader to Lie groups and Lie algebras, therepresentation theory of groups, symplectic and Poisson geometry, and the applicationsof complex analysis in surface theory.

We use Lie groups as important examplesof smooth manifolds and we expose symplectic and Poisson geometry in their closerelation with mechanics and the theory of integrable systems. This book is a translation with minor revisions and corrections of ,6 which is based on lecture notes7 that arose froma one-semester course given in Novosibirsk State University, and on some advancedminicourses. The contents and the concise style of exposition are due to the expectation that thebook will be suitable for a full semester course at the upper undergraduate or beginninggraduate level.

During the course at Novosibirsk State University we included a com-plete treatment of Chapters 1, 2, and 5. Chapters 3 and 4 give a much more detailedexposition of some further material which was touched upon in the course. Curves and surfaces. Note that, unless specified to the contrary, repeated upper and lower indices implysummation. For example. Although this notation rarely appears in textbooks, it is widely used in scientific pub-lications, especially in physical applications of differential geometry.

It is one of themore practical ways in which the book will be useful for students with wider interestsin such applications of differential geometry. The Bibliography is shorter than the list that appeared in the Russian edition, whichcontains many publications unavailable in English translation. Hence we have addedsupplementary reading sources in English.

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For further bibliography we refer to [4]. Let Rn be the n-dimensional Euclidean space with coordinates x1, : : : , xn.

- Lectures on Differential Geometry (EMS Series of Lectures in Mathematics);
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For brevity, we denote an n-dimensional vector space over the field R also by Rn,assuming that it is always clear from the context which object is meant. We use thenotation. About the Author : Iskander A.

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