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INTRODUCTORY DIFFERENTIAL GEOMETRY FOR PHYSICISTS INTRODUCTORY DIFFERENTIAL GEOMETRY FOR PHYSICISTS This book has its origin in a set of lectures given by the late Professor A. Visconti who could not see it through to its final form. As it is, it ought to be useful to a wide variety of readers, since the exposition has been deliberately kept close to geometrical and physical in tuition. INTRODUCTION "The axiomatisation and algebraisation of mathematics, which has been taking place in the last 50 years, has made many mathematical texts so illegible that the old danger of complete loss of contact with physics and with natural sciences has become a reality" V. I. Arnold At first sight, such a statement may seem exaggerated. I believe how ever that it is a faithful description of reality. Indeed a characteristic of mathematicians and physicists of the XIXth century was the existence of a common language. There was consequently no linguistic barrier between them, the only obstacles to comprehension being the ones that arose from an incomplete mastery of the subject matter or from the requirements of rigour. The landscape changes at the beginning of the XXth century. The development of set theory and the resolution of paradoxes that arose there forced the mathematicians to work with ever increasing precision and led eventually to the necessity of a new language. The ideal solution would have been to coin new names for new objects; however the vocabulary is iii iv Introduction not infinitely extensible, and the new objects had to be described by old words. The language was changing and the physicists were confused by the use of terms which did not mean any more what they had meant for the last three centuries. In such a way a barrier started arising. This barrier was even more important because of the fact that the points of departure of the mathematical reasoning were not any more the ones that had been found spontaneously starting from the immediate in terpretation of experience. The new framework did allow the reaching of traditional conclusions which had been verified many times by experience, but this was often at the price of a certain lengthiness and of intellectual contorsions. The gap between physics and mathematics, which was appar ent already at the end of the XIXth century became an abyss after 1950. It is good to remark, in order to excuse a certain inertia, that this new language introduced an additional difficulty in the application of the "Correspondence Principle". This principle is a criterion of validity for a new discipline (for example, Special Relativity) which extends and general izes a discipline that is older {for instance, Newtonian Mechanics); it states that there exists a common domain where the predictions of the first are in accordance with the statements of the second. For instance, the intrinsic formulation of Differential Geometry (which will be described in this work) will be shown at Level 2 to extend the taxonomic formulation described at Level 1. All the valid results of the first Level should be found as statements of the second. The verification of this assertion is however not always as straightforward as one might wish! The present work has no other ambition than to be useful: its aim is to show to the Physicist that his discussions are sometimes unfinished in their logical structure, and to make the Mathematician aware of certain advan tages to the scientific community if he changed his methods of exposition so as to become more easily understandable to his colleague the Physicist. The plan of the present work is extremely simple; it is divided into three levels corresponding to three chapters: Level 0 - the nearest to intu ition and to geometrical experience - is a short summary of the theory of curves and surfaces as it was developed in the XIXth century. In Level 0, metric concepts play a central part, indeed the most elementary experi- Introduction v ment involving a curved surface consists of measuring lengths of arcs of curves of the surface with a simple string. Starting from there, one is led straight away to the fundamental notions of Christoffel symbols, curvature, Riemann tensor and geodesics. The landscape changes at Level 1 (Chap ter II): the concept of the variance of a field (i.e. description of how"a field changes under a point transformation) allows the classification of 'tensor fields, then the introduction - independent of any metrical considerations - of Christoffel symbols or affine connection coefficients lead to the notion of absolute or covariant derivative. Given an hypersurface, suppose that a vector field describes under certain conditions (parallel transpprt) a closed curve of that surface: it may happen that its initial value differs from the final one when the closed curve has been described. That difference can be used to define the curvature of the hypersurface. All these properties have nothing to do with the metrical properties of the surface, the study of which constitutes a separate chapter of that Level 1 (Riemannian geometry). At Level 2 (Chapter III) the point of view is again different from the previous one: the fundamental concept here is the one of manifold. On a manifold, one introduces the notion of tangent vector attached to a given point and one defines a tangent vector as a vector of the manifold under study. One considers also linear forms attached to any point of a manifold, and using the technique of tensorial products one is finally able to define tensors (at a point) of different orders and types. After all these preliminaries one defines an affine connection on a manifold and the affine connection coef ficients appear in the local expression of the connection. Again, metrical properties and riemannian geometry are introduced in a final step. But the main characteristic of the notions introduced at that Level 2 lies in their intrinsic presentation which avoids, as far as possible (in contrast to Lev els 0 and 1), all recourse to special frames leading to particular coordinates of a point of the considered manifold or to components of the mathematical objects defined there. There are advantages in this procedure, first intellectual ones: concepts loosely defined acquire a distinct and clear meaning and secondly formal ones: the avalanche of indices common to the methods of Levels 0 and 1 is by now stopped, formulas can be written in a compact form which can vi Introduction be used to simplify certain involved calculations. Indeed such an approach often appears clear and simple but this is not always the case. Some intu itive and fundamental concepts, which were near to everyday experiments presented within the framework of Level 0, become, of course, more general but also more abstract and the average physicist may find some difficulty in understanding them correctly when they are presented in that novel perspective. The final Chapter IV is dedicated to the study of fibre spaces, and that constitutes another and new presentation of differential geometry. It had been for some years familiar to mathematicians, but particle physicists (the theoretical ones!) began using the language in their study of gauge theories. This chapter of mathematics received there a renewed attention from their part and the application of fibre space techniques to physics (and to general relativity which can be also considered as a gauge theory) is by now very extensive. In the present textbook we give a short and elementary survey of the domain of fibre spaces: the interested reader should consult some of the references given in the bibliography. Exercises and problems have been added at the end of Levels 0, 1, 2 (Chapters I, II, III): each problem has its own title which summarizes the subject under study. Each problem is followed by "Hints" or by a "solution" more or less complete depending on its difficulty. Problems of Level 0 deal with the theory of surfaces using traditional notations, the study of the curvature of surfaces has been emphasized. We also devoted some attention to variational problems which are important for physical applications. Exercises and problems of Level 1 (Chapter II) deal with tensor anal ysis using the standard presentation and notations which are familiar to all students of general relativity. A set of the problems of that level has been devoted to physical applications ranging from the study of Lorentz transformation and Maxwell equations to the Robertson-Walker metric of general relativity. The exercises and problems of Level 2 are devoted to the study of the intrinsic presentation of differential geometry. Problems dealing with