


This book is a faithful record of Lectures which thd author delivered in the first semester of a twoyear course in geometry at the MathematicsMechanics Faculty of Moscow State University to students studying mathematics. The contents of these Lectures were determined by the curriculum, by the established traditions of the Faculty's Department of Higher Geometry and Topology, by the needs of the second semester's course and by the author's personal aims. The sequence of presentation was governed by the necessity of agreement with the concurrently delivered courses in algebra and analysis, by the requirements of the assistants conducting seminars and by other similar considerations of no fundamental consequence but of paramount practical importance. For example, the decision to consider some question or other at one of the last Lectures was dictated by the impracticability of consolidating the material of those Lectures with the aid of exercises. The contents of the very last Lecture were determined by the fact that owing to the postponement of some Lectures because of the intervening holidays it would often fall on the examination period and is sometimes not delivered at all for lack of time, etc. etc. Only two features of the book seem to deserve special mention. The first is that from the outset the exposition is based on axioms and geometric visualization is made use of solely for propaedeutic purposes. For obvious reasons, from the many possible systems of axioms the "vectorpoint" one developed by H. Weyl has been chosen. This accounts for the unusually early introduction of the general concept of ector space. Experience has shown that as a rule students learn this material without difficulty. The other, more controversial, feature of the book is a systematic development and use of bivectors and trivectors. This makes it possible to separate distinctly the affine part of the theory from its metric part and provides a background for a general theory of multivectors in the second semester. Each "Lecture" in the book is really a two hours' discourse, as a rule. This explains why a previous topic often gives way to a new one in the middle of a Lecture. One exception is the last, 28th, Lecture which is a combination of two different versions of the concluding Lecture. Because of the specific character of oral and written forms of presentation, the "isochronous" Lectures have turned out to be of different lengths in the book. Their number is accounted for by the fact that although the curriculum assigns 36 Lectures to a course in analytic geometry, in practice it has to be ended as early as the 28th Lecture or earlier. I also wish to express my gratitude to TP. Aldatova for her prompt and excellent typing of the original manuscript. M.M. Postnikov After the Russian edition of this book appeared some of my fellow Lecturers asserted that many of the Lectures in the book are far too long to be physically delivered during the allowed two teaching periods. By the right of friendship I had to remind them that a Lecturer must prepare for his Lectures—even if he has been lecturing for over a dozen years— and make in advance an elaborate, practically minutebyminute plan of every Lecture. It is necessary to consider eforehand the rhythm of the Lecture to be delivered—what portions of it are to be read slowly, almost at dictation speed, and what may be said quicker—and its pattern of intonation—where to raise the voice and where to lower it. One also needs a joke somewhere about the middle of the Lecture to rouse the tired students and it should be prepared yet at home, and in every detail, up to a play of facial muscles. It goes without saying that one must plan in advance what to write on the blackboard and in what order and where, and when to delete anything, and coordinate all this with everything else. It is surprising how all this extends the limits of Lecture time and how much it is then possible to say in an outwardly unhurried and thorough manner, with numerous repetitions and explanations. Some reviewers have reproached me for a systematic use of bivectors and trivectors saying that one may well do without them. Some wellknown physicist, Max Planck, I think, once said that new ideas (he meant scientific ideas but this can be fully applied to methodical ideas as well), could win only when their opponents have retired from the stage as a result of a natural change of generations. An ex cellent example illustrating this thesis is the introduction of vectors into the courses of analytic geometry half a century ago. Now only a few people remember the fierce discussions concerning this matter and the present generation does not know how many a lance was broken and how much ink split in attempts to prove that vectors were a harmful thing because replacing three equations in coordinates by one vector equation they saved paper but proportionally hampered comprehension. The last of the authoritative opponents of vectors in the USSR died soon after the war but some ten years more passed before diffidently excusatory reservations disappeared altogether from vectorial presentations of geometry (as well as from mechanics and physics where, however, this happened a little earlier). Now bivectors and trivectors are awaiting their turn. I have taken the opportunity to introduce some minor improvements in the text. The most serious one is perhaps a simpler construction of the complexification of an affine space in Lecture 19. It is true that it contains a certain element of arbitrariness (which was what restrained me at first) but experience has shown that this arbitrariness is perfectly harmless. Besides, the last Lecture has been divided into two since two versions of the concluding Lecture were combined in it for purely technical, internal editorial reasons. So the book contains 29 Lectures now. As far as I can judge with my poor knowledge of English the translation is well done and conveys all the nuances of my thought. M.M. Postnikov May 1, 1980 