现代几何结构和场论（影印版）_（俄罗斯）诺维科夫（S. P. Novikov），（俄罗斯）泰曼诺夫（I.A. Taimanov）著_9787040469189

本书将黎曼几何现代形式的基础表示为微分流形的几何及其上最重要的结构。作者的处理方法是：黎曼几何的所有构造都源于一个可以让我们计算切向量之标量积的流形。按此方式，作者展示了黎曼几何对于现代数学几个基础领域及其应用的巨大影响。 ● 几何是纯数学与自然科学首先是物理学之间的一个桥梁。自然界基本规律严格表示为描述各种物理量的几何场之间的关系。 ● 对几何对象整体性质的研究导致了拓扑学深远的发展，这包括了纤维丛的拓扑与几何。 ● 描述许多物理现象的哈密顿系统的几何理论导致了辛几何和泊松几何的发展。本书讲述的场论和高维变分学将数学与理论物理统一了起来。 ● 复几何和代数流形将黎曼几何和现代复分析、代数和数论统一了起来。本书的预备知识包括几门基础的本科课程，如高等微积分、线性代数、常微分方程以及拓扑要义。

　　As far back as in the late 1960s one of the authors of this book started preparations to writing a series of textbooks which would enable a modern young mathematician to learn geometry and topology.By that time， quite a number of problems of training nature were collected from teaching experi-ence.These problems （mostly topological） were included into the textbooks [DNF1l-[NF] or published as a separate collection [NMSF].The program mentioned above was substantially extended after we had looked at text-books in theoretical physics （especially， the outstanding series by Landau and Lifshits， considerable part of which， e.g.， the books [LL1， LL2l， involve geometry in its modern sense）， as well as from discussions with special-ists in theoretical mechanics， especially L.I.Sedov and V.P.Myasnikov，in the Mechanics Division of the Mechanics and Mathematics Department of Moscow State University， who were extremely int'erested in establish-ing courses iri modern geometry needed first of all in elasticity and other branches of mechanics.Remarkably， designing a modern course in geometry began in 1971 within the Mechanics， rather than the Mathematics Division of the department because this was where this knowledge was really needed.Mathematicians conceded to it later.Teaching these courses resulted inpublication of lecture notes （in duplicated form）：
　　S.P.Novikov， Rieman，nian geometry and tensor an，alysis.Parts I and II， Moscow State University， 1972/73.
　　Subsequently these courses were developed and extended， including， iri particular， elements of topology， and were published as：
　　S.P.Novikov and A.T.Fomenko， Riemannian， geometry and tensor an，alysis.Part Ⅲ， Moscow State University， 1974.
　　After that， S.P.Novikov wrote the program of a course in the fundamen-tals of modern geometry and topology.It was realized in a series of books [DNF1l-[NFl， written jointly with B.A.Dubrovin and A.T.Fomenko.Afterwards the topological part was completed by the book [N1]， which contained a presentation of the basic ideas of classical topology as they have formed by the late 1960s-early 1970s.The later publication [N2] also in-cluded some recent advances in topology， but quite a number of deep new areas （such as， e.g.， modern symplectic and contact topology， as well as new developments in the topology of 4-dimensional manifolds） were not covered yet.We recommend the book [AN].We can definitely say that even now there is no comprehensible textbook that would cover the main achieve-ments in the classical topology of the 1950s-1970s， to say nothing of the later period.Part II of the book [1] and the book [2] are insufficient， other books are sometimes unduly abstract； as a rule， they are devoted to spe-cial subjects and provide no systematic presentation of the progress made during this period， very important in the history of topology.Some well-written books （e.g.， [M1]-[MS]） cover only particular areas of the theory.The book [BT] is a good supplement to [DNF1， DNF2]， but its coverage is still insufficient.
　　Nevertheless， among our books， Part II of [DNF1] is a relatively good textbook containing a wide range of basic theory of differential topology in its interaction with physics.Nowadays this book could be modernized by essentially improving the technical level of presentation， but as a whole this book fulfills its task， together with the books [DNF2] and [Nl]， intended for a more sophisticated reader.
　　As for Part I， i.e.， the basics of Riemannian geometry， it has become clear during the past 20 years that this book must be substantially revised，as far as the exposition of basics and more complete presentation of modern ideas are concerned.To this end， the courses [T] given by the second author，I.A.Taimanov， at Novosibirsk University proved to be useful.We joined our efforts in writing a new course using all the material mentioned above.
　　We believe that the time has come when a wide community of mathe-maticians working in geometry， analysis， and related fields will finally turn to the deep study of the contribution to mathematics made by theoretical physics of the 20th century.This turn was anticipated already 25 years ago，but its necessity was not realized then by a broad mathematical community.The advancements in this direction made in our books such as [DNF1] had not elicited a proper response among mathematicians for a long time.In our view， the situation is different nowadays.Mathematicians understand much better the necessity of studying mathematical tools used by physicists.Moreover， it appears that the state of the art in theoretical physics itself is After that， S.P.Novikov wrote the program of a course in the fundamen-tals of modern geometry and topology.It was realized in a series of books [DNF1l-[NFl， written jointly with B.A.Dubrovin and A.T.Fomenko.Afterwards the topological part was completed by the book [N1]， which contained a presentation of the basic ideas of classical topology as they have formed by the late 1960s-early 1970s.The later publication [N2] also in-cluded some recent advances in topology， but quite a number of deep new areas （such as， e.g.， modern symplectic and contact topology， as well as new developments in the topology of 4-dimensional manifolds） were not covered yet.We recommend the book [AN].We can definitely say that even nowthere is no comprehensible textbook that would cover the main achieve-ments in the classical topology of the 1950s-1970s， to say nothing of the later period.Part II of the book [1] and the book [2] are insufficient， other books are sometimes unduly abstract； as a rule， they are devoted to spe-cial subjects and provide no systematic presentation of the progress made during this period， very important in the history of topology.Some well-written books （e.g.， [M1]-[MS]） cover only particular areas of the theory.The book [BT] is a good supplement to [DNF1， DNF2]， but its coverage is still insufficient.
　　Nevertheless， among our books， Part II of [DNF1] is a relatively good textbook containing a wide range of basic theory of differential topology in its interaction with physics.Nowadays this book could be modernized by essentially improving the technical level of presentation， but as a whole this book fulfills its task， together with the books [DNF2] and [Nl]， intended for a more sophisticated reader.
　　As for Part I， i.e.， the basics of Riemannian geometry， it has become clear during the past 20 years that this book must be substantially revised，as far as the exposition of basics and more complete presentation of modern ideas are concerned.To this end， the courses [T] given by the second author，I.A.Taimanov， at Novosibirsk University proved to be useful.We joined our efforts in writing a new course using all the material mentioned above.
　　We believe that the time has come when a wide community of mathe-maticians working in geometry， analysis， and related fields will finally turn to the deep study of the contribution to mathematics made by theoretical physics of the 20th century.

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