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本書將黎曼幾何現(xiàn)代形式的基礎(chǔ)表示為微分流形的幾何及其上最重要的結(jié)構(gòu)。作者的處理方法是：黎曼幾何的所有構(gòu)造都源于一個可以讓我們計算切向量之標(biāo)量積的流形。按此方式，作者展示了黎曼幾何對于現(xiàn)代數(shù)學(xué)幾個基礎(chǔ)領(lǐng)域及其應(yīng)用的巨大影響。 ● 幾何是純數(shù)學(xué)與自然科學(xué)首先是物理學(xué)之間的一個橋梁。自然界基本規(guī)律嚴(yán)格表示為描述各種物理量的幾何場之間的關(guān)系。 ● 對幾何對象整體性質(zhì)的研究導(dǎo)致了拓撲學(xué)深遠的發(fā)展，這包括了纖維叢的拓撲與幾何。 ● 描述許多物理現(xiàn)象的哈密頓系統(tǒng)的幾何理論導(dǎo)致了辛幾何和泊松幾何的發(fā)展。本書講述的場論和高維變分學(xué)將數(shù)學(xué)與理論物理統(tǒng)一了起來。 ● 復(fù)幾何和代數(shù)流形將黎曼幾何和現(xiàn)代復(fù)分析、代數(shù)和數(shù)論統(tǒng)一了起來。本書的預(yù)備知識包括幾門基礎(chǔ)的本科課程，如高等微積分、線性代數(shù)、常微分方程以及拓撲要義。

　　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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