By Shiing-Shen Chern, Wei-Huan Chen, K. S. Lam

This e-book is a translation of an authoritative introductory textual content in accordance with a lecture sequence added by means of the popular differential geometer, Professor S S Chern in Beijing collage in 1980. the unique chinese language textual content, authored by means of Professor Chern and Professor Wei-Huan Chen, used to be a distinct contribution to the maths literature, combining simplicity and economic climate of strategy with intensity of contents. the current translation is geared toward a large viewers, together with (but no longer restricted to) complex undergraduate and graduate scholars in arithmetic, in addition to physicists attracted to the various purposes of differential geometry to physics. as well as a radical remedy of the basics of manifold idea, external algebra, the outside calculus, connections on fiber bundles, Riemannian geometry, Lie teams and relocating frames, and complicated manifolds (with a succinct creation to the idea of Chern classes), and an appendix at the dating among differential geometry and theoretical physics, this booklet incorporates a new bankruptcy on Finsler geometry and a brand new appendix at the historical past and up to date advancements of differential geometry, the latter ready especially for this version by means of Professor Chern to carry the textual content into views.

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**Example text**

X V, to 2 by C(V1,.. ,V,; 2 ) . Suppose f,g E C(V1,.. ,V,; Z ) , a E IF. For any vi E &, 1 5 i 5 r, define + g and af are also in L(V1,.. , V,; 2 ) . Obviously, the set Then f C(V1,.. , V,) is a vector space over IF with respect t o the two operations. The structure of the space C ( V ;2 ) is simpler. Choose a basis { a l , . . ,an} of V and a basis { b l , . . ,bm} of 2. Then f E C ( V ;2 ) is determined by its action on the basis { u i } . Suppose m f(ai)= C f i j b j , 15 i 5 n. 11) j=1 Then f corresponds to the n x m matrix (fij).

Suppose M and N are both n-dimensional smooth manifolds, and f : M -+N is a smooth m a p . If at a point p E M , the tangent m a p f* : T,(M) --+ T f ( p ) ( Nis) a n isomorphism, t h e n there exists a neighborhood U of p in M such that V = f ( U ) is a neighborhood of f ( p ) in N and f l u : U -+ V is a diffeomorphism. Proof. 3) is a smooth map. Obviously the determinant of the Jacobian of f at point f ( p ) is nonzero. 1, there exist neighborhoods 0 c cp(U0) and V c +(Vo) of cp(p) and $ ( q ) in Rn, respectively, such that f \- : 0 -+ is a U diffeomorphism.

Suppose is a basis of Vi. Then 1 5 ji 5 ni, i = 1 , . . @V,)= d i m V l - d i m V 2 - ... -dimV,. 27) a;:) 8 a (2) j, @ . . 8 a;:), is a basis of V1 @ . . @ V,. @V,=C(VT, . . ,V,*;IF). 3. Suppose h : V1 x . . x V, --+V1@.. , for any ui E V,, 1 5 i 5 s, h(u1, . . @us. 28) Then for any f E L(V1,. . , V,; Z ) , there exists a unique linear m a p g E C ( V l 8 . . ~V,--+Z. 1, and should be carried out by the reader. 47 $2-2: Tensors 52-2 Tensors We have discussed some general tensor product concepts in the last section.