Lectures on Differential Geometry by Shiing-Shen Chern, Wei-Huan Chen, K. S. Lam

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

This booklet is a translation of an authoritative introductory textual content in accordance with a lecture sequence added via the well known differential geometer, Professor S S Chern in Beijing collage in 1980. the unique chinese language textual content, authored by way of Professor Chern and Professor Wei-Huan Chen, was once a distinct contribution to the math literature, combining simplicity and economic climate of technique with intensity of contents. the current translation is aimed toward a large viewers, together with (but now not constrained to) complex undergraduate and graduate scholars in arithmetic, in addition to physicists drawn 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 intricate manifolds (with a succinct advent to the idea of Chern classes), and an appendix at the courting among differential geometry and theoretical physics, this ebook encompasses 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 specifically for this version by means of Professor Chern to carry the textual content into views.

Show description

Read Online or Download Lectures on Differential Geometry PDF

Similar geometry books

Geometry of Complex Numbers (Dover Books on Mathematics)

Illuminating, greatly praised ebook on analytic geometry of circles, the Moebius transformation, and 2-dimensional non-Euclidean geometries. "This e-book will be in each library, and each specialist in classical functionality conception can be accustomed to this fabric. the writer has played a different carrier by means of making this fabric so very easily available in one publication.

Geometric Tomography (Encyclopedia of Mathematics and its Applications)

Geometric tomography offers with the retrieval of data a few geometric item from facts referring to its projections (shadows) on planes or cross-sections through planes. it's a geometric relative of automatic tomography, which reconstructs a picture from X-rays of a human sufferer. the topic overlaps with convex geometry and employs many instruments from that region, together with a few formulation from indispensable geometry.

First Steps in Differential Geometry: Riemannian, Contact, Symplectic (Undergraduate Texts in Mathematics)

Differential geometry arguably bargains the smoothest transition from the traditional college arithmetic series of the 1st 4 semesters in calculus, linear algebra, and differential equations to the better degrees of abstraction and evidence encountered on the top department by means of arithmetic majors. this day it's attainable to explain differential geometry as "the examine of constructions at the tangent space," and this article develops this perspective.

Additional resources for Lectures on Differential Geometry

Sample text

C sgn (7 0 u) . (7 o uES(r) T . AT(z). 28) Therefore S,(T'(V)) c P'(V), A,(T'(V)) c A'(V). Furthermore, it is easy t o show that a symmetric tensor is invariant under the symmetrizing mapping and an alternating tensor is invariant under the alternating mapping. Therefore P T ( V )= S,(P'(V)), A'(V) = A,(A'(V)). Thus P'(V) = S,(TT(V)), A'(V) = A,(T'(V)). 0 The above discussion about symmetric and alternating contravariant tensors can be applied analogously t o covariant tensors. The set of all symmetric covariant tensors of order r is denoted by P'(V*),and the set of all alternating covariant tensors of order r by A r ( V * ) .

If V is an n-dimensional vector space over IF, then V' is also an ndimensional vector space over IF. To see this, suppose {ul , .. ,a,} is a basis of V , and n v = CV%Ei v, f E V'. i= 1 Then i=l Therefore the linear function f is determined by its values f ( u i ) , 1 5 i 5 n, on the basis. We may define linear functions a" E V ' , 1 5 i 5 n , such that Then a*i(w) = vi. 4) says that any element in V * can be expressed as a linear combination of {a*i, 1 5 i 5 n}. It is easy to see that the expression is unique, and therefore {a*', 1 5 i 5 n } is a basis of V * ,which is called the dual basis of {ai, 1 5 i 5 n}.

Therefore V can be viewed as a vector space formed by all IF-valued linear functions on V * . In other words, V is the dual space of V * . Now we generalize the discussion above. Assume that V ,W,2 are all finitedimensional vector spaces over the field IF. l. 1 + a2w2) =alf(w1) + a2f(w2). J , a h 1 a 2 2 . 2 ) = a l f ( v ,W l ) a 2 f ( v w2). , + Similarly we can define an r-linear map f : V1 x . . x V, are vector spaces over IF. 9) + 2 where V1,. . 1 become IF-valued linear functions, IF-valued bilinear functions, and IF-valued r-linear functions, respectively.

Download PDF sample

Rated 4.83 of 5 – based on 10 votes