By P Li, Gang Tian, Shiu-Yuen Cheng

This quantity is a set of study papers on nonlinear partial differential equations and comparable components, representing many points of latest advancements in those parts. specifically, the next are incorporated: nonlinear conservation legislation; semilinear elliptic equations, nonlinear hyperbolic equations; nonlinear parabolic equations; singular restrict difficulties; and research of actual and numerical recommendations. vital components reminiscent of numerical research, rest conception, multiphase idea, kinetic thought, combustion thought, dynamical platforms and quantum box conception also are lined The lifestyles and arithmetic of Shiing-Shen Chern / R.S. Palais and C.-L. Terng -- My Mathematical schooling / S.S. Chern -- A precis of My medical existence and Works / S.S. Chern -- S.S. Chern as Geometer and good friend / A. Weil -- a few Reflections at the Mathematical Contributions of S.S. Chern / P.A. Griffiths -- Shiing-Shen Chern as pal and Mathematician / W.-L. Chow -- Abzahlungen fur Gewebe -- On necessary Geometry in Klein areas -- an easy Intrinsic facts of the Gauss-Bonnet formulation for Closed Riemannian Manifolds -- at the Curvatura Integra in a Riemannian Manifold -- attribute sessions of Hermitian Manifolds -- Sur une Classe Remarquable de Varietes dans l'espace Projectif a N Dimensions -- A Theorem on Orientable Surfaces in 4-dimensional area / S.S. Chern and E. Spanier -- at the Kinematic formulation within the Euclidean house of N Dimensions -- On a Generalization of Kahler Geometry -- at the overall Curvature of Immersed Manifolds / S.S. Chern and R.K. Lashof

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**Example text**

We call P a universal principal G-b\Uldle if the latter map is bijective. The heart of the classification theorem is the fact that universal G-bundles do exist . In fact it can be shown that a principal G-b\Uldle is \Uliversal provided its total space is contractible, and there are even a number of methods for explicitly constructing such bundles. We will denote by UG some choice of universal principal G-b\Uldle. Its base space will be denoted by BG and is called the classifying space for G. (Although BG is not unique, its homotopy type is) .

T,) course P(al[Z],.. ,u,[Z]) is also in R". = trace(Z) E C[tl, . , t,] then of In fact, R" = C[u1,. . , P ( t l , . . ,t,) H P(al[Z],. . , u,[Z]) is a ring isomorphism. From this fact, together with Ehresmann's explicit description of the homology of complex Grassmannians, Chern was easily able to verify that the Chern- Weil homomorphism is in fact an isomorphism of Rd with Char(U(n)). l, = uk (& Z) . Then we get a of dimension 2k, called the k t h Chern class, and these n classes cl, .

Is called the torsion of the connection, and what Chern exploited was the fact that he could in certain cases define "intrinsic" N(G) connections by putting conditions on T. For example, the Levi-Civita connection can be characterized as the unique N(O(n)) connection on N(P) such that T = O. In fact, in [43] Chern showed that if the Lie algebra L( G) satisfied a certain simple algebraic condition ("property C") then it was always possible to define an intrinsic N(G) connection in this way, and he proved that any compact G satisfies property C.