By P Li, Gang Tian, Shiu-Yuen Cheng
This quantity is a suite of analysis papers on nonlinear partial differential equations and similar components, representing many elements of latest advancements in those parts. particularly, 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. very important parts similar to numerical research, leisure thought, multiphase idea, kinetic idea, combustion idea, dynamical structures and quantum box concept 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 clinical existence and Works / S.S. Chern -- S.S. Chern as Geometer and pal / A. Weil -- a few Reflections at the Mathematical Contributions of S.S. Chern / P.A. Griffiths -- Shiing-Shen Chern as buddy 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 periods 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 area 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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Extra resources for A mathematician and his mathematical work : selected papers of S.S. Chern
Although there were many published proofs of this, Chern reproved it for himself by a new method that was very natural from a moving frames perspective. Moreover, unlike the published proofs, Chern's had the potential to generalize to higher dimensions. To explain Chern's method, we start by applying the standard moving frames approach to n-dimensional oriented Riemannian manifolds M, then specialize to n = 2. The orientation together with the Riemannian structure give an SO(n) structure for M .
Palais and Chuu-Lian Terng As Chern was starting his research career, a major challenge facing geometry was to find what this seemingly disparate class of examples had in common, and thereby discover a general framework for the Equivalence Problem. Cartan saw this clearly, and had already made important steps in that direction with his general machinery of ''moving frames". His approach was to reduce a general equivalence problem to one of a special class of equivalence problems for differential forms.
We will consider only the case of a Lie group G . Since the theory is essentially the same for a Lie group and one of its maximal compact subgroups, we will also assume that G is compact. A "space" will mean a paracompact topological space, and a G-space will mean a space, P, together with a continuous right action of G on P. We will write Rg for the homeomorphism p 1-+ P9. , if for all pin P, Rg(p) f- p unless 9 is the identity element e of G. More specifically, P is called a principal Gbundle over a space X if we are given some fixed homeomorphism of X with the orbit space PIG, or equivalently if there is given a "projection map" II : P --+ X such that the G orbits of P are exactly the "fibers" II-I (x) of the map II.
A mathematician and his mathematical work : selected papers of S.S. Chern by P Li, Gang Tian, Shiu-Yuen Cheng