By P Li, Gang Tian, Shiu-Yuen Cheng
This quantity is a suite of study papers on nonlinear partial differential equations and comparable parts, representing many points 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 options. very important components akin to numerical research, rest idea, multiphase thought, kinetic thought, combustion concept, dynamical platforms and quantum box conception also are lined The existence 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 imperative 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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Additional resources for A mathematician and his mathematical work : selected papers of S.S. Chern
Finally, Chern realized that in this setting one could describe geometrically the invariants for a G-structure given by Cartan's general method; in fact they can all be calculated from the curvature forms of the intrinsic connection. Note that this covers one of the most important examples of a Gstructure; namely the case G = O(n), corresponding to Riemannian geometry. The intrinsic connection is of course the "Levi-Civita connection" . , to find explicitly a complete set of local invariants for a Riemannian metric.
If M is a smooth, compact, oriented n-dimensional manifold then e(F(M» E Hn(M) when evaluated on the fundamental class of M is just X(M). One can thus interpret the Generalized GaussBonnet Theorem as saying that). = Pf(w) represents e(F(M}) in de Rham cohomology. This inspired Chern to look for a general technique for representing characteristic classes by de Rham classes. This was in 1944-1945, while Chern was in Princeton, and he discussed this problem frequently with his friend Andre Wei! who encouraged him in this search.
Chern feels that the time is now ripe to recast all the beautiful global results of Riemannian geometry of the past several decades in the Finsler context, and he points out that thinking of Riemannian geometry as a special case of Finsler geometry was already advocated by David Hilbert in his twenty-third problem a t the turn of the last century. R. Acad. Sci. Paris, t. 314, SQrieI, p. 1, 1993). He has also recently spelled out the general program in a paper that is as yet unpublished, "Riemannian geometry as a special case of Finsler geometry".
A mathematician and his mathematical work : selected papers of S.S. Chern by P Li, Gang Tian, Shiu-Yuen Cheng