By P Li, Gang Tian, Shiu-Yuen Cheng

This quantity is a suite of study papers on nonlinear partial differential equations and similar components, representing many facets of latest advancements in those parts. particularly, the subsequent are incorporated: nonlinear conservation legislation; semilinear elliptic equations, nonlinear hyperbolic equations; nonlinear parabolic equations; singular restrict difficulties; and research of tangible and numerical ideas. vital components equivalent to numerical research, rest concept, multiphase thought, kinetic thought, combustion thought, dynamical platforms and quantum box idea 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 pal / 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 essential Geometry in Klein areas -- an easy Intrinsic evidence 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 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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**Example text**

If we use a different connection w' on P with curvature matrix n' then we get a different closed form Q(w') on M with IP(Q(w')) = Q(n'). What is the relation between Q(w') and Q(w)? Weil provided Chern with the necessary lemma: they differ by an exact form, so that [Q(w)] is a well-defined element of H+ (M), independent of the connection. We 58 The Life and Mathematics of Shiing-Shen Chern will denote it by Q(P). (Weil's lemma can be derived as a corollary of the fact that Q(Q) is closed. 298).

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.

But for the finite dimensional problems of geometry it is preferable to stick with these finite dimensional models) . By replacing the real numbers respectively by the complex numbers and the quaternions, Chern and Sun proved analogo1lS results for the other classical groups U(n) and Sp(n). They went on to note that if G is any compact Lie group, then by taking a faithful representation of G in some O(n), yen, N + n) becomes a prinCipal G bundle by restriction, and the corresponding orbit space Yen, N +n)/G becomes a classifying space Be for compact ANR's of dimension ~ k.