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The DD6 Symposium was, like its predecessors DD1 to DD5 both a
research symposium and a summer seminar and concentrated on
differential geometry. This volume contains a selection of the
invited papers and some additional contributions. They cover recent
advances and principal trends in current research in differential
geometry.
This book gives a treatment of exterior differential systems. It
will in clude both the general theory and various applications. An
exterior differential system is a system of equations on a manifold
defined by equating to zero a number of exterior differential
forms. When all the forms are linear, it is called a pfaffian
system. Our object is to study its integral manifolds, i. e.,
submanifolds satisfying all the equations of the system. A
fundamental fact is that every equation implies the one obtained by
exterior differentiation, so that the complete set of equations
associated to an exterior differential system constitutes a
differential ideal in the algebra of all smooth forms. Thus the
theory is coordinate-free and computations typically have an
algebraic character; however, even when coordinates are used in
intermediate steps, the use of exterior algebra helps to
efficiently guide the computations, and as a consequence the
treatment adapts well to geometrical and physical problems. A
system of partial differential equations, with any number of inde
pendent and dependent variables and involving partial derivatives
of any order, can be written as an exterior differential system. In
this case we are interested in integral manifolds on which certain
coordinates remain independent. The corresponding notion in
exterior differential systems is the independence condition:
certain pfaffian forms remain linearly indepen dent. Partial
differential equations and exterior differential systems with an
independence condition are essentially the same object."
In the period since the original four volumes of Phillip
Griffiths's Selecta were published (Selected Works of Phillip A.
Griffiths with Commentary), Parts 1-4, Collected Works, Volume 18),
Griffiths has continued to produce beautiful and important work.
The current two-part publication brings Griffiths's Selecta up to
date by including the majority of his recent articles, as well as
two older papers on differential geometry whose length had
precluded their inclusion in the original Selecta. The papers are
organized along the three main topics: Differential Geometry and
Hodge Theory (Part 5) and Algebraic Cycles (Part 6). In addition to
his papers, Griffiths has been an author of a number of research
monographs. To give the reader an overview of what these monographs
contain, introductions to some of these are also included.
Finsler geometry generalizes Riemannian geometry in the same sense
that Banach spaces generalize Hilbert spaces. This book presents an
expository account of seven important topics in Riemann-Finsler
geometry, ones which have recently undergone significant
development but have not had a detailed pedagogical treatment
elsewhere. Each article will open the door to an active area of
research, and is suitable for a special topics course in
graduate-level differential geometry. The contributors consider
issues related to volume, geodesics, curvature, complex
differential geometry, and parametrized jet bundles, and include a
variety of instructive examples.
This book presents an expository account of six important topics in
Riemann-Finsler geometry suitable for in a special topics course in
graduate level differential geometry. These topics have recently
undergone significant development, but have not had a detailed
pedagogical treatment elsewhere. Each article will open the door to
an active area of geometrical research. Rademacher gives a detailed
account of his Sphere Theorem for non-reversible Finsler metrics.
Alvarez and Thompson present an accessible discussion of the
picture which emerges from their search for a satisfactory notion
of volume on Finsler manifolds. Wong studies the geometry of
holomorphic jet bundles, and finds that Finsler metrics play an
essential role. Sabau studies protein production in cells from the
Finslerian perspective of path spaces, employing both a local
stability analysis of the first order system, and a KCC analysis of
the related second order system. Shen's article discusses Finsler
metrics whose flag curvature depends on the location and the
direction of the flag poles, but not on the remaining features of
the flags. Bao and Robles focus on Randers spaces of constant flag
curvature or constant Ric
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