|
Showing 1 - 4 of
4 matches in All Departments
This treatment of differential geometry and the mathematics
required for general relativity makes the subject accessible, for
the first time, to anyone familiar with elementary calculus in one
variable and with some knowledge of vector algebra. The emphasis
throughout is on the geometry of the mathematics, which is greatly
enhanced by the many illustrations presenting figures of three and
more dimensions as closely as the book form will allow.
This treatment of differential geometry and the mathematics required for general relativity makes the subject of this book accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow. The imaginative text is a major contribution to expounding the subject of differential geometry as applied to studies in relativity, and will prove of interest to a large number of mathematicians and physicists. Review from L'Enseignement Mathématique
The main motivation for this book lies in the breadth of
applications in which a statistical model is used to represent
small departures from, for example, a Poisson process. Our approach
uses information geometry to provide a c- mon context but we need
only rather elementary material from di?erential geometry,
information theory and mathematical statistics. Introductory s-
tions serve together to help those interested from the applications
side in making use of our methods and results. We have available
Mathematica no- books to perform many of the computations for those
who wish to pursue their own calculations or developments. Some 44
years ago, the second author ?rst encountered, at about the same
time, di?erential geometry via relativity from Weyl's book [209]
during - dergraduate studies and information theory from Tribus
[200, 201] via spatial statistical processes while working on
research projects at Wiggins Teape - searchandDevelopmentLtd-cf.
theForewordin[196]and[170,47,58]. H- ing started work there as a
student laboratory assistant in 1959, this research environment
engendered a recognition of the importance of international c-
laboration, and a lifelong research interest in randomness and
near-Poisson statistical geometric processes, persisting at various
rates through a career mainly involved with global di?erential
geometry. From correspondence in the 1960s with Gabriel Kron [4,
124, 125] on his Diakoptics, and with Kazuo Kondo who in?uenced the
post-war Japanese schools of di?erential geometry and supervised
Shun-ichi Amari's doctorate [6], it was clear that both had a much
wider remit than traditionally pursued elsewhere.
Many geometrical features of manifolds and fibre bundles modelled
on Frechet spaces either cannot be defined or are difficult to
handle directly. This is due to the inherent deficiencies of
Frechet spaces; for example, the lack of a general solvability
theory for differential equations, the non-existence of a
reasonable Lie group structure on the general linear group of a
Frechet space, and the non-existence of an exponential map in a
Frechet-Lie group. In this book, the authors describe in detail a
new approach that overcomes many of these limitations by using
projective limits of geometrical objects modelled on Banach spaces.
It will appeal to researchers and graduate students from a variety
of backgrounds with an interest in infinite-dimensional geometry.
The book concludes with an appendix outlining potential
applications and motivating future research.
|
|