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Dieses Buch bietet ein Panorama der Schicksale österreichischer
Mathematikerinnen und Mathematiker, deren Leben von der NS-Zeit
beeinflusst wurde. Zu Beginn wird in einem Ăśberblick das
allgemeine geistige und politische Klima und die Entwicklung des
Staates Österreich und besonders der universitären Institutionen
geschildert. Der Zeitraum umfasst den ersten Weltkrieg bis zur
Erholung der Republik Ă–sterreich nach dem 2. Weltkrieg.
Geographisch geht der Blick darĂĽber hinaus und erfasst auch
Mathematiker in den „im Reichsrathe vertretenen Königreichen und
Ländern“ sowie den zur anderen Reichshälfte der Doppelmonarchie
gehörenden deutschsprachigen Gebieten. Dazu gehören auch die
kriegsgefangenen französischen Mathematiker, gelegentlich
tschechische Mathematiker aus Brünn oder Prag, Gäste aus Polen
oder Ungarn, schlieĂźlich auch Mathematiker aus der slowenischen
Schule der altösterreichischen Lehrbuchautoren. Im Brennpunkt der
Betrachtung stehen die Menschen, die Mathematik treiben, das
Nebeneinander von individuellen, aber untereinander und mit
Ständestaat- und NS-Institutionen verflochtenen Biografien.
The aim of this work is threefold: First it should be a
monographical work on natural bundles and natural op erators in
differential geometry. This is a field which every differential
geometer has met several times, but which is not treated in detail
in one place. Let us explain a little, what we mean by naturality.
Exterior derivative commutes with the pullback of differential
forms. In the background of this statement are the following
general concepts. The vector bundle A kT* M is in fact the value of
a functor, which associates a bundle over M to each manifold M and
a vector bundle homomorphism over f to each local diffeomorphism f
between manifolds of the same dimension. This is a simple example
of the concept of a natural bundle. The fact that exterior
derivative d transforms sections of A kT* M into sections of A
k+1T* M for every manifold M can be expressed by saying that d is
an operator from A kT* M into A k+1T* M."
The literature on natural bundles and natural operators in
differential geometry, was until now, scattered in the mathematical
journal literature. This book is the first monograph on the
subject, collecting this material in a unified presentation. The
book begins with an introduction to differential geometry stressing
naturality and functionality, and the general theory of connections
on arbitrary fibered manifolds. The functional approach to
classical natural bundles is extended to a large class of
geometrically interesting categories. Several methods of finding
all natural operators are given and these are identified for many
concrete geometric problems. After reduction each problem to a
finite order setting, the remaining discussion is based on
properties of jet spaces, and the basic structures from the theory
of jets are therefore described here too in a self-contained
manner. The relations of these geometric problems to corresponding
questions in mathematical physics are brought out in several places
in the book, and it closes with a very comprehensive bibliography
of over 300 items. This book is a timely addition to literature
filling the gap that existed here and will be a standard reference
on natural operators for the next few years.
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