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Preface
The chapter on bundles and connections starts with a thorough treatment
of the Frolicher-Nijenhuis bracket via the study of all graded derivations
of the algebra of differential forms. This bracket is a natural extension
of the Lie bracket from vector fields to tangent bundle valued differential
forms; it is one of the basic structures of differential geometry. We begin
our treatment of connections in the general setting of fiber bundles (without
structure group). A connection on a fiber bundle is just a projection onto
the vertical bundle. Curvature and the Bianchi identity are expressed with
the help of the Frolicher-Nijenhuis bracket. The parallel transport for such
a general connection is not defined along the whole of the curve in the base
in general if this is the case, the connection is called complete. We
show that every fiber bundle admits complete connections. For complete
connections we treat holonomy groups and the holonomy Lie algebra, a
subalgebra of the Lie algebra of all vector fields on the standard fiber. Then
we present principal bundles and associated bundles in detail together with
the most important examples. Finally we investigate principal connections
by requiring equivariance under the structure group. It is remarkable how
fast the usual structure equations can be derived from the basic properties
of the Frolicher-Nijenhuis bracket. Induced connections are investigated
thoroughly we describe tools to recognize induced connections among
general ones. If the holonomy Lie algebra of a connection on a fiber bundle
with compact standard fiber turns out to be finite-dimensional, we are able
to show that in fact the fiber bundle is associated to a principal bundle and
the connection is an induced one. I think that the treatment of connections
presented here offers some didactical advantages: The geometric content of
a connection is treated first, and the additional requirement of equivariance
under a structure group is seen to be additional and can be dealt with later
so the student is not required to grasp all the structures at the same time.
Besides that it gives new results and new insights. This treatment is taken
from [146].
The chapter on Riemann geometry contains a careful treatment of connec-
tions to geodesic structures to sprays to connectors and back to connections
considering also the roles of the second and third tangent bundles in this.
Most standard results are proved. Isometric immersions and Riemann sub-
mersions are treated in analogy to each other. A unusual feature is the
Jacobi flow on the second tangent bundle. The chapter on isometric ac-
tions starts off with homogeneous Riemann manifolds and the beginnings of
symmetric space theory; then Riemann G-manifolds and polar actions are
treated.
The final chapter on symplectic and Poisson geometry puts some emphasis
on group actions, momentum mappings and reductions.
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