The roots of this book lie in a series of lectures that I presented
at the University of Ioannina, in the summer of 1997. The central
theme is the geometry of Lie groups and homogeneous spaces. These
are notions which are widely used in differential geometry, algebraic
topology, harmonic analysis and mathematical physics. There is no
doubt that there are several books on Lie groups and Lie algebras,
which exhaust these topics thoroughly. Also, homogeneous spaces
are occasionally tackled in more advanced textbooks of differential
The present book is designed to provide an introduction to sev-
eral aspects of the geometry of Lie groups and homogeneous spaces,
without becoming too detailed. The aim was to deliver an exposition
at a relatively quick pace, where the fundamental ideas are empha-
sized. Several proofs are provided, when it is necessary to shed light
on the various techniques involved. However, I did not hesitate to
mention more difficult but relevant theorems without proof, in ap-
propriate places. There are several references cited, that the reader
can consult for more details.
The audience I have in mind is advanced undergraduate or grad-
uate students. A first course in differential geometry would be desir-
able, but is not essential since several concepts are presented. Also,
researchers from neighboring fields will have the chance to discover a
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