Immersions and submersions, which are special tools in Differential
Geometry, also play a fundamental role in Riemannian Geometry, expecially when
the involved manifolds carry an additional structure (of contact, Hermitian,
quaternionic type, etc.). Even if submersions are, in a certain sense, a
counterpart of immersions, the corresponding theories are quite different, also
from a historical point of view.
The theory of isometric immersions, started with the work of Gauss
on surfaces in the Euclidean 3-space, is classical and widely explained in
many books, whereas the theory of Riemannian submersion goes back to
four decades ago, when B. O'Neill and A. Gray, independently, formulated
the basis of such theory, which has hugely been developed in the last two
decades. Nowadays several works are still in progress. For instance, a new
point of view on Riemannian submersions appears in a paper by H. Karcher
in 1999.
Obviously the content of this book is not exhaustive, anyway, the results
presented are enough to solve problems concerning many areas, like
Theoretical Physics and the theory of Einstein, EinsteinWeyl spaces. This theory
falls into a more general context, extensively treated in the Besse's book
Einstein Manifolds and, more recently, in Surveys in differential geometry:
essays on Einstein manifolds, edited by C. Le Brun and M. Wang.
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