Extrinsic geometry describes properties of foliations on Riemannian
manifolds which can be expressed in terms of the second fundamental form
of the leaves. The authors of Topics in Extrinsic Geometry of
Codimension-One Foliations achieve a technical tour de force, which
will lead to important geometric results.
The Integral Formulae, introduced in chapter 1, is a useful for
problems such as: prescribing higher mean curvatures of foliations,
minimizing volume and energy defined for vector or plane fields on
manifolds, and existence of foliations whose leaves enjoy given
geometric properties. The Integral Formulae steams from a Reeb formula,
for foliations on space forms which generalize the classical ones. For a
special auxiliary functions the formulae involve the Newton
transformations of the Weingarten operator.
The central topic of this book is Extrinsic Geometric Flow (EGF) on
foliated manifolds, which may be a tool for prescribing extrinsic
geometric properties of foliations. To develop EGF, one needs
Variational Formulae, revealed in chapter 2, which expresses a change
in different extrinsic geometric quantities of a fixed foliation under
leaf-wise variation of the Riemannian Structure of the ambient manifold.
Chapter 3 defines a general notion of EGF and studies the evolution of
Riemannian metrics along the trajectories of this flow(e.g., describes
the short-time existence and uniqueness theory and estimate the maximal
existence time).Some special solutions (called Extrinsic Geometric
Solutions) of EGF are presented and are of great interest, since they
provide Riemannian Structures with very particular geometry of the
leaves.
This work is aimed at those who have an interest in the differential
geometry of submanifolds and foliations of Riemannian manifolds.
