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In the chapter we study the metrics g t satisfying the Extrinsic Geometric Flow equation (see Sect. 3.2 Sections 3.4 and 3.5 collect results about existence and uniqueness of solutions (Theorems 3.1 and 3.2) and their proofs. The key role in proofs play hyperbolic PDEs and the generalized companion matrix studied in Sect. 3.3. In Sect. 3.6, we estimate the maximal existence time. In Sect. 3.7 we use the first derivative of functionals (when they are monotonous) to show convergence of metrics in a weak sense (Theorem 3.3). In Sect. 3.8 we study soliton solutions of the geometric flow equation (Theorems 3.4 and 3.5), and characterize them in the cases of umbilical foliations and foliations on surfaces (Theorems 3.6–3.8). Section 3.9 is devoted to applications and examples, including the geometric flow produced by the extrinsic Ricci curvature tensor (Theorem 3.9).

This chapter provides conditions either necessary or sufficient for a given quantity (either a scalar function or a vector field) to be the mean curvature of a given foliation with respect to some Riemannian metric. The particular case of this quantity being identically zero (tautness) has been described separately. In the codimension-one case, the only obstructions for a scalar function to be the mean curvature of a foliation arise from Stokes’ Theorem and a well known formula by H. Rummler relating mean curvature with the exterior derivative of the leaf volume form.

In this introductory chapter several notions from differential geometry (vector fields, differential forms and tensor fields etc.) are discussed in brief. The reader is assumed to have the necessary background in topology, manifolds and differential geometry. Some knowledge of differential equations would also be helpful. For the reader’s convenience, we provide also the basics of foliation theory. This chapter contains a concise survey of notions and facts concerning these topics. We present structures with constant partial Ricci curvature (playing the key role in the book) and define several functions which interpolate between the weighted sectional and Ricci curvature. Toponogov’s problem on totally geodesic foliations with positive mixed sectional curvature is discussed with relation to the weighted curvature.

This is an essay about osculation, that is, the tangency of highest order of different types, of different objects (hyperplanes, spheres, cyclides, and others), and arbitrary hypersurfaces.

In this chapter we give an overview of integral formulas, and some of their consequences, appearing in the study of extrinsic geometry of foliations and distributions on Riemannian manifolds.

Part I: Geometry.- The Ricci flow on some generalized Wallach spaces (N.A. Abiev, A. Arvanitoyeorgos, Y.G. Nikonorov, P. Siasos).- Gaussian mean curvature flow for submanifolds in space forms (A. Borisenko, V. Rovenski).- Cantor laminations and exceptional minimal sets in codimension one foliations (G. Hector).- Integral formulas in foliations theory (K. Andrzejewski, P. Walczak, V. Rovenski).- On prescribing the mixed scalar curvature of a foliations (V. Rovenski, L. Zelenko).- The partial Ricci flow for foliations (V. Rovenski).- Osculation in general (P. Walczak).- On stability of totally geodesic unit vector fields on three-dimensional Lie groups (A. Yampolsky).- Part II: Applications.- Rotational liquid film interacted with ambient gaseous media (I. Gaissinski, Y. Levy, V. Rovenski, V. Sherbaum).- On cycles and other geometric phenomena in phase portraits of some nonlinear dynamical systems (V. Golubyatnikov, Yu. A. Gaidov).- Remez-type inequality for smooth functions (Y. Iomdin).