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A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called the mixed curvature is also related to the extrinsic geometry of a Riemannian almost $k$-product manifold. Our inequality also contains mixed scalar curvature type invariants related to B.-Y Chen's $\delta$-invariants. Applications are given for isometric immersions of multiply twisted and warped products (we improve some known optimal inequalities by replacing the sectional curvature with our invariant) and to problems of non-immersion and non-existence of compact leaves of foliated submanifolds., Comment: 10 pages

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.

Este articulo esta dedicado a la critica de la teoria del desarrollo intelectual de Piaget, realizada por Gareth B. Matthews (1929-2011). Este filosofo estadounidense analiza los conceptos de Piaget (1896-1980) desde la perspectiva de la pregunta sobre el significado y la posibilidad de filosofar con ninos y ninas. La discusion de Matthews con Piaget tambien puede ser interesante hoy en dia debido al hecho de que la teoria del desarrollo intelectual en etapas ha dado forma a la educacion moderna y es un factor que consolida el enfoque esceptico de la filosofia de ninas y ninos. Este articulo reconstruye e interpreta los argumentos de Matthews, que demuestran de manera consistente y sistematica que Piaget no entendio el fenomeno de la filosofia infantil. Hablando del filosofar infantil, Matthews se refiere a la tendencia de los ninos a formular preguntas filosoficamente significativas, a notar los problemas que los grandes filosofos han estado tratando durante siglos, la actitud especifica de ninas y ninos hacia el mundo y su capacidad de problematizar la experiencia. Todas estas caracteristicas observables del pensamiento de ninas y ninos fueron interpretadas por Piaget como deficits intelectuales. Matthews argumenta en contra de tal interpretacion. La discusion se refiere a cuestiones tales como: el problema de determinar el criterio de la madurez intelectual, la importancia de la creatividad y la fantasia en el pensamiento, el desarrollo de conceptos y habilidades linguisticas y las diferencias entre el uso del lenguaje por parte de ninos y adultos. El analisis de la critica de la teoria del desarrollo intelectual de Piaget lleva a la conclusion de que la psicologia infantil necesita cierta apertura a este aspecto del pensamiento de ninas y ninos, ignorado por Piaget. Todavia hay una tarea importante para que psicologos y filosofos comprendan mejor el fenomeno del filosofar infantil.

Let M be a smooth three-dimensional manifold equipped with a vector field T transverse to a plane field $${\mathcal {D}}$$ —the kernel of a one form $$\omega $$ such that $$\omega (T)=1$$ . Recently, in Rovenski and Walczak (A Godbillon–Vey type invariant for a 3-dimensional manifold with a plane field, 2017. arXiv:1707.04847 ), we constructed a three-form analogous to that defining the Godbillon–Vey class of a foliation, showed how does this form depend on $$\omega $$ and T, and deduced Euler–Lagrange equations of the associated functional. In this paper, we continue our study when distributions/foliations and forms are defined outside a “singularity set” (a finite union of pairwise disjoint closed submanifolds of codimension $$\ge 2$$ ) under additional assumption of convergence of certain integrals. We characterize critical pairs $$(\omega ,T)$$ and foliations for different types of variations, find sufficient conditions for critical pairs when variations are among foliations, consider applications to transversely holomorphic flows, calculate the index form and present examples of critical foliations among Reeb foliations and twisted products.

We provide an entropy estimate from below for a finitely generated group of transformation of a compact metric space which contains a ping-pong game with several players located anywhere in the group.

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb's formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that total mean curvatures of arbitrary order for a codimension-one foliated Riemannian manifold of constant curvature don't depend on a foliation). All of that is done by a comparison of extrinsic and intrinsic curvatures of the two Riemannian structures which arise in a natural way from a given Randers structure., 17 pages. arXiv admin note: text overlap with arXiv:1602.00610

In this article, we provide a short review of general results on harmonic measures on foliated manifolds and discuss briefly about uniquely ergodic foliations.

We derive and apply a new integral formula for a closed Riemannian manifold equipped with a pair of complementary orthogonal distributions (plane fields). The integrand depends on the second fundamental forms and integrability tensors of the distributions, their covariant derivatives, and of the Ricci curvature of the ambient manifold. Also, we discuss some applications of this formula and of another formula of this sort, the one obtained earlier by the second author, and show that both formulae may hold when the distributions are defined only outside a “reasonable” closed subset of the manifold under consideration.

The problem with the waste from electric and electronic equipment (WEEE), consisting of about 80% of television sets and computers containing a cathode ray tube (CRT) have just begun to deal with. Cathode ray tubes (CRTs) are the video display components of televisions and computer monitors (EPA, 1995). Arising problem related with recovery and recycling of CRT glass stimulates examinations aimed at making cullet-based ceramic materials of reinforced mechanical strength. Glasses as cone and neck contain principally different lead contents another dangerous elements instead panel glass has other heavy metals (Ba, Sr, etc.) that forbid their recycling in the glass industry for the production of containers, domestic glassware and glass fiber. As it is well known concrete is an excellent material for immobilization of these metals, what was the reason to undertake this research. The use of CRT (Cathode Ray tube) waste glasses in concrete has attracted a lot of interest worldwide due to the increased disposal costs and environmental concerns. Because of silica large quantities CRT glass in the theory is a pozzolanic in nature, thus it can be used as a sand replacement in light concrete. The research was performed with use of concrete mortars. As a reference concrete mortar a mix of cement, sand, expanded clay aggregate and water was used. The purpose of this work was to investigate the influence of additives e.g. CRT glass cullet as an aggregate, and fluidized fly ash on concrete properties. Based on obtained results both CRT glass cullet and fluidize fly ash improved compressive and flexural strength in respect of standard concrete mortar without these additives. The use of CRT glass as a sand replacement caused the increased of compressive strength of concrete mortar of about 16% and its flexural strength of about 14%.

This experimental study aimed to design an autoclaved aerated concrete (AAC) with different kind of glass wastes as an alternative fine aggregates for mass housing projects that will meet the requirements in order to help contribute to the industry in saving the environment. Moreover we want to encourage the government to find solutions regarding the disposal to landfills of glassy waste materials and provide new knowledge to the contractors and developers on how to improve the construction industry methods and services by using glassy waste to sustain good product performance and meet recycling goals. Different kinds of glass wastes in the form of sub grain packaging glass cullet, CRT (Cathode Ray Tube) panel glass waste and calsi glass, were used as a sand replacement in AAC production. Large proportion of the post-consumer glass is recycled into the packaging stream again, but some sub grain packaging glass cullet does not meet the strict criteria for packaging glass thus is sent to landfill. CRT is classified as a hazardous glass waste and there are several issues surround CRTs which create barriers to increased recovery of the glass. In countries e.g. Latin America and Asia where still CRT glass is being produced even when shredder, the glass stream is a mixture of panel and funnel glass, making it difficult for reuse in new CRTs. Based on our knowledge also calsi glass as a processed blast furnace slag is not used in packaging production. The viability of current and novel applications of mentioned glasses is examined here. Although aerated concrete was initially envisaged as a good insulation material, there has been renewed interest in its structural characteristics in view of its lighter weight, savings in material and potential for large scale utilization of wastes. Due to this fact we undertake the studies of an influence of different amounts of glass wastes onto properties of aerated concrete in terms of physical (microstructure, density), mechanical (compressive and tensile strengths) and chemical characteristic. Use of milled waste glass in concrete as partial re-placement of sand represents an important step towards development of sustainable (environmentally friendly, energy-efficient and economic concrete-based infrastructure systems.

In recent years the evaluation of U-value for buildings materials has been seen. Since 1st January 2014 U-value can’t be higher than 0,25 [W/m2K], but since 2017 this value will be 0,23 and 0,20 from 2021. Therefore, a good solution to fulfill mentioned conditions is using buildings material with better thermos insulation. One of the best construction materials, which have very low λ value [W/mK] is an autoclaved aerated concrete (AAC). It's the reason why AAC is the most popular buildings material from years. AAC could be produced with using various aggregates, like sand or fly ash. Test results of thermal conductivity clearly shown that AAC based on siliceous fly ash have better λ value than sand AAC in the same density. Polish energy policy is based on coal and production AAC based on siliceous fly ash is a great solution to utilization this waste. Autoclaved aerated concrete has better λ value than other available construction materials in polish market. Lower density have good impact on environment, because lower weight caused possibility to transport more products by the same truck and also it means less waste from buildings made from AAC in the future. This waste can be used again in normal production process of AAC. Due to these facts were undertaken studies of possibility of production AAC in 350 [kg/m3] density in PGS process technology. The PGS process technology is a very specific because in a production cement is not used. As a binder are used only: quick lime, gypsum and some part of fly ash. The most important properties of AAC were tested, like: compressive strength, density, λ value or phase composition. Production AAC in lower density could be the next step to improve that it's an environmental friendly material for energy-efficient constructions.

We consider a 3-dimensional smooth manifold $M$ equipped with an arbitrary, \textit{a priori} non-integrable, distribution (plane field) ${\cal D}$ and a vector field $T$ transverse to ${\cal D}$. Using a 1-form $\omega$ such that ${\cal D} = \ker\,\omega$ and $\omega(T)=1$ we construct a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and show how does this form depend on $\omega$ and~$T$. For a compatible Riemannian metric on $M$, we express this 3-form in terms of the curvature and torsion of normal curves and the non-symmetric second fundamental form of ${\cal D}$. We deduce Euler-Lagrange equations of associated functionals: for variable $({\cal D},T)$ on $M$, and for variable Riemannian or Randers metric on $(M,{\cal D})$. We show that for a geodesic field $T$ (e.g., for a contact structure) such $({\cal D},T)$ is critical, characterize critical pairs when ${\cal D}$ is integrable, and prove that these critical pairs are not extrema., Comment: 20 pages

International audience; In 1872, G. Darboux defined a family of curves on surfaces of $\mathbb{R}^3$ which are preserved by the action of the Mobius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric $\Lambda^4$ contained in the 5-dimensional Lorentz space $\mathbb{R}^5_1$ naturally associated to the surface. We construct a new conformal object: the Darboux plane-field $\mathcal{D}$ and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that $\mathcal{D}$ is integrable when the surface is a special canal.

We estimate expansion growth types (in the sense of Egashira) of certain distal groups of homeomorphisms and manifold diffeomorphisms.The estimate implies zero entropy (in the sense of Ghys, Langevin and the author) and existence of invariant measures for such groups. We prove also existence of invariant measures for pseudogroups satisfying some conditions of distality type.

International audience; In 1872, Gaston Darboux defined a family of curves on surfaces in the 3-dimensional Euclidean space $E^3$ which are preserved by the action of the Möbius group and share many properties with geodesics. Here, we study the Darboux curves from a dynamical viewpoint on special canal surfaces, quadrics and some Darboux cyclides. We also describe the generic behavior of Darboux curves near ridge points (zig-zag and beak-to-beak).

In this paper, we study hypersurfaces with constant rth mean curvature S r . We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an rth Jacobi field of a hypersurface with S r + 1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation.

Canal surfaces defined as envelopes of 1-parameter families of spheres, can be characterized by the vanishing of one of the conformal principal curvatures. We distinguish special canals which are characterized by the fact that the non-vanishing conformal principal curvature is constant along the characteristic circles and show that they are conformally equivalent to either surfaces of revolution, or to cones over plane curves, or to cylinders over plane curves, so they are isothermic.

In this article, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan and the second author) for foliations of codimension 1 or unit vector fields and obtain an infinite series of such formulae involving invariants of the Weingarten operator of a unit vector field, of the Jacobi operator in its direction, and their products. We write several such formulae explicitly, on locally symmetric spaces as well as on arbitrary Riemannian manifolds where they involve also covariant derivatives of the Jacobi operator. We work also with foliations of codimension 1 (or vector fields) which admit “good” (in a sense) singularities.

In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds. Using these transformations, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan, Walczak, etc.) for foliations of codimension one. We obtain integral formulae involving rth mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation, and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second order mean curvature.

(2008). Holomorphic Maps and Pencils of Circles. The American Mathematical Monthly: Vol. 115, No. 8, pp. 690-700.

In this article, we begin a systematic study of conformal properties of codimension-1 foliations. We first define and study local conformal invariants. A case of particular interest is that of harmonic foliations of the plane. Then we study existence of totally umbilical and “Dupin” foliations on compact 3-manifolds of constant curvature.

We classify surfaces in 3-dimensional space forms which have all the local conformal invariants constant and show that compact hyperbolic 3-manifolfs admit no foliations by such surfaces.

We study extrinsic geometry of a codimension-one foliation ${\cal F}$ of a closed Finsler space $(M,F)$, in particular, of a Randers space $(M,\alpha+\beta)$. Using a unit vector field $\nu$ orthogonal (in the Finsler sense) to the leaves of ${\cal F}$ we define a new Riemannian metric $g$ on $M$, which for Randers case depends nicely on $(\alpha,\beta)$. For that $g$ we derive several geometric invariants of ${\cal F}$ (e.g. the Riemann curvature and the shape operator) in terms of $F$, then under natural assumptions on $\beta$ which simplify derivations, we express them in terms of corresponding invariants arising from $\alpha$ and $\beta$. Using our approach (2012), we produce the integral formulae for ${\cal F}$ on $(M, F)$ and $(M, \alpha+\beta)$, which relate integrals of mean curvatures with those involving algebraic invariants obtained from the shape operator of a foliation, and the Riemann curvature in the direction $\nu$. They generalize the formulae by Brito, Langevin and Rosenberg, which show that total mean curvatures (of arbitrary order $k$) for codimension-one foliations on a closed $(m+1)$-dimensional manifold of constant curvature $K$ don't depend on a choice of ${\cal F}$., Comment: 22 pages

One of substantial problems in dynamical systems is to characterize the topological types of minimal sets. In this paper, we are concerned with locally connected minimal sets of surface homeomorphisms. In the case of a homeomorphism of a closed orientable surface dieren t from the torus, we obtain that its locally connected minimal set is either a nite set or a nite disjoint union of simple closed curves. On the other hand, there is a homeomorphism of the torus with a locally connected minimal set which is neither a nite set nor a nite disjoint union of simple closed curves nor the whole torus. We will show that, if such a minimal set exists, then the minimal set is unique and it satises conditions similar to those of the Sierpi nski curve.