Search

In this article, we introduce and study a new structure on a Riemannian manifold: a distribution represented as the sum of k > 2 pairwise orthogonal distributions. We define the mixed scalar curvature of this structure and prove integral formulas generalizing classical and recent results on foliations and distributions generating the tangent bundle of a manifold. Examples with one-dimensional distributions, paracontact manifolds, hypersurfaces in space forms, etc., illustrate the results.

We prove vanishing theorems for the kernel of the Sampson Laplacian, acting on symmetric tensors on a Riemannian manifold and estimate its first eigenvalue on negatively pinched Riemannian manifolds. Some applications of these results to conformal Killing tensors are presented.

Mathematicians have shown interest in manifolds endowed with several distributions, e.g., webs composed of different regular foliations and multiply warped products, as well as distributions having variable dimensions (e.g., singular Riemannian foliations). In this paper, we extend our previous study of the mixed scalar curvature of two orthogonal singular distributions for the case of $k>2$ singular (or regular) pairwise orthogonal distributions, prove an integral formula with this kind of curvature, and illustrate it by characterizing autoparallel singular distributions.

We consider the Sampson Laplacian acting on covariant symmetric tensors on a Riemannian manifold. This operator is an example of the Lichnerowicz-type Laplacian. It is of fundamental importance in mathematical physics and appears in many problems in Riemannian geometry including the theories of infinitesimal Einstein deformations, the stability of Einstein manifolds and the Ricci flow. We study the Sampson Laplacian using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting Weitzenbock decomposition and further of estimating its lowest eigenvalue. In addition, we also survey a series of results that we obtained earlier.

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 paper we introduce new metric structures on g -foliations that are less rigid than the well-known structures: almost contact and 3-Sasakian structures as well as f -structures with parallelizable kernel and almost para- ϕ -structures with complemented frames. We discuss the properties of the new structures in order to demonstrate similarities with the corresponding classical structures. Then using the flow of metrics on a g -foliation, we build deformation retraction of our structures with positive partial Ricci curvature onto the subspace of the aforementioned classical structures.

We continue our study of the mixed Einstein–Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler–Lagrange equations (which in the case of space-time are analogous to those in Einstein–Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler–Lagrange equations of the mixed Einstein–Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric connections.

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $$(m,\rho )$$ -quasi-Einstein structure on a contact metric manifold. First, we prove that if a K-contact or Sasakian manifold $$M^{2n+1}$$ admits a closed $$(m,\rho )$$ -quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $$2n(2n+1)$$ , and for the particular case—a non-Sasakian $$(k,\mu )$$ -contact structure—it is locally isometric to the product of a Euclidean space $${\mathbb {R}}^{n+1}$$ and a sphere $$S^n$$ of constant curvature 4. Next, we prove that if a compact contact or H-contact metric manifold admits an $$(m,\rho )$$ -quasi-Einstein structure, whose potential vector field V is collinear to the Reeb vector field, then it is a K-contact $$\eta $$ -Einstein manifold.

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.

In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton’s Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.

In this paper, we extend our previous study regarding a Riemannian manifold endowed with a singular (or regular) distribution, generalizing Bochner&rsquo, s technique and a statistical structure. Following the construction of an almost Lie algebroid, we define the central concept of the paper: The Weitzenb&ouml, ck type curvature operator on tensors, prove the Bochner&ndash, Weitzenb&ouml, ck type formula and obtain some vanishing results about the null space of the Hodge type Laplacian on a distribution.

In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $\eta$-Ricci solitons and $\eta$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $\eta$-Ricci soliton is Einstein metric if either it is $\eta$-Einstein or the potential vector field $V$ is an infinitesimal contact transformation or $V$ is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $\eta$-Ricci solitons and gradient $\eta$-Ricci solitons, which illustrate our results., Comment: 10 pages

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.

The mixed gravitational field equations have been recently introduced for codimension one foliated manifolds, e.g. stably causal and globally hyperbolic spacetimes. These Euler-Lagrange equations for the total mixed scalar curvature (as analog of Einstein-Hilbert action) involve a new kind of Ricci curvature (called the mixed Ricci curvature). In the work, we derive Euler-Lagrange equations of the action for any spacetime, in fact, for a pseudo-Riemannian manifold endowed with a non-degenerate distribution. The obtained equations are presented in the classical form of Einstein field equation with the new Ricci type curvature instead of Ricci curvature

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

Miniature air blast atomizers are intended for use in small jet engines. The typical miniature atomizer with rotating air outside the pressure swirl nozzle was investigated by theoretically and experimentally. The equations which describe the flow of rotating hollow liquid jet in a gaseous environment, where there is difference between outer and inner pressures of the liquid film were developed. The effects of surface tension, gravity forces, and inner-outer pressure difference on liquid film modes and its disintegration were considered. A set of the relevant equations was solved numerically. The initial conditions for the equations: pressure distribution inside and outside of the ‘onion’ as function of outer rotating air and liquid parameters, were obtained experimentally. The numerical results were verified by the experimental study. It was shown that exists two shapes of liquid film nearby nozzle exit – ‘onion’ and open film. Shape of the liquid film for the given nozzle pressure drop depends mainly on air pressure difference outside and inside the film. Two resulting spray modes are distinguished in droplet characteristics drastically. The results can be applied for estimation of the operational parameters of air-blast atomizers that operate with very low liquid pressure drop.

The work is devoted to the differential geometry of singular distributions (i.e., of varying dimension) on a Riemannian manifold. We define such distributions as images of the tangent bundle with smooth endomorphisms. We introduce an operator of the divergence type and prove a new divergence theorem. Then we derive a Codazzi type equation for a pair of transverse singular distributions, generalizing the known one for an almost product structure. Tracing our Codazzi type equation gives an expression of a modified divergence of the sum of mean curvature vectors in terms of invariants of distributions including the modified mixed scalar curvature, which allows us to obtain some splitting results. Applying our divergence type theorem to the above expression with a modified mixed scalar curvature, we obtain an integral formula for a pair of transversal singular distributions; the formula generalizes the well-known one with many applications.

We study the properties of Ricci curvature of g -manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the transverse manifold of the characteristic foliation are investigated. In particular, sufficient conditions are found under which the g -manifold can be a Ricci soliton or a gradient Ricci soliton. Finally, we obtain a amazing (non-existence) higher dimensional generalization of the Boyer-Galicki theorem on Einstein K-manifolds for a special class of abelian g -manifolds.

The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In this case, there are the exterior differential and codifferential defined on the vector space of these differential one-forms. Then a symmetric bilinear form is said to be harmonic if it is closed and coclosed as a one-form with values in the cotangent bundle of a Riemannian manifold. In the present paper we prove that the kernel of the little known Bourguignon Laplacian is a finite-dimensional vector space of harmonic symmetric bilinear forms on a compact Riemannian manifold. We also prove that every harmonic symmetric bilinear form on a compact Riemannian manifold with non-negative sectional curvature is invariant under parallel translations. In addition, we investigate the spectral properties of the little studied Bourguignon Laplacian.