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.

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.

In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D=TF⊕span(N), and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation.

A Riemannian almost paracomplex manifold is a 2n-dimensional Riemannian manifold (M,g), whose structural group O(2n,R) is reduced to the form O(n,R)×O(n,R). We define the scalar curvature π of this manifold and consider relationships between π and the scalar curvature s of the metric g and its conformal transformations.

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).

A Riemannian manifold endowed with k&gt, 2 orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for k=2) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections.

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 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.

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.

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

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.

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.

In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold $(M,g)$ equipped with a vector field $X$. We define several functions ($q$th Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The novel concepts of the "mixed curvature-dimension" condition and "synthetic dimension of a distribution" allow us to update the estimate of the diameter of a compact Riemannian foliation and to prove new splitting theorems for almost-product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. In the case of positive (and nonnegative) weighted mixed sectional curvature we explore the weighted generalization of Toponogov's conjecture on totally geodesic foliations., Comment: 12 PAGES

We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenb&ouml, ck type curvature operator on tensors, prove the Weitzenb&ouml, ck type decomposition formula, and derive the Bochner&ndash, Weitzenb&ouml, ck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.

In the present paper, we study twisted and warped products of Riemannian manifolds. As an application, we consider projective submersions of Riemannian manifolds, since any Riemannian manifold admitting a projective submersion is necessarily a twisted product of some two Riemannian manifolds.

We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also considered., Comment: 15 pages. Final version. To appear in Geometriae Dedicatta. Theorem 6 in the previous version removed, details on some of the proofs added, overall exposition improved

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

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.

We study the flow of metrics on a foliation (called the Partial Ricci Flow), ∂ t g = −2 r(g), where r is the partial Ricci curvature; in other words, for a unit vector X orthogonal to the leaf, r(X, X) is the mean value of sectional curvatures over all mixed planes containing X. The flow preserves total umbilicity, total geodesy, and harmonicity of foliations. It is used to examine the question: Which foliations admit a metric with a given property of mixed sectional curvature (e.g., constant)? We prove local existence/uniqueness theorem and deduce the evolution equations (that are leaf-wise parabolic) for the curvature tensor. We discuss the case of (co)dimension-one foliations and show that for the warped product initial metric the solution for the normalized flow converges, as t → ∞, to the metric with \(r = \Phi \,\hat{g}\), where \(\Phi \) is a leaf-wise constant.

Section 2.1 discusses the vectors and their use in analytic geometry. In Section 2.2 we study rigid motions of \({\mathbb{R}}^{n}\) (n≥2), including the complex and quaternionic approaches. Section 2.3 is devoted to the geometry on a sphere (induced by Euclidean geometry of the space), it and also introduces the stereographic projection. The study of polyhedra is organized in Section 2.4 in into the a sequence of themes: the notion of a polyhedron, Platonic solids, symmetries of geometrical figures, star-shaped polyhedra, and Archimedean solids. Section 2.5 (Appendix) surveys matrices and groups.

where (M, g) is a fixed compact oriented n-dimensional Riemannian manifold with the curvature tensor R, D ranges over the space of all codimension-one distributions (plane fields) on M , m = 1, 2, . . . n−1 (especially, m even), N the unit normal of D, and σm(D) is the m-symmetric function of the shape operator AN : D → D. For example, the minimal value of I2(D) can be used for estimation from below of the energy E(N) of N , because [1], E(N) ≥ 1 2n−2 ∫

How does a curve look in the neighborhood of a singular point? Recall that a formal definition of a singular and regular point on a curve (see Section 5.1) depends on a class of parametrization.

Section 1.1 is a short introductory excursion into the basic rules of using MAPLE for symbolic transformations and two-dimensional plots. In Sections 1.2 and 1.3, we plot graphs of some elementary and special functions. In Section 1.3, we play with transformations and obtain animated graphs (or “movies”) using the command animate. In Section 1.4, MAPLE helps us to investigate functions using derivatives.