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Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev exponent on the boundary., Final version, to appear in the Journal of Functional Analysis. arXiv admin note: text overlap with arXiv:0906.0927

In this paper, we prove two Liouville theorems for harmonic maps and apply them to study the topology of manifolds with positive spectrum and stable minimal hypersurfaces in Riemannian manifolds with non-negative bi-Ricci curvature.

The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth $n$-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called $p$-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on $p$-dimensional planes, $1\leq p\leq n$. Such characterization roughly consists on a convexity condition of the $p$-Renyi entropy along $L^{2}$-Wasserstein geodesics, where the role of reference measure is played by the $p$-dimensional Hausdorff measure. As application we establish a new Brunn-Minkowski type inequality involving $p$-dimensional submanifolds and the $p$-dimensional Hausdorff measure., Comment: Final version, published by Advances in Mathematics

For complete noncompact Riemannian manifolds ( M n , g ) with harmonic curvature, we prove that g is Einstein under an inequality involving L n 2 -norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant. Furthermore, we achieve that M n is a constant curvature space under such inequality and finite L 2 -norm of the Weyl curvature.

We consider generalized ( κ , μ ) {(\kappa,\mu)} -paracontact metric manifolds satisfying certain flatness conditions on the ℳ {\mathcal{M}} -projective curvature tensor. Specifically, we study ξ- ℳ {\mathcal{M}} -projectively flat and ℳ {\mathcal{M}} -projectively flat generalized ( κ , μ ) {(\kappa,\mu)} -paracontact metric manifolds and, further, ϕ- ℳ {\mathcal{M}} -projectively symmetric generalized ( κ ≠ - 1 , μ ) {(\kappa\neq-1,\mu)} -paracontact metric manifolds. We prove that they are characterized by certain structures whose properties are discussed in some detail.

We study the trajectories of differential inclusions on complete Riemannian manifolds. The topological properties of the reachable set and invariance of trajectories with respect to a closed set are investigated. In order to use required geometric tools, we introduce a linear growth condition by using the distance function on Riemannian manifold. This condition provides us a compact set containing all trajectories with the same initial value and end time. Finally, we apply our results to study a nonsmooth version of control affine systems, where the vector fields are only continuous or locally Lipschitz.

We prove $R$-bisectoriality and boundedness of the $H^\infty$-functional calculus in $L^p$ for all $1, Comment: 23 pages, submitted for publication

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on noncommutative two tori, the quantum part of the curvature consists of actions of the modular derivation through two local curvature functions. Explicit expressions for those functions are obtained for all even dimensions (greater than two). In dimension four, the one variable function shows striking similarity to the analytic functions of the characteristic classes appeared in the Atiyah-Singer local index formula, namely, it is roughly a product of the $j$-function (which defines the $\hat A$-class of a manifold) and an exponential function (which defines the Chern character of a bundle). By performing two different computations for the variation of the Einstein-Hilbert action, we obtain a deep internal relations between two local curvature functions. Straightforward verification for those relations gives a strong conceptual confirmation for the whole computational machinery we have developed so far, especially the Mathematica code hidden behind the paper., Comment: 44 pages, 11 figures, some minor updates from the previous version

We consider the generalization of the Navier-Stokes equations from $\mathbb R^n$ to the Riemannian manifolds. There are inequivalent formulations of the Navier-Stokes equations on manifolds due to the different possibilities for the Laplacian operator acting on vector fields on a Riemannian manifold. We present several distinct arguments that indicate that the form of the equations proposed by Ebin and Marsden in 1970 should be adopted as the correct generalization of the Navier-Stokes to the Riemannian manifolds., Comment: 16 pages; published version. See version 1 for the details of the non-relativistic limit, which were omitted in the published version

We consider \(\lambda \) is the principle eigenvalue of the complex Laplacian on a compact Hermitian manifold M. We prove that \(\lambda \ge C\) where C depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M, and a norm, expressed in curvature, that determines how much M fails to be Kahler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng (on curvature tensors of Hermitian manifolds, 2016. arXiv:1602.01189). We combine these results to obtain the main estimate. We also discuss several special cases in which one can obtain a lower bound solely in terms of the Riemannian geometry.

A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the $$L^2$$ -norm of the Weyl curvature. When the Riemannian 4-manifold in question is a Kahler surface, we provide a rough classification of solutions, followed by detailed results regarding each case in the classification. The most mysterious case prominently involves 3-dimensional CR manifolds.

We obtain isoperimetric inequalities under bounded integral norms of Ricci curvature and mean curvature. Also we generalize the results to metric-measure spaces.

In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact manifolds.

In this paper, we study the nature of Lorentzianα-Sasakian manifolds admitting M-projective curvature tensor. We show that M-projectively flat and irrotational M-projective curvature tensor of Lorentzian α-Sasakian manifolds are locally isometric to unit sphere Sn(c) , wherec = α2. Next we study Lorentzianα-Sasakian manifold with conservative M-projective curvature tensor. Finally, we find certain geometrical results if the Lorentzianα-Sasakian manifold satisfying the relation M(X,Y)⋅R=0.

We give a sufficient condition for the essential self-adjointness of a perturbed biharmonic-type operator acting on sections of a Hermitian vector bundle on a geodesically complete Riemannian manifold with Ricci curvature bounded from below by a (possibly unbounded) non-positive function depending on the distance from a reference point. We also establish the separation property in the case when the corresponding operator acts on functions.

We prove the nonexistence of Hopf real hypersurfaces in complex two-plane Grassmannians such that the covariant derivatives with respect to Levi-Civita and kth generalized Tanaka–Webster connections in the direction of the Reeb vector field applied to the Riemannian curvature tensor coincide when the shape operator and the structure operator commute on the \(\mathcal Q\)-component of the Reeb vector field.

Recently the present author introduced the notion of generalized quasi-conformal curvature tensor which bridges conformal curvature tensor, concircular curvature tensor, projective curvature tensor and conharmonic curvature tensor. The object of the present paper is to find out curvature conditions for which Ricci solitons in Sasakian manifolds are sometimes shrinking and some other time remain expanding.

In this paper, we obtain two Myers type compactness theorems for a Finsler manifold with a positive weighted Ricci curvature bound and a reasonable condition on the distortion or the S curvature.

In the present article we consider a Lie group G equipped with a left invariant Riemannian metric g. Then, by using complete and vertical lifts of left invariant vector fields we induce a left invariant Riemannian metric \(\widetilde{g}\) on the tangent Lie group TG. The Levi-Civita connection and sectional curvature of \((TG,\widetilde{g})\) are given, in terms of Levi-Civita connection and sectional curvature of (G, g). Then, we present Levi-Civita connection, sectional curvature and Ricci tensor formulas of \((TG,\widetilde{g})\) in terms of structure constants of the Lie algebra of G. Finally, some examples of tangent Lie groups of strictly negative and non-negative Ricci curvatures are given.

This paper considers optimization problems on Riemannian manifolds and analyzes the iteration-complexity for gradient and subgradient methods on manifolds with nonnegative curvatures. By using tools from Riemannian convex analysis and directly exploring the tangent space of the manifold, we obtain different iteration-complexity bounds for the aforementioned methods, thereby complementing and improving related results. Moreover, we also establish an iteration-complexity bound for the proximal point method on Hadamard manifolds.

We present a classification of complete locally irreducible Riemannian manifolds with nonnegative curvature operator, which admit a nonzero and nondecomposable harmonic form with its square-integrable norm. We prove a vanishing theorem for harmonic forms on complete generic Riemannian manifolds with nonnegative curvature operator. We obtain similar results for closed and co-closed conformal Killing forms.

R. Hamilton defined Ricci flow as evolution of Riemannian metrics satisfying a weak parabolic partial differential equation. We have considered and studied Ricci flow as an integral curve of a certain vector field in the manifold of Riemannian metrics. In this paper, we study properties of this integral curve and find some results on the stability of the solution.

We start this study with last multipliers and the Liouville equation for a symmetric and non-degenerate tensor field Z of (0, 2)-type on a given Riemannian geometry (M, g) as a measure of how far away is Z from being divergence-free (and hence \(g^C\)-harmonic) with respect to g. The some topics are studied also for the Riemannian curvature tensor of (M, g) and finally for a general tensor field of (1, k)-type. Several examples are provided, some of them in relationship with Ricci solitons. Inspired by the Riemannian setting, we introduce last multipliers in the abstract framework of Dirichlet forms and symmetric Markov diffusion semigroups. For the last framework, we use the Bakry-Emery carre du champ associated to the infinitesimal generator of the semigroup.

In this paper, we consider the Monge-Ampere type equations with the Neumann boundary conditions on Riemannian manifolds. Our main result is the gradient estimates for the degenerate elliptic solution, which is the extension of the main theorem in Jiang et al. (2016) from R n to Riemannian manifolds.

We investigate curvatures of normal almost contact Riemannian 3-manifolds. In particular, we show that Kenmotsu 3-manifolds of constant scalar curvature are of constant curvature −1.

We prove the Lichnerowicz type lower bound estimates for finite connected graphs with positive Ricci curvature lower bound.

In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\varrg$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen.

The object of the present paper is to study Ricci semisymmetric contact metric manifolds. As a consequence of the main result we deduce some important corollaries. Besides these we study contact metric manifolds satisfying the curvature condition Q,R = 0, where Q and R denote the Ricci operator and curvature tensor respectively. Also we study the symmetric properties of a second order parallel tensor in contact metric manifolds. Finally, we give an example to verify the main result.

In this paper, we investigate Ricci pseudo-symmetric and Ricci generalized pseudo-symmetric P-Sasakian manifolds. Next we study P-Sasakian manifolds satisfying the curvature condition S ? R = 0. Finally, we give an example of a 5-dimensional P-Sasakian manifold to verify some results.

In this paper, we obtain sharp inequalities on Riemannian manifolds admitting a Riemannian submersion and give some characterizations using these inequalities. We improve Chen-Ricci inequality for Riemannian submersion and present some examples which satisfy this inequality.

In this paper, we study the following gradient system on a complete Riemannian manifold M, {-x?(t) = grad'(x(t)) x(0) = x0, where ? : M ? R is a C1 function with Argmin ? ? ?. We prove that the gradient flow x(t) converges to a critical point of ? if ? is pseudo-convex, or if ? is quasi-convex and M is Hadamard. As an application to minimization, we consider a discrete version of the system to approximate a minimum point of a given pseudo-convex function ?.

The object of the present paper is to study P-Sasakian manifolds admitting a semi-symmetric non-metric connection whose concircular curvature tensor satisfies certain curvature conditions

This is a survey article on recent progress (in [Oh3], [OS]) of the theory of weighted Ricci curvature in Finsler geometry. Optimal transport theory plays an impressive role as is developed in the Riemannian case by Lott, Sturm and Villani.

Along an idea of von Renesse, couplings of the Brownian motion on a Riemannian manifold and their extensions are studied. We construct couplings as a limit of coupled geodesic random walks whose components approximate the Brownian motion respectively. We recover Kendall and Cranston's result under lower Ricci curvature bounds instead of sectional curvature bounds imposed by von Renesse. Our method provides applications of coupling methods on spaces admitting a sort of singularity.

The monograph deals with the theory of conformal, geodesic, holomorphically projective, F-planar and others mappings and transformations of manifolds with affine connection, Riemannian, Kahler and Riemann-Finsler manifolds. Concretely, the monograph treats the following: basic concepts of topological spaces, the theory of manifolds with affine connection (particularly, the problem of semigeodesic coordinates), Riemannian and Kahler manifolds (reconstruction of a metric, equidistant spaces, variational problems in Riemannian spaces, SO(3)-structure as a model of statistical manifolds, decomposition of tensors), the theory of differentiable mappings and transformations of manifolds (the problem of metrization of affine connection, harmonic diffeomorphisms), conformal mappings and transformations (especially conformal mappings onto Einstein spaces, conformal transformations of Riemannian manifolds), geodesic mappings (GM; especially geodesic equivalence of a manifold with affine connection to an equiaffine manifold), GM onto Riemannian manifolds, GM between Riemannian manifolds (GM of equidistant spaces, GM of Vn(B) spaces, its field of symmetric linear endomorphisms), GM of special spaces, particularly Einstein, Kahler, pseudosymmetric manifolds and their generalizations, global geodesic mappings and deformations, GM between Riemannian manifolds of different dimensions, global GM, geodesic deformations of hypersurfaces in Riemannian spaces, some applications of GM to general relativity, namely three invariant classes of the Einstein equations and geodesic mappings, F-planar mappings of spaces with affine connection, holomorphically projective mappings (HPM) of Kahler manifolds (fundamental equations of HPM, HPM of special Kahler manifolds, HPM of parabolic Kahler manifolds, almost geodesic mappings, which generalize geodesic mappings, Riemann-Finsler spaces and their geodesic mappings, geodesic mappings of Berwald spaces onto Riemannian spaces.

In this paper we study two entropic dynamical models from the viewpoint of information geometry. We study the geometry structures of the associated statistical manifolds. In order to analyse the character of the instability of the systems, we obtain their geodesics and compute their Jacobi vector fields. The results of this work improve and extend a recent advance in this topics studied in [13], Accepted for publication in Chaos, Solitons & Fractals

An expression for the coarse Ricci curvature of a weighted tree with a random walk on the vertex set is obtained. As a corollary, it is shown that the structure of a binary tree can be reconstructed up to isomorphism from the matrix of pairwise Ricci curvatures of its vertices.

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.

We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature (PIC). Then we review an application of the classification theorem, which is from Chen and Zhu’s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature.

Synthetic theory of Ricci curvature bounds is reviewed, from the conditions which led to its birth, up to some of its latest developments.

In this paper, we investigate eigenvalues of the eigenvalue problem with Dirichlet boundary condition of the drifting Laplacian on an n -dimensional, complete noncompact Riemannian manifold. Some estimates for eigenvalues are obtained. By utilizing Cheng and Yang recursion formula, we give a sharp upper bound of the k th eigenvalue. As we know, product Riemannian manifolds, Ricci solitons and self-shrinkers are some important Riemannian manifolds. Therefore, we investigate the eigenvalues of the drifting Laplacian on those Riemannian manifolds. In particular, by some theorems of classification for Ricci solitons, we can obtain some eigenvalue inequalities of drifting Laplacian on the Ricci solitons with certain conditions.

In this paper, we investigate K-contact Einstein manifolds satisfying the conditions RC = Q(S,C), where C is the conformal curvature tensor and R the Riemannian curvature tensor. Next we consider K-contact Einstein manifolds satisfying the curvature condition C.S = 0, where S is the Ricci tensor. Also we study K-contact Einstein manifolds satisfying the condition S.C = 0. Finally, we consider K-contact Einstein manifolds satisfying Z .C = 0, where Z is the concircular curvature tensor.

Bu makalede ݄߮ ߟ‐paralel tensör alanlı hemen hemen ߙ‐Kenmotsu manifoldların projektif, konformal ve konsirküler flat durumlarındaki geometrisi incelendi. Makalenin sonunda ߙ‘ya bağlı olarak hemen hemen α‐Kenmotsu manifoldlar üzerinde örnekler inşa edildi. We study the geometry of almost ߙ‐Kenmotsu manifolds when they are projectively, conformally (Weyl) and concircularly flat with η‐parallel tensor ݄߮ and we give illustrating examples on almost α‐ Kenmotsu manifolds with respect to ߙ.