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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.

In this paper, we derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L 2 norm of the mean curvature vector in a Riemannian manifold.

Let ( M n , g ) ( n ≥ 4 ) be an n-dimensional complete Einstein manifold. Denote by W the Weyl curvature tensor of M. We prove that ( M n , g ) is isometric to a spherical space form if ( M n , g ) has positive scalar curvature and unit volume, and the L p ( p ≥ n 2 ) -norm of W is pinched in [ 0 , C ) , where C is an explicit positive constant depending only on n, p and S, which improves the isolation theorems given by [24] , [14] , [17] . This paper also states that W goes to zero uniformly at infinity if for p ≥ n 2 , the L p -norm of W of M with non-positive scalar curvature and positive Yamabe constant is finite. Assume that M has negative scalar curvature and the L α -norm of W is finite. As application, we prove that M is a hyperbolic space form if the L p -norm of W is sufficiently small, which generalizes an L n 2 -norm of W pinching theorem in [19] .

It was conjectured in the 80s that every critical metric of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume must be Einstein. We prove that such a conjecture is true under a second-order vanishing condition on the Weyl tensor.

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.

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.

In this paper, we derive a DDVV-type inequality for submianifolds in a Riemannian manifold of nearly quasi-constant curvature. Moreover, two inequalities involving the Casorati curvature and the scalar curvature are obtained.

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.

This study investigates entire radially symmetric graphs in the warped product M × f R , where M is a Riemannian manifold with a pole and f : M → R + is the warping function. The main results demonstrate the existence and uniqueness of entire radially symmetric graphs with prescribed mean curvature.

Let M n (n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an (n + p)-dimensional locally symmetric Riemannian manifold N n+p . We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu [15].

We introduce a conformal Sasakian manifold and we find the inequality involving Ricci curvature and the squared mean curvature for semi-invariant, almost semi-invariant, $\theta $-slant, invariant and anti-invariant submanifolds tangent to the Reeb vector field and the equality cases are also discussed. Also the inequality involving scalar curvature and the squared mean curvature of some submanifolds of a conformal Sasakian space form are obtained.

This note is a study of nonnegativity conditions on curvature preserved by the Ricci flow. We focus on a specific class of curvature conditions which we call non-coercive: These are the conditions for which nonnegative curvature and vanishing scalar curvature does not imply flatness. We show, in dimensions greater than 4, that if a Ricci flow invariant nonnegativity condition is satisfied by all Einstein curvature operators with nonnegative scalar curvature, then this condition is just the nonnegativity of scalar curvature. As a corollary, we obtain that a Ricci flow invariant curvature condition, which is stronger than a nonnegative scalar curvature, cannot be strictly satisfied by curvature operators (other than multiples of the identity) of compact Einstein symmetric spaces. We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.

For a compact Riemannian manifold M immersed into a higher dimensional manifold which can be chosen to be a Euclidean space, a unit sphere, or even a projective space, we successfully give several upper bounds in terms of the norm of the mean curvature vector of M for the first non-zero eigenvalue of the p-Laplacian (1 < p < +∞) on M. This result can be seen as an extension of Reilly’s bound for the first non-zero closed eigenvalue of the Laplace operator.

The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces., 120 pages, 12 figures, (v2) minor revision; (v3) new sections on Finsler manifolds, slow growth distributions, Heisenberg group; (v4) major revision, new extended section on 3D contact structures, constant curvature, improved results about existence of ample geodesics on SR structures, 2 new appendices, many minor revisions; (v5) 1 new appendix, minor revisions

We establish a sharp inequality for a doubly warped product submanifold in a Riemannian manifold of quasi-constant curvature.

We analyze an upper bound on the curvature of a Riemannian manifold, using "root-Ricci" curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of root-Ricci curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our root-Ricci bound implies Gunther's inequality on the candle function of a manifold, thus bringing that inequality closer in form to the complementary inequality due to Bishop.

The Schouten tensor A of a Riemannian manifold ( M , g ) provides the important σ k -scalar curvature invariants, that are the symmetric functions in the eigenvalues of A, where, in particular, σ 1 coincides with the standard scalar curvature Scal ( g ) . Our goal here is to study compact manifolds with positive Γ 2 -curvature, i.e., when σ 1 ( g ) > 0 and σ 2 ( g ) > 0 . In particular, we prove that a 3-connected non-string manifold M admits a positive Γ 2 -curvature metric if and only if it admits a positive scalar curvature metric. Also we show that any finitely presented group π can always be realised as the fundamental group of a closed manifold of positive Γ 2 -curvature and of arbitrary dimension greater than or equal to six.

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold ( M n , g ) endowed with the complementary orthogonal distributions D 1 and D 2 . We provide conditions for symmetric ( 0 , 2 ) -tensors T of a simple form (defined on M ) to admit metrics g , conformal to g , that solve the partial Ricci equations. The solutions are given explicitly. Using above solutions, we also give examples to the problem of prescribing the mixed scalar curvature related to D i . In aim to find “optimally placed” distributions, we calculate the variations of the total mixed scalar curvature (where again the partial Ricci curvature plays a key role), and give examples concerning minimization of a total energy and bending of a distribution.

According to Tashiro–Obata, on a Riemannian manifold (M, g) with its Ricci curvature bounded positively from below, the first eigenvalue of the Laplacian on functions satisfies a simple inequality in terms of the scalar curvature, and equality characterizes the Riemannian sphere. We discuss a similar inequality for a certain elliptic operator on a manifold with conjugate connections. As application we characterize hyperellipsoids in Blaschke’s unimodular-affine hypersurface theory.

In this paper, we prove the existence of warping functions on Riemannian warped product manifolds with fiber manifold of class (B) having some prescribed scalar curvatures.

In this paper, we prove Chen’s general inequalities and Chen–Ricci inequalities for submanifolds of a Riemannian manifold of nearly quasi-constant curvature. We also establish inequalities between the warping function and the squared mean curvature for warped product submanifolds in a Riemannian manifold of nearly quasi-constant curvature.

We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity., 9 pages

We find a convenient expression for the value of the covariant curvature 4-tensor of an arbitrary Riemannian manifold on a quadruple of its Killing vector fields. With its use, we in particular obtain a simple deduction of the well-known formula to calculate the sectional curvature of a homogeneous Riemannian space.

If $\pi :M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature.

In this paper, the authors consider a family of smooth immersions Ft: Mn → Nn+1 of closed hypersurfaces in Riemannian manifold Nn+1 with bounded geometry, moving by the Hk mean curvature flow. The authors show that if the second fundamental form stays bounded from below, then the Hk mean curvature flow solution with finite total mean curvature on a finite time interval [0, Tmax) can be extended over Tmax. This result generalizes the extension theorems in the paper of Li (see “On an extension of the Hk mean curvature flow, Sci. China Math., 55, 2012, 99–118”).

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M given by R-n/2(g) := integral(M) vertical bar R(g)vertical bar(n//2) dv(g) where R(g), dv(g) denote the Riemannian curvature and volume form corresponding to g. We show that there are locally symmetric spaces which are unstable critical points for this functional.

We obtain -continuous paths of explicit Riemannian metrics , , whose scalar curvatures decrease, where is a flat metric, i.e. a metric with vanishing curvature. Most of them can exist on tori of dimension . Some of them yield scalar curvature decrease on a ball in the Euclidean space.

Let $M$ be a compact smooth Riemannian manifold of finite dimension $n+1$ with boundary $\partial M$and $\partial M$ is a compact $n$-dimensional submanifold of $M$. We show that for generic Riemannian metric $g$, all the critical points of the mean curvature of $\partial M$ are nondegenerate.

It has been conjectured that, on a compact 3-dimensional orientable manifold, a critical point of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume is Einstein. In this paper we prove this conjecture under a condition that ker , which generalizes the previous partial results.

In this paper we shall show that a compact gradient generalized m -quasi-Einstein metric ( M n , g , ∇ f , λ ) with constant scalar curvature must be isometric to a standard Euclidean sphere S n with the potential f well determined.

A linear transformation of velocity for a mechanical system is said to quasi-linearize the equations of motion of the system if it eliminates all terms quadratic in the velocity. It is well-known that controller/observer synthesis becomes tractable when the dynamics of a mechanical system are in quasi-linearized form. In this technical note, we show that the quasi-linearization property is equivalent to the property that the Lie algebra of Killing vector fields is pointwise equal to the tangent space to the configuration manifold with the Riemannian metric induced by the mass tensor of the mechanical system. A sufficient condition for this property is that the Riemannian manifold be locally symmetric. We further show that a necessary and sufficient condition for quasi-linearizability on 2-D Riemannian manifolds is that the scalar curvature is constant. The above results extend the zero Riemannian curvature condition that has been extensively applied since its introduction in 1992. Moreover, the local symmetricity condition and the constant scalar curvature condition can be easily verified using differentiation.

Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature form of the Hopf fibration. We also estimate the number of conjugate points of a sub-Riemannian extremal in terms of the bounds of the sectional curvature and the curvature form. It presents a typical example for the study of curvature maps and comparison theorms for a general corank 1 sub-Riemannian structure with symmetries done by C.Li and I.Zelenko., Comment: 6 pages. arXiv admin note: substantial text overlap with arXiv:0908.4397

Finite volume scheme for regularised Riemannian mean curvature flow equation is discussed. Stability estimates and the uniqueness of the numerical solution are listed. Convergence of the numerical scheme to the discrete solution is listed as well. Numerical results are presented in the final section.

Some basic inequalities, involving the scalar curvature and the mean curvature, for a pseudo-Riemannian submanifold of a pseudo-Riemannian manifold are obtained. We also find inequalities for spacelike submanifolds. Equality cases are also discussed. (C) 2016 Elsevier B.V. All rights reserved.

Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case.

The mixed scalar curvature is one of the simplest curvature invariants of a foliated Riemannian manifold. We explore the problem of prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibred manifold, we reduce the problem to solution of a leafwise elliptic equation, which has three stable solutions -- only one of them corresponds to the case of a foliated Riemannian manifold., 28 pages

In this paper we study the closedness properties of generalized cur- vature 2-forms, which are said to be Riemannian, Conformal, Projective, Concircular and Conharmonic curvature 2-forms, associated to each generalized curvature tensors on a Riemannian manifold. Corresponding to each curvature tensors, such generalized curvature 2-forms are the associated curvature 2-forms. In particular, we focus on the closedness of difierential 2-forms associated to the divergence of generalized curvature tensors, which is weaker than the notion of harmonic curvature. In this case, we give an algebraic condition involving the Riemann curvature tensor and the Ricci tensor arising from an old identity due to Lovelock.

In this paper, we obtain the mcci curvature and the scalar curvature on SU (3) with some left invariant Riemannian metric. And then we get a necessary and sufficient condition for the scalar curvature (resp. the Ricci curvature) on the Riemannian manifold SU (3) to be positive.

In this paper, we study a complete -Riemannian manifold whose curvature bounded from below. Let be a compact totally geodesic submanifold of . Then, for any , we can make use of the first variation formula and the second variation formula of distance to prove that is bounded.

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold M. We prove that if the criti- cal point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an n-dimensional Rie- mannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.

We find a one-parameter family of Riemannian metrics on for for some number with the following property: is the Euclidean metric on , the scalar curvatures of are strictly decreasing in t in the open unit ball and is isometric to the Euclidean metric in the complement of the ball.

In this paper, we consider an n-dimensional compact Riemannian manifold (M,g) of constant scalar curvature and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ together with a condition on Ricci curvature of M, that the Ricci curvature in the direction of a certain vector field is greater than or equal to (n − 1)λ, forces M to be isometric to the n-sphere Sn(λ).

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations onRiemannian warped product manifold M =(a, ∞ )× f N with prescribed scalar curvature functions.Key words : Warped Product, Scalar Curvature, Partial Differential Equation 1. Introduction In a recent study [7-9] , M.C. Leung has studied theproblem of scalar curvature functions on Riemannianwarped product manifolds and obtained partial resultsabout the existence and nonexistence of Riemannianwarped metric with some prescribed scalar curvaturefunction. He has studied the uniqueness of positive solu-tion to equation(1.1)where is the Laplacian operator for an n-dimen-sional Riemannian manifold (N, g 0 ) and d n = n−2/4(n−1). Equation (1.1) is derived from the conformal defor-mation of Riemannian metric [1,4-6,8,9] .Similarly, let (N, g 0 ) be a compact Riemanniandimensional manifold. We consider the (n+1)−dimen-sional Riemannian warped product manifold M =(a, ∞)× f N with the metric g = dt

@@ t gi jD 2Ri j; t2T0; T/; on a smooth, compact n-dimensional Riemannian manifold M. If the flow has uniformly bounded scalar curvature and develops Type I singularities at T , we show that suitable blow-ups of the evolving metrics converge in the pointed Cheeger‐Gromov sense to a Gaussian shrinker by using Perelman’s W-functional. If the flow has uniformly bounded scalar curvature and develops Type II singularities at T , we show that suitable scalings of the potential functions in Perelman’s entropy functional converge to a positive constant on a complete, Ricci flat manifold. We also show that if the scalar curvature is uniformly bounded along the flow in certain integral sense then the flow either develops a Type II singularity at T or it can be smoothly extended past time T .

We study a geometric problem that originates from theories of nonlinear elasticity: given a non-flat n-dimensional Riemannian manifold with boundary, homeomorphic to a bounded subset of ℝ n , what is the minimum amount of deformation required in order to immerse it in a Euclidean space of the same dimension? The amount of deformation, which in the physical context is an elastic energy, is quantified by an average over a local metric discrepancy. We derive an explicit lower bound for this energy for the case where the scalar curvature of the manifold is non-negative. For n = 2 we generalize the result for surfaces of arbitrary curvature.

Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for which this set is noncompact. These manifolds have umbilic boundary, but they are not conformally equivalent to the unit ball., 44 pages. Some typos corrected. This is a more detailed version of a paper accepted to Journal of Differential Equations

Sekigawa proved in 1977 that a 3-dimensional Riemannian manifold which is curvature homogeneous up to order 1 in the sense of I.M. Singer is always locally homogeneous. We deal here with the modification of the curvature homogeneity which is said to be “of type (1, 3)”. We give example of a 3-dimensional Riemannian manifold which is curvature homogeneous up to order 1 in the modified sense but still not locally homogeneous.

We study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor on compact manifolds with boundary; which reduces to a Dirichlet problem. We prove the existence of the solution under some suitable conditions. In particular, we prove that every smooth compact n-dimensional manifold with boundary, with n ≥ 3, admits a complete Riemannian metric g whose Ricci curvature Ricg and scalar curvature R g satisfy This result generalizes Aviles and McOwen's in the scalar curvature case.