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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] .

There have been many works on the problem of finding a conformal metric on the standard sphere S n , n ≥ 3 , when the prescribed scalar curvature function is flat near its critical points with order of flatness β . All the existence results up to this research concern the case 1 β n . The present paper deals with the case β ≥ n . We provide a complete analysis of the asymptotic expansion of the associated gradient vector field for any β > 1 . Moreover, we establish (in this first part) existence results concerning some cases of β ≥ n .

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.

Motivated by Lohkamp's conjecture on curvature deformation in [13] , we present a local smooth decrease of scalar curvature by big scale on a sphere as follows. Given any positive numbers N, a , b with a b π , we obtain a C ∞ -continuous path of Riemannian metrics g t , 0 ≤ t ≤ 1 , on the 4-dimensional sphere S 4 , with g 0 being the round metric of constant curvature 1, such that the scalar curvatures s ( g t ) are strictly decreasing in t on the open ball B b g 0 ( p ) of g 0 -radius b centered at a point p, s ( g 1 ) − N on B a g 0 ( p ) and g t = g 0 on the complement of the ball B b g 0 ( p ) . This result goes beyond what can be done with Corvino's local first-order deformation theory of scalar curvature [5] . Albeit done on a sphere, the argument here seems generalizable to a larger class of metrics.

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study and characterize the ( α , β ) -metrics of scalar flag curvature. When the dimension of the manifold is greater than 2, we classify Randers metrics of weakly isotropic flag curvature (that is, Randers metrics of scalar flag curvature with isotropic S-curvature). Further, we characterize and classify ( α , β ) -metrics of scalar flag curvature with isotropic S-curvature. Finally, we conclude that the non-trivial regular ( α , β ) -metrics of scalar flag curvature with isotropic S-curvature on an n-dimensional manifold M ( n ≥ 3 ) must be Randers metrics.

Riemannian cubics are curves that generalise cubic polynomials to arbitrary Riemannian manifolds, in the same way that geodesics generalise straight lines. Considering that geodesics can be extended indefinitely in any complete manifold, we ask whether Riemannian cubics can also be extended indefinitely. We find that there are always exceptions in Riemannian manifolds with strictly negative sectional curvature. On the other hand, we show that Riemannian cubics can always be extended in complete locally symmetric Riemannian manifolds of non-negative curvature.

We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied by the scalar curvature. We also glue asymptotically Euclidean metrics to Schwarzschild ones and the same for asymptotically Delaunay metrics, keeping bounds on the scalar curvature, if any. This extends the Corvino gluing near infinity to non-constant scalar curvature metrics.

We consider the existence of contact forms of prescribed Webster scalar curvature on a (2n+1)-dimensional CR compact manifold locally conformally CR equivalent to the standard unit sphere S2n+1 of Cn+1. We give some existence results, using dynamical and topological methods involving the study of the critical points at infinity of the associated noncompact variational problem.

We show any pseudo-Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and ⋆-scalar curvature.

The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H-1 (M. R) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space H-1 (M,R) are stable norms of a Riemannian metric on M. If the dimension of M is at least three, I. Babenko and F. Balacheff proved in [I. Babenko, F Balacheff, Sur la forme de la boule unite de la norme stable uniclimensionnelle, Manuscripta Math. 119 (3) (2006) 347-358] that every polyhedral norm ball in H I (M, R), whose vertices are rational with respect to the lattice of integer classes in HI(M,R), is the stable norm ball of a Riemannian metric on M. This metric can even be chosen to be conformally equivalent to any given metric. In [I. Babenko, F. Balacheff, Sur la forme de la boule unite de la norme stable uniclimensionnelle, Manuscripta Math. 119 (3) (2006) 347-358], the stable norm induced by the constructed metric is computed by comparing the metric with a polyhedral one. Here we present an alternative construction for the metric. which remains in the geometric framework of smooth Riemannian metrics. (C) 2009 Elsevier B.V. All rights reserved.

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space S 1 n + 1 ( c ) , n ⩾ 3 , with constant normalized scalar curvature R satisfying n − 2 n c ⩽ R ⩽ c totally umbilical?

Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional orientable Riemannian manifold N. Suppose that the Ricci curvature of N is bounded below by a positive constant k. We show that 2 λ 1 > k − ( n − 1 ) max M | H | where λ 1 is the first eigenvalue of the Laplacian of M and H is the mean curvature of M.

For a compact Riemannian manifold M , we obtain an explicit upper bound of the volume entropy with an integral of Ricci curvature on M and a volume ratio between two balls in the universal covering space.

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds ( M , g ) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal.

It is well known that if the tangent bundle TM of a Riemannian manifold (M,g) is endowed with the Sasaki metric gs, then the flatness property on TM is inherited by the base manifold [Kowalski, J. Reine Angew. Math. 250 (1971) 124–129]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace gs by the most general Riemannian “g-natural metric” on TM (see [Kowalski and Sekizawa, Bull. Tokyo Gakugei Univ. (4) 40 (1988) 1–29; Abbassi and Sarih, Arch. Math. (Brno), submitted for publication]). In this direction, we prove that if (TM,G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M,g) possesses the same property, respectively. We also give explicit examples of g-natural metrics of arbitrary constant scalar curvature on TM.

Let 2≤p≤m−2 . We classify all p -Osserman algebraic curvature tensors on R m . We also show that the only p -Osserman Riemannian metrics are the metrics of constant sectional curvature.

This paper studies contraction constants of non-zero degree mappings from compact spin Riemannian manifolds onto the standard Riemannian sphere. Assuming uniform lower bound for the scalar curvature, we find a sharp lower bound for the dilation constants in terms of the dimension of the sphere. In the best case, we prove rigidity

We will characterize the fundamental groups of compact manifolds of (almost) nonnegative Ricci curvature and also the fundamental groups of manifolds that admit bounded curvature collapses to manifolds of nonnegative sectional curvature. Actually it turns out that the known necessary conditions on these groups are sufficient as well. Furthermore, we reduce the Milnor problem—are the fundamental groups of open manifolds of nonnegative Ricci curvature finitely generated?—to manifolds with abelian fundamental groups. Moreover, we prove for each positive integer n that there are only finitely many non-cyclic, finite, simple groups acting effectively on some complete n -manifold of nonnegative Ricci curvature. Finally, sharping a result of Cheeger and Gromoll [6], we show for a compact Riemannian manifold (M,g 0 ) of nonnegative Ricci curvature that there is a continuous family of metrics (g λ ),λ∈[0,1] such that the universal covering spaces of (M,g λ ) are mutually isometric and (M,g 1 ) is finitely covered by a Riemannian product N×T d , where T d is a torus and N is simply connected.

We prove that a minimal immersion of a complete Riemannian manifold M into another complete noncompact Riemannian manifold N of positive curvature must have an unbounded image provided that M has scalar curvature bounded away from −∞. This extends the unboundedness theorems of Gromoll and Meyer for complete geodesics and of Galloway and Rodriguez for parabolic minimal surfaces. Furthermore, we prove that in case M is of codimension 1, only the Ricci curvature and not necessarily the full sectional curvature of the ambient space N need be positive in order for the same conclusion to hold.

Combining vanishing arguments with certain Gromov-Lawson techniques some sharp estimates on the scalar curvature are found. More precisely, any compact spin (n + 4k)-dimensional manifold which admits a distance decreasing non-zero Â-degree map onto the standard n-dimensional sphere must have a point at which the scalar curvature is stictly less than n(n + 1)/(n + 4k)(n + 4k − 1), otherwise, the map is an isometric submersion.

Some new, improved, curvature depending lower bounds for the first eigenvalue of the Dirac operator on compact Riemannian manifolds are proved. If certain curvature conditions are satisfied, then these lower bounds are also useful in cases where the scalar curvature has zeros or attains negative values. This implies stronger vanishing theorems for harmonic spinors.

This paper is devoted to the partial classification of homogeneous Einstein 6-manifolds with positive scalar curvature.